On January 29, 2001, the New York Stock exchange ended its 200-year tradition of quoting stock prices in fractions and switched to decimals.

It was said that pricing stocks the same way other consumer items were priced would make it easier for investors to understand and compare stock prices. Foreign exchanges had been trading in decimals for decades. Supporters of the change claimed that trading volume, the number of shares of stock traded, would increase and improve efficiency.

But switching to decimals would have another effect of *narrowing the spread*. The *spread* is the difference between the best price offered by buyers, called the *bid*, and the price requested by sellers called the *ask*. Stock brokers make commissions as a percentage of the spread which, using fractions, could be anywhere upwards from 12 cents per share.

When the New York Stock Exchange began back in 1792, the dollar was based on the Spanish *real*, (pronounced ray-al), also called *pieces of eight* as these silver coins were often cut into quarters or eighths to make change. This is what led to stock prices first denominated in eighths. Thus, the smallest spread that could occur would be 1/8 of a dollar, or 12.5 cents. That may seem like small change, but buying 1000 shares for $1 per share with a $0.125 spread is a $125.00 commission. Not bad for a quick trade!

Decimalization of stock pricing allowed for spreads as small as 1 cent. Since the number of shares traded on stock market exchanges have skyrocketed, with trillions of shares traded daily, stock broker commissions have not suffered. And the ease with which investors can quickly grasp the price of stock shares has contributed to the opening of markets for all classes of people.

In this chapter, we’ll learn about how to compute and solve problems with decimals, and see how they relate to fractions.

## What is 1/5 as a decimal?

Converting 1/5 to a decimal is quite possibly one of the easiest calculations you can make. In this (very short) guide, we'll show you how to turn any fraction into a decimal in 3 seconds of less! Here we go!

Want to quickly learn or show students how to convert 1/5 to a decimal? Play this very quick and fun video now!

First things first, if you don't know what a numerator and a denominator are in a fraction, we need to recap that:

Here's the little secret you can use to instantly transform any fraction to a decimal: **Simply divide the numerator by the denominator:**

That's literally all there is to it! 1/5 as a decimal is 0.2.

I wish I had more to tell you about converting a fraction into a decimal but it really is that simple and there's nothing more to say about it.

If you want to practice, grab yourself a pen and a pad and try to calculate some fractions to decimal format yourself. If you're *really feeling lazy* you can use our calculator below instead!

## 5.1: Decimals

**UNDERSTANDING THE STANDARD**

**ESSENTIAL UNDERSTANDINGS**

**ESSENTIAL KNOWLEDGE AND SKILLS**

· The structure of the Base-10 number system is based upon a simple pattern of tens in which each place is ten times the value of the place to its right. This is known as a ten-to-one place value relationship.

· A decimal point separates the whole number places from the places less than one. Place values extend infinitely in two directions from a decimal point. A number containing a decimal point is called a *decimal number* or simply a *decimal*.

· read the whole number to the left of the decimal point, if there is one

· read the decimal point as “and”

· read the digits to the right of the decimal point just as you would read a whole number and

· say the name of the place value of the digit in the smallest place.

· Decimals may be written in a variety of forms:

· Written: Twenty-three and four hundred fifty-six thousandths

· Expanded: (2 ´ 10) + (3 ´ 1) + (4 ´ 0.1) +

· To help students identify the ten-to-one place value relationship for decimals through thousandths, use Base-10 manipulatives, such as place value mats/charts, decimal squares, Base-10 blocks, and money.

· Decimals can be rounded to the nearest whole number, tenth or hundredth in situations when exact numbers are not needed.

· Strategies for rounding decimal numbers to the nearest whole number, tenth and hundredth are as follows:

· Look one place to the right of the digit to which you wish to round.

· If the digit is less than 5, leave the digit in the rounding place as it is, and change the digits to the right of the rounding place to zero.

· If the digit is 5 or greater, add 1 to the digit in the rounding place and change the digits to the right of the rounding place to zero.

· Create a number line that shows the decimal that is to be rounded.

· The position of the decimal will help children conceptualize the number’s placement relative for rounding. An example is to round 5.747 to the nearest hundredth:

· Understand that decimals are rounded in a way that is similar to the way whole numbers are rounded.

· Understand that decimal numbers can be rounded to estimate when exact numbers are not needed for the situation at hand.

**The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to**

## 5.1: Decimals

To start off, note that 5 1/9 is a mixed number, also known as a mixed fraction. It has a whole number and a fractional number. We have labeled the parts of the mixed number below so it is easier to follow along.

5 = Whole number

1 = Numerator

9 = Denominator

To get 5 1/9 in decimal form, we basically convert the mixed number to a fraction and then we divide the numerator of the fraction by the denominator of the fraction.

Here are the detailed math steps we use to convert 5 1/9 mixed number to decimal form:

**Step 1:** Multiply the whole number by the denominator:

**Step 2:** Add the product you got in Step 1 to the numerator:

**Step 3:** Divide the sum from Step 2 by the denominator:

That's it folks! The answer to 5 1/9 in decimal form is displayed below:

**Mixed Number in Decimal Form**

5 1/9 in decimal form is not all we can do! Here you can convert another mixed number to decimal form.

**What is 5 2/1 in decimal form?**

Here is the next mixed number on our list that we have converted into decimal form.

## Fractions in word problems:

Cyka made 6 19/20 cups of punch punch at two different types of juice in it. If the punch had 4 1/5 cups of one type of juice how many cups of the other type of juice did it have?

Farmer Peter paints 12 chicken coop. He started painting this day morning. Now he only has 1/4 of the chicken coop left to paint this afternoon. How many chicken coops did farmer Peter paint this morning?

Father has 12 1/5 meters long wood. Then I cut the wood into two pieces. One part is 7 3/5 meters long. Calculate the length of the other wood?

3 pounds subtract 1/3 of a pound.

There are 40 pupils in a certain class. 3/5 of the class are boys. How many are girls?

Express in mm: 5 3/10 cm - 2/5 mm

Mrs. Lazo bought 9 1/8 m curtain cloth. She used 3 5/6 m to make a curtain for their bedroom. How many meters of cloth were not used?

Martin is making a model of a Native American canoe. He has 5 1/2 feet of wood. He uses 2 3/4 feet for the hull and 1 1/4 feet for a paddle. How much wood does he have left? Martin has feet of wood left.

Pediatrician this month of 20 working days takes 8 days holidays. What is the probability that on Monday it will be at work?

What is the difference between 4 2/3 and 3 1/6?

Ananya has a bunny. She bought 4 7/8 pounds of carrots. She fed her bunny 1 1/4 pounds of carrots the first week. She fed her bunny 5/6 pounds of carrots the second week. All together, how many pounds of carrots did she feed her bunny? 1. Draw a tape diag

Federal law requires that all residential toilets sold in the United States use no more than 1 3/5 gallons of water per flush. Before this legislation, conventional toilets used 3 2/5 gallons of water per flush. Find the amount of water saved in one year

Heather has 2 cups of powdered sugar. She sprinkles 3/5 of the sugar onto a plate of brownies and sprinkles the rest into a plate of lemon cookies. How much sugar does Heather sprinkle on the brownies? How much sugar does Heather sprinkle on the lemon coo

## Converting decimal and numeric data

For **decimal** and **numeric** data types, SQL Server considers each combination of precision and scale as a different data type. For example, **decimal(5,5)** and **decimal(5,0)** are considered different data types.

In Transact-SQL statements, a constant with a decimal point is automatically converted into a **numeric** data value, using the minimum precision and scale necessary. For example, the constant 12.345 is converted into a **numeric** value with a precision of 5 and a scale of 3.

Converting from **decimal** or **numeric** to **float** or **real** can cause some loss of precision. Converting from **int**, **smallint**, **tinyint**, **float**, **real**, **money**, or **smallmoney** to either **decimal** or **numeric** can cause overflow.

By default, SQL Server uses rounding when converting a number to a **decimal** or **numeric** value with a lower precision and scale. Conversely, if the SET ARITHABORT option is ON, SQL Server raises an error when overflow occurs. Loss of only precision and scale isn't sufficient to raise an error.

Prior to SQL Server 2016 (13.x), conversion of **float** values to **decimal** or **numeric** is restricted to values of precision 17 digits only. Any **float** value less than 5E-18 (when set using either the scientific notation of 5E-18 or the decimal notation of 0.0000000000000000050000000000000005) rounds down to 0. This is no longer a restriction as of SQL Server 2016 (13.x).

## 5.1.2 Decimals & Fractions

Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.

*For example*: Possible names for the number 0.0037 are:

3 thousandths + 7 ten thousandths

a possible name for the number 1.5 is 15 tenths.

Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.

Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.

*For example*: Which is larger 1.25 or $frac<6><5>$?

*Another example*: In order to work properly, a part must fit through a 0.24 inch wide space. If a part is $frac<1><4>$ inch wide, will it fit?

Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.

*For example*: When comparing 1.5 and $frac<19><12>$, note that 1.5 $=1frac<1><2>=1frac<6><12>=frac<18><12>$, so 1.5 ltfrac<19><12>$.

Round numbers to the nearest 0.1, 0.01 and 0.001.

*For example*: Fifth grade students used a calculator to find the mean of the monthly allowance in their class. The calculator display shows 25.80645161. Round this number to the nearest cent.

### Overview

##### Standard 5.1.2 Essential Understandings

The study of rational numbers now includes decimal representations to millionths as well as fractions. Fifth graders extend their understanding of the base ten numeration system and place value concepts to include millionths. *For example*: Possible names for the number 0.0037 are: 37 ten thousandths 3 thousandths + 7 ten thousandths and a possible name for the number 1.5 is 15 tenths.

Students determine .1 more/less, .01 more/less and .001 more/less than a given number. They are able to compare and order fractions and decimals and locate them on a number line.

Students develop an understanding of conversion between fractions and decimals. Work with equivalent fractions continues as students encounter fractions with denominators of 15, 16, 20, 25, 50 and 100. These understandings are used in solving real-world and mathematical situations.

#### All Standard Benchmarks

Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.

Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.

Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.

Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.

Round numbers to the nearest 0.1, 0.01 and 0.001.

Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.

*For example*: Possible names for the number 0.0037 are: 37 ten thousandths 3 thousandths + 7 ten thousandths a possible name for the number 1.5 is 15 tenths.

Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.

##### 5.1.2.3

Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.

Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.

Round numbers to the nearest 0.1, 0.01 and 0.001.

*For example*: Fifth grade students used a calculator to find the mean of the monthly allowance in their class. The calculator display shows 25.80645161. Round this number to the nearest cent.

##### What students should know and be able to do [at a mastery level] related to these benchmarks:

- convert between fraction and decimal representations of a number.
- know decimal names for common fractions such as ¼ as 0.25, ⅓ as 0.33 (repeating), ½ as 0.5 and ⅕ as 0.2 etc. to facilitate ordering, comparing fractions and decimals and converting to percents.
- find 0.1, 0.01, and 0.001 more or less than a number.
- locate and order fractions and decimals on a number line.
- order a set of numbers that includes fractions, decimals, and mixed numbers.
- locate fractions and decimals, including mixed numbers and improper fractions, on a number line.
- extend their understanding of the base ten numeration system and place value concepts to include millionths.
*For example*: Possible names for the number 0.0037 are: 37 ten thousandths 3 thousandths + 7 ten thousandths a possible name for the number 1.5 is 15 tenths. - round a number to the nearest 0.1, 0.01, 0.001.
- recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions.
- easily translate between proper and improper fractions and mixed numbers.

##### Work from previous grades that supports this new learning includes:

- represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives.

- use models to determine equivalent fractions.
- locate fractions on a number line.
- use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions.
- use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations.
- develop a rule for addition and subtraction of fractions with like denominators.
- read and write decimals with words and symbols.
- use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths.
- compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks. For example, they could determine a fraction between ⅓ and ¼ or a decimal between .9 and .91.
- read and write tenths and hundredths in decimal and fraction notations using words and symbols.
- know the fraction and decimal equivalents for halves and fourths.
- round decimals to the nearest tenth.
- develop understanding of fraction equivalence. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions.
- extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions.
- estimate the relative size of fractions and decimals by using benchmarks such as 0, ½, and 1 and beyond.

#### NCTM Standards

##### Understand numbers, ways of representing numbers, relationships among numbers, and number systems

- understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals
- recognize equivalent representations for the same number and generate them by decomposing and composing numbers
- develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers
- use models, benchmarks, and equivalent forms to judge the size of fractions
- recognize and generate equivalent forms of commonly used fractions, decimals, and percents
- explore numbers less than 0 by extending the number line and through familiar applications
- describe classes of numbers according to characteristics such as the nature of their factors.

#### Common Core State Standards

#### Understand the place value system.

**5.NBT.1**. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

**5.NBT.2**. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

**5.NBT.3**. Read, write, and compare decimals to thousandths.

**5.NBT.3a**.Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

**5.NBT.3b**. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

**5.NBT.4**. Use place value understanding to round decimals to any place.

### Misconceptions

##### Student Misconceptions and Common Errors

- the numerator and denominator are separate whole numbers.
- whole number relationships can be applied to fractions or decimals. For example, believing 0.26 is greater than 0.8 because 26 is greater than 8.
- whole numbers are always larger than fractions including mixed numbers.
- smaller is bigger with fractions - 1/7 is greater than 2/7- the smallest numerator is the largest piece or ½ is smaller than ⅓ because 2 is smaller than 3
- the difference between the denominator and the numerator indicates how close the fraction is to one. For example, ¾ and ⅔ are both one away from the whole so they are the same in size. Or, ½ and 5/7 - ½ must be larger since it is 1 away from the whole and 5/7 is 2 pieces away from the whole.
- decimals are just like whole numbers everything you do with whole numbers you do with decimals.
- the more digits to the right of the decimal point the bigger the number.
- decimals are completely different from fractions.

### Resources

##### Teacher Notes

- Students may need support in further development of previously studied concepts and skills
*.* - Equivalent fractions are created by multiplying by 1 (2/2, 3/3, 4/4), etc.
- Simplifying fractions requires dividing by 1 (2/2, 3/3, 4/4), etc.
- When comparing fractions only teaching students to find common denominators instead of building understanding using benchmark fraction of 0, 1/2 and 1 to estimate size reduces the opportunity for students to develop number sense.
- Annexing or placing zeroes to make the decimals being compared the same number of digits is misguided and does not allow students to focus on place value.
- Reading decimals correctly such as 0.26 as "twenty six hundredths" instead of "point two six" supports place value understanding.
- Use number lines to determine appropriate placement of decimals such as .9, .09, .19 etc.
- Careful use of language when modeling equivalent fractions is important. For example, using fractions strips to model changing ¾ to the equivalent fraction 6/8, focus the discussion on the change that occurs in the numerator and denominator when multiplying by 1 whole (2/2) vs. dividing each of the fourths into halves to get eighths. This creates confusion for students into believing the operation is division rather than multiplication.
- The symmetrical balance for whole numbers and decimals is the ones place not the decimal point.
- According to MCA III Test Specifications denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 15, 16 and 20. When recognizing and generating equivalent fractions, decimals, mixed numbers and improper fractions denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 25, 50 and 100.
- The Rational Number Project (Initial Fraction Ideas) provides researched based strategies and lessons supporting conceptual understanding of fractions including connections to operations with fractions.

- The Rational Number Project (Fraction Operations and Initial Decimal Ideas) provides researched based strategies and lessons supporting conceptual understanding of fractions and decimals including connections to operations with fractions and decimals

- Good
**questions**, and good**listening**, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

**Getting Started**

What do you need to find out?

What do you know now? How can you get the information? Where can you begin?

What terms do you understand/not understand?

What similar problems have you solved that would help? **While Working**

How can you organize the information?

Can you make a drawing (model) to explain your thinking? What are other possibilities?

What would happen if.

Can you describe an approach (strategy) you can use to solve this?

What do you need to do next?

Do you see any patterns or relationships that will help you solve this?

How does this relate to.

Why did you.

What assumptions are you making?**Reflecting about the Solution**

How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?

How can you convince me your answer makes sense?

What did you try that did not work? Has the question been answered?

Can the explanation be made clearer?**Responding** (helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part .

(Adapted from *They're Counting on Us,* California Mathematics Council, 1995)

##### NCTM Illuminations

- Lessons include: "Developing Fractions--the length model" and Developing Fractions--the set model." Activities include: Equivalent Fractions Fraction Models and the Fraction Game

- This game could be used when working with the equivalence of decimals, fractions, and percentages. -

##### Additional Instructional Resources

Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). *Navigating through number and operations in grades 3-5*. Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). *Good questions: Great ways to differentiate mathematics instruction.* New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). *Elementary and middle school mathematics: Teaching developmentally.* (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). *Teaching student-centered mathematics grades 3-5.* Boston, MA: Pearson Education.

**rational number:** A number expressible in the form *a*/*b* or - *a*/*b* for some fraction *a*/*b*. The rational numbers include the integers.

**numerator:** The number that is written above the line in a fraction. It tells how many of the whole you have or how an y parts are being considered.

**denominator:** The number below the line in a fraction. It shows how many how many equal pieces the whole has been divided into.

**mixed number:** A number that has both a whole number part and a fractional part such as 2 ⅓. Mixed numbers represent values greater than 1.

**improper fraction:** A fraction in which the numerator is greater than the denominator such as 11/3. Improper fractions represent values greater than one.

*"Vocabulary literally is the*

** key tool for thinking."**

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

*Integration*: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

*Repetition*: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

*Meaningful* *Use:* Multiple and varied opportunities to use the words in context. Theseopportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

##### Strategies for vocabulary development

*Students do not learn vocabulary words by memorizing and practicing definitions***.** The following strategies keep vocabulary visible and accessible during instruction.

**Mathematics Word Bank**: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

**Labeled pictures and charts**: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

**Frayer Model**: The Frayer Model connects words, definitions, examples and non-examples.

**Example/Non-example Charts**: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

**Vocabulary Strips**: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

Encouraging students to verbalize thinking by drawing, talking, and writing, increases opportunities to use the mathematics vocabulary words in context.

**Additional Resources for Vocabulary Development**

Murray, M. (2004). *Teaching mathematics vocabulary in context*. Portsmouth, NH: Heinemann.

Sammons, L. (2011). *Building mathematical comprehension: Using literacy strategies to make meaning.* Huntington Beach, CA: Shell Education.

##### Professional Learning Communities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

What are the key ideas related to decimal understanding at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?

How would you know a student understands the decimal system when using number from .0001 to .1?

What representations should a student be able to make for the number 365.4729 if they understand place value?

What experiences do students need in order to develop an understanding of rounding decimals to the nearest tenth, hundredth and thousandth?

When checking for student understanding of decimals, what should teachers

- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?

Examine student work related to a place value task involving decimals What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

What are the key ideas related to fraction understanding at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?

What representations should a student be able to make for the fraction ______?

When checking for student understanding of fractions at the fifth grade level, what should teachers

- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?

Examine student work related to a task involving fractions. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

What is meant by equivalent representations? How can teachers help students understand equivalent representations?

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the fifth grade level?

##### Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). *Math misconceptions prek-grade 5: From misunderstanding to deep understanding.* Portsmouth, NH: Heinemann.

Chapin, S., and Johnson, A. (2006). *Math matters: Understanding the math you teach, grades K-8*. (2 nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). *Classroom discussions: Using math talk to help students learn (Grades K-6)*. Sausalito, CA: Math Solutions.

Fosnot, C., & Dolk, M. (2002). *Young mathematicians at work: Multiplication and division.* Portsmouth, NH: Heinemann.

Hyde, Arthur. (2006). *Comprehending math adapting reading strategies to teach mathematics,* *K-6.* Portsmouth, NH: Heinemann.

Lester, F. (2010). *Teaching and learning mathematics: Transforming research for elementary school teachers.* Reston, VA: National Council of Teachers of Mathematics.

Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). *Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5.* Reston, VA.: National Council of Teachers of Mathematics.

Parrish, S. (2010). *Number talks: Helping children build mental math and computation strategies grades K-5.* Sausalito. CA: Math Solutions.

Sammons, L., (2011). *Building mathematical comprehension: Using literacy strategies to make meaning.* Huntington Beach, CA: Shell Education.

Schielack, J. (2009). *Focus in grade 3, teaching with curriculum focal points.* Reston, VA: National Council of Teachers of Mathematics.

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). *Math misconceptions prek-grade 5: From misunderstanding to deep understanding.* Portsmouth, NH: Heinemann.

Bender, W. (2009). *Differentiating math instruction: Strategies that work for k-8 classrooms!* Thousand Oaks, CA: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008). *Supporting English language learners in math class, grades k-2.* Sausalito, CA: Math Solutions Publications.

Burns, Marilyn. (2007). *About teaching mathematics: A k-8 resource* (3rd ed.). Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed). (1998). *Leading the way: Principals and superintendents look at math instruction.* Sausalito, CA: Math Solutions.

Caldera, C. (2005). *Houghton Mifflin math and English language learners*. Boston, MA: Houghton Mifflin Company.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). *Children's mathematics cognitively guided instruction.* Portsmouth, NH: Heinemann.

Cavanagh, M. (2006). *Math to learn: A mathematics handbook.* Wilmington, MA: Great Source Education Group, Inc.

Chapin, S., & Johnson, A. (2006). *Math matters: Understanding the math you teach, grades K-8*. (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). *Classroom discussions: Using math talk to help students learn (Grades K-6)*. Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). *Math for all: Differentiating instruction k-2.* Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). *How students learn: Mathematics in the classroom.* Washington, DC: National Academies Press.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). *Developing essential understanding of number & numeration pre-k-grade 2*. Reston, VA: National Council of Teachers of Mathematics.

Felux, C., & Snowdy, P. (Eds.). ( 2006). *The math coach field guide: Charting your course*. Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009). *Focus in grade 2 teaching with curriculum focal points.* Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006). *Comprehending math adapting reading strategies to teach mathematics,* *K-6.* Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). *Adding it up: Helping children learn mathematics.* Washington, DC: National Academies Press.

Leinwand, S. (2000). *Sensible mathematics: A guide for school leaders*. Portsmouth, NH: Heinemann.

Lester, F. (2010). *Teaching and learning mathematics: Transforming research for elementary school teachers.* Reston, VA: National Council of Teachers of Mathematics.

Murray, M. (2004). *Teaching mathematics vocabulary in context.* Portsmouth, NH: Heinemann.

Murray, M., & Jorgensen, J. (2007). *The differentiated math classroom: A guide for teachers k-8.* Portsmouth, NH: Heinemann*.*

National Council of Teachers of Mathematics. (2000). *Principles and standards for school mathematics.* Reston, VA: NCTM.

Parrish, S. (2010). *Number talks: Helping children build mental math and computation strategies grades K-5.* Sausalito. CA: Math Solutions*.*

Reeves, D. (2007). *Ahead of the curve: The power of assessment to transform teaching and learning.* Indiana: Solution Tree Press.

Sammons, L. (2011). *Building mathematical comprehension: Using literacy strategies to make meaning.* Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., . & Zbiek, R. M. (2006). *Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence.* Reston, VA: NCTM.

Seeley, C. (2009). *Faster isn't smarter: Messages about math teaching and learning in the 21st century.* Sausalito, CA: Math Solutions.

Small, M. (2009). *Good questions: Great ways to differentiate mathematics instruction.* New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). *Elementary and middle school mathematics: Teaching developmentally.* (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. A., & Lovin, L. H. (2006). *Teaching student-centered mathematics grades K-3.* Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). *Content focused coaching: Transforming mathematics lessons.* Portsmouth, NH: Heinemann*.*

Wickett, M., & Burns, M. (2003). *Teaching arithmetic: Lessons for extending division, grades 4-5.* Sausalito, CA: Math Solutions.

### Assessment

**A.** 0.20815**B.** 0.30256**C.** 0.40571**D.** 0.50098

Solution : B 0.30256

Benchmark 5.1.2.1

MCA III Item Sampler

- Johan's race time was 45.03 seconds. Kyle's race time was 0.1 second less than Johan's time. What was Kyle's race time?

**A.** 44.03 seconds**B.** 44.93 seconds**C.** 45.13 seconds**D.** 45.14 seconds

Solution: B 44.93. seconds

Benchmark 5.1.2.2

MCA III Item Sampler

**A.** 0.45**B.** 0.458**C.** 0.459**D.** 0.4583

Solution: B 0.458

Benchmark 5.1.2.5

MCA III Item Sampler

Solution: B. K and L

Benchmark: 5.1.2.3

MCA III Item Sampler

Solution: A 0.04

Benchmark 5.1.2.4

MCA III Item Sampler

Solution: Will vary.

Benchmark: 5.1.2.3

Solution: 1 2/6, 1 ⅓, 1.333

Benchmark: 5.1.2.4

Solution: 0.33 0.2 0.375 1.5

Benchmark: 5.1.2.4

Solution: 1 3/10 2/3 3/4 1/20 1/2

Benchmark 5.1.2.4

### Differentiation

- Structure consistent computational fluency activities utilizing physical models such as number lines and base ten blocks to help reconstruct multiplication/division facts when needed.
- Actively engage students in learning situations that focus on both concept and skill development (Place value millions to millionths) Provide explicit systematic instruction that includes opportunities for students to ask and answer questions and think aloud when making decisions while solving problems. Be sure that students understand the place value symmetry for whole numbers and decimals is
**one**and not the decimal point. - Instructional settings should include direct instruction work before and after the mathematics lesson such as I Do (teacher demonstrates), You do, (student models) or vocabulary instruction whole group (students receive core instruction with classmates in regular classroom and small group situations (such as partner work) that are well structured and have clear expectations. Make use of technology as appropriate.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as rational numbers, numerator, denominator, mixed number, and improper fractions.

- Carefully connect prior knowledge (place value thousands to thousandths to new learning of large numbers (millions to millionths) to read, write, compare and round decimals.
- Carefully connect prior knowledge (fractions, fraction benchmarks, and fraction models) to compare fractions with unlike denominators.
- Pose meaningful problems set in familiar situations.
- Incorporate visual models such as the number lines, fraction bars, and decimal grids.

##### Concrete - Representational - Abstract Instructional Approach

(Adapted from *The Access Center: Improving Access for All K-8 Students)*

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

**Concrete - Representational - Abstract**

The **Concrete Stage** is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The **Representational Stage** is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The **Abstract Stage** is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.

##### Additional Resources

Bender, W. (2009). *Differentiating math instruction: Strategies that work for k-8 classrooms!* Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). *Math for all: Differentiating instruction grades 3-5.* Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). *The differentiated math classroom: A guide for teachers k-8.* Portsmouth, NH: Heinemann*.*

Small, M. (2009). *Good questions: Great ways to differentiate mathematics instruction.* New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). *Elementary and middle school mathematics: Teaching developmentally.* (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). *Teaching student-centered mathematics grades 3-5.* Boston, MA: Pearson Education.

Teachers need to demonstrate and model the use of manipulatives (place value blocks) or representations such as bar and area models when connecting language and concepts as students work with fractions and decimals.

- Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.

- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

The fraction __________ is the same as the decimal __________________.

The decimal __________ is the same as the fraction _________________.

The decimal _____________ means ___________________________________.

The fraction _____________ means ___________________________________.

- When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.

##### Additional ELL Resources:

Bresser, R., Melanese, K., & Sphar, C. (2008). *Supporting English language learners in math class, grades k-2.* Sausalito, CA: Math Solutions Publications*.*

Bender, W. (2009). *Differentiating math instruction: Strategies that work for k-8 classrooms!* Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). *Math for all: Differentiating instruction grades 3-5.* Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). *The differentiated math classroom: A guide for teachers k-8.* Portsmouth, NH: Heinemann*.*

*Good questions: Great ways to differentiate mathematics instruction.* New York, NY: Teachers College Press.

### Parents/Admin

##### Administrative/Peer Classroom Observation

explaining thinking for ordering fractions and decimals.

asking clarifying questions which illustrate

student thinking. Helping students use benchmark numbers as referents when comparing and ordering fractions and decimals.

finding equivalent representations of fractions and decimals.

providing a variety of models of fractions and decimals as they develop conceptual and procedural understanding of equivalent fractions and decimals.

using appropriate mathematics vocabulary.

developing vocabulary throughout instruction.

finding .001 more/less, .01 more/less and .1 more/less than a number.

providing representations for finding .001 more/less, .01 more/less and .1 more/less than a number.

rounding to the nearest .1 .01, and .001.

providing number lines with appropriate scales as representations for rounding.

### What should I look for in the mathematics classroom?

**(Adapted from SciMathMN,1997)**

**What are students doing?**

- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.

**What are teachers doing?**

- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.

### Additional Resources

##### For Mathematics Coaches

Chapin, S. and Johnson, A. (2006). *Math matters: Understanding the math you teach: Grades k-8*. (2nd ed.). Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). *How students learn: Mathematics in the classroom.* Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). *The math coach field guide: Charting your course*. Sausalito, CA: Math Solutions.

Sammons, L., (2011). *Building mathematical comprehension: Using literacy strategies to make meaning.* Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). *Content focused coaching: Transforming mathematics lessons.* Portsmouth, NH: Heinemann*.*

##### For Administrators

Burns, M. (Ed). (1998). *Leading the way: Principals and superintendents look at math instruction.* Sausalito, CA: Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). *Adding it up: Helping children learn mathematics.* Washington, DC*:* National Academies Press.

Leinwand, S. (2000). *Sensible mathematics: A guide for school leaders*. Portsmouth, NH: Heinemann.

Lester, F. (2010). *Teaching and learning mathematics: Transforming research for school administrators.* Reston, VA: National Council of Teachers of Mathematics.

Seeley, C. (2009). *Faster isn't smarter: Messages about math teaching and learning in the 21st century.* Sausalito, CA: Math Solutions.

#### Parent Resources

##### Mathematics handbooks to be used as home references:

Cavanagh, M. (2004). *Math to Know: A mathematics handbook.* Wilmington, MA: Great Source Education Group, Inc.

Cavanagh, M. (2006). *Math to learn: A mathematics handbook.* Wilmington, MA: Great Source Education Group, Inc.

##### Helping your child learn mathematics

Provides activities for children in preschool through grade 5

#### Help Your Children Make Sense of Math

##### Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

Good **questions**, and good **listening**, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

*Getting Started*

What do you need to find out?

What do you know now? How can you get the information? Where can you begin?

What terms do you understand/not understand?

What similar problems have you solved that would help?

*While Working*

How can you organize the information?

Can you make a drawing (model) to explain your thinking? What are other possibilities?

What would happen if . . . ?

Can you describe an approach (strategy) you can use to solve this?

What do you need to do next?

Do you see any patterns or relationships that will help you solve this?

How does this relate to.

Can you make a prediction?

Why did you.

What assumptions are you making?

*Reflecting about the Solution*

How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?

How can you convince me your answer makes sense?

What did you try that did not work?

Has the question been answered?

Can the explanation be made clearer?

** Responding**(helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part.

Adapted from *They're counting on us,* California Mathematics Council, 1995.

Here is a simple way of ordering the given list of numbers in ascending and descending order. In this online ordering decimals calculator, enter a list of random numbers and submit to know the ascending order and descending order of the numbers.

**Use of Ordering Decimals from Least to Greatest**: Arranging the numbers in ascending and descending orders will be helpful for students and professionals and mathematicians to apply the ordered result in various applications.

**Ordering Decimals Calculator from Least to Greatest:** Enter the decimal numbers in the input field, the calculator will compare the numbers and update you the numbers in ascending order (arranging numbers from least to greatest) and descending order (arranging numbers from largest to smallest) respectively. Students can solve the ordering decimals related problems easily using this calculator. This ordering decimals calculator helps you to know the ascending order and the descending order of the given numbers list in just fraction of a second and saves your time and make you calculations simple.

### Example:

Consider a set of numbers : 3.4,9.3,12.5,7.4,22.2,89.4

#### Solution,

Total numbers in the set is 6.

Ascending Order (Least to Greatest) is 3.4, 7.4, 9.3, 12.5, 22.2, 89.4

Descending Order (greatest to Least) is 89.4, 22.2, 12.5, 9.3, 7.4, 3.4

** Ordering Decimals** from least to greatest and vice versa made easier here.

## Contents

Many numeral systems of ancient civilizations use ten and its powers for representing numbers, probably because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called *decimal fractions* or *decimal numbers*, for forming the *decimal numeral system*.

For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 [7] the decimal separator is the dot " . " in many countries, [4] [8] but also a comma " , " in other countries. [5]

For representing a non-negative number, a decimal numeral consists of

- either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer, a m a m − 1 … a 0
a_ ldots a_<0>> - or a decimal mark separating two sequences of digits (such as "20.70828")

If *m* > 0 , that is, if the first sequence contains at least two digits, it is generally assumed that the first digit *a*_{m} is not zero. In some circumstances it may be useful to have one or more 0's on the left this does not change the value represented by the decimal: for example, 3.14 = 03.14 = 003.14 . Similarly, if the final digit on the right of the decimal mark is zero—that is, if *b*_{n} = 0 —it may be removed conversely, trailing zeros may be added after the decimal mark without changing the represented number [note 1] for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200 .

For representing a negative number, a minus sign is placed before *a*_{m} .

The *integer part* or *integral part* of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the *fractional part*, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example .1234 , instead of 0.1234 ). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

More generally, a decimal with *n* digits after the separator represents the fraction with denominator 10 *n* , whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation.

Expressed as a fully reduced fraction, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are

Decimal numerals do not allow an exact representation for all real numbers, e.g. for the real number π . Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real π , being less than 10 −5 off so decimals are widely used in science, engineering and everyday life.

More precisely, for every real number x and every positive integer n , there are two decimals *L* and *u* with at most *n* digits after the decimal mark such that *L* ≤ *x* ≤ *u* and (*u* − *L*) = 10 −*n* .

Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with *n* digits after the decimal mark, as soon as the absolute measurement error is bounded from above by 10 −*n* . In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

For a real number x and an integer *n* ≥ 0 , let [*x*]_{n} denote the (finite) decimal expansion of the greatest number that is not greater than *x* that has exactly n digits after the decimal mark. Let *d*_{i} denote the last digit of [*x*]_{i} . It is straightforward to see that [*x*]_{n} may be obtained by appending *d*_{n} to the right of [*x*]_{n−1} . This way one has

and the difference of [*x*]_{n−1} and [*x*]_{n} amounts to

which is either 0, if *d*_{n} = 0 , or gets arbitrarily small as *n* tends to infinity. According to the definition of a limit, *x* is the limit of [*x*]_{n} when *n* tends to infinity. This is written as x = lim n → ∞ [ x ] n < extstyle x=lim _

which is called an **infinite decimal expansion** of *x* .

Any such decimal fraction, i.e.: *d*_{n} = 0 for *n* > *N* , may be converted to its equivalent infinite decimal expansion by replacing *d*_{N} by *d*_{N} − 1 and replacing all subsequent 0s by 9s (see 0.999. ).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of [*x*]_{n} , and the other containing only 9s after some place, which is obtained by defining [*x*]_{n} as the greatest number that is *less* than x , having exactly *n* digits after the decimal mark.

### Rational numbers Edit

Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a *repeating decimal*. For example,

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally). [10] For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, [11] [12] especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic). [13]

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of 10

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten fingers/digits. [16] Standardized weights used in the Indus Valley Civilization (c. 3300–1300 BCE ) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the *Mohenjo-daro ruler* – was divided into ten equal parts. [17] [18] [19] Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system, [20] as did the Cretan hieroglyphs (c. 1625−1500 BCE ) of the Minoans whose numerals are closely based on the Egyptian model. [21] [22] The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (c. 18th century BCE−1450 BCE) and Linear B (c. 1375−1200 BCE) – the number system of classical Greece also used powers of ten, including, Roman numerals, an intermediate base of 5. [23] Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 10 8 [23] and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery. [24] Hittite hieroglyphs (since 15th century BCE) were also strictly decimal. [25]

Some non-mathematical ancient texts such as the Vedas, dating back to 1700–900 BCE make use of decimals and mathematical decimal fractions. [26]

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000. [27] The world's earliest positional decimal system was the Chinese rod calculus. [28]

### History of decimal fractions Edit

Decimal fractions were first developed and used by the Chinese in the end of 4th century BCE, [29] and then spread to the Middle East and from there to Europe. [28] [30] The written Chinese decimal fractions were non-positional. [30] However, counting rod fractions were positional. [28]

J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century. [32] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. [33] The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century. [32] Al Khwarizmi introduced fraction to Islamic countries in the early 9th century a Chinese author has alleged that his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing. [28] This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by al-Uqlidisi and by al-Kāshī in his work "Arithmetic Key". [28] [34]

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. [35]

### Natural languages Edit

A method of expressing every possible natural number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many Indo-Aryan and Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10. [36]

The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").

A straightforward decimal rank system with a word for each order (10 十 , 100 百 , 1000 千 , 10,000 万 ), and in which 11 is expressed as *ten-one* and 23 as *two-ten-three*, and 89,345 is expressed as 8 (ten thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen".

Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as *ten with one* and 23 as *two-ten with three*.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability. [37]

Enter two or more decimals separated by "commas"

Given numbers are 1.2,1.5,1.8. The highest number of digits after the decimal point in the given case is 1

Thus, in order to get rid of the decimal point we need to multiply them with 10. On doing so, they are as follows

On finding the LCM of 12,15,18 we get the Least Common Multiple as 180

### Least Common Multiple (LCM) of 12,15,18 By Common Division

∴ So the LCM of the given numbers is 2 x 3 x 2 x 5 x 3 = 180

Divide the result you got with the number you multiplied to make it as integer in the first step. In this case, we need to divide by 10 as we used it to make the given numbers into integers.

On dividing the LCM 180/10 we get 18

Thus the Least Common Multiple of 1.2,1.5,1.8 is 18

### Least Common Multiple of 12,15,18 with GCF Formula

We need to calculate greatest common factor of 12,15,18 and common factors if more than two numbers have common factor, than apply into the LCM equation.

common factors(in case of two or more numbers have common factors) = 6

GCF(12,15,18) x common factors =3 x 6 = 18

LCM(12,15,18) = ( 12 × 15 × 18 ) / 18

### LCM of Decimals Calculation Examples

Here are some samples of LCM of Decimals calculations.

### Frequently Asked Questions on Decimal LCM of 1.2, 1.5, 1.8

1. What is the LCM of 1.2, 1.5, 1.8?

Answer: LCM of 1.2, 1.5, 1.8 is 18.

2. How to Find the LCM of 1.2, 1.5, 1.8?

Answer: Least Common Factor(LCM) of 1.2, 1.5, 1.8 = 18

Step 1: First calculate the highest decimal number after decimal point.

Step 2: Then multiply all numbers with 10.

Step 3: Then find LCM of 12,15,18. After getting LCM devide the result with 10 the value that is previously multiplied.