Decimals : Decimals : Expanded form and number-line

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Decimals : Expanded form and number-line

what you'll learn...

overview

This page covers expanded form of decimals and simple overview of placing decimals in number-line.

Expanded form provides the place value of each digits of the number.
Eg: $12.34 = 10 + 2 + 3 / 10 + 4 / 100$ $12.34 = 10+2+3//10+4//100$

Decimals are placed with the integers and maintains the ordinal property. Thus the decimals are placed in the number-line.

warm up to learn

We studied expanded forms in whole numbers. The number $734$ $734$ in expanded form is " $700$ $700$ $+ 30 + 4$ $+30+4$ ".

The word 'expand' means: to make something large by giving more details.

expanded form of decimal

The expanded form of $32.625$ $32.625$ is " $30 + 2 + 6 / 10 + 2 / 100 + 5 / 1000$ $30+2 + 6//10 + 2//100 + 5//1000$ ".

Expanded form : provides the place value of each digits of the number.
Eg: $12.34 = 10 + 2 + 3 / 10 + 4 / 100$ $12.34 = 10+2+3//10+4//100$

What is the expanded form of $0.0023$ $0.0023$ ?
The answer is " $2 / 1000 + 3 / 10000$ $2//1000+3//10000$ ".

ordered sequence

In whole numbers, we learned that the ordinal-property of the numbers is defined by $0 < 1 < 2 < 3 < \dots$ $0<1<2<3< cdots$ .

In integers, we learned that the ordinal-property of the numbers is extended to $\dots < - 3 < - 2 < - 1 <$ $cdots <-3<-2<-1<$ $0 < 1 < 2 < 3 < \dots$ $0<1<2<3< cdots$ .

In fractions, we learned that the fractions are placed within the integer numbers and maintains the ordinal property.

Decimals do have ordinal property -- ordinal property for decimals is same as that of fractions.

The ordinal property is best captured by the number-line.

• Number-line of whole numbers starts from $0$ $0$ and extends in one direction. It consists of points at positions $0, 1, 2, \dots$ $0,1,2, cdots$ .

• Number-line of integers extends in both the directions. It consists of points at positions $\dots, - 2, - 1, 0, 1, 2, \dots$ $cdots, -2,-1,0,1,2, cdots$

• Number-line of fractions is same as that of integers, and fractions are placed on points on the line segments between the integers. "the number-line of decimals is same as that of fractions as decimals are an alternate representation of fractions"

Consider the number line of decimals. Where is the decimal $0.5$ $0.5$ located in the number-line?

" $0.5$ $0.5$ is between integers $0$ $0$ and $1$ $1$ ". The line-segment between $0$ $0$ and $1$ $1$ is split into $10$ $10$ equal pieces. $0.5$ $0.5$ is at the position of $5$ $5$ th piece as shown in the figure.

Consider the number line of decimals.

The decimal $2.34$ $2.34$ located in the number-line is located between integers $2$ $2$ and $3$ $3$ at $34$ $34$ th position of $100$ $100$ pieces". The line-segment between $2$ $2$ and $3$ $3$ is split into $100$ $100$ equal pieces. $2.34$ $2.34$ is at the position of $34$ $34$ th piece.

summary

Expanded form : provides the place value of each digits of the number.
Eg: $12.34 = 10 + 2 + 3 / 10 + 4 / 100$ $12.34 = 10+2+3//10+4//100$

Decimals do have ordinal property -- ordinal property for decimals is same as that of fractions.

Oridnal property of integers is given by $\dots < - 3 < - 2 < - 1 <$ $cdots <-3<-2<-1<$ $0 < 1 < 2 < 3 < \dots$ $0<1<2<3< cdots$

The fractions are placed within the integers and maintains the ordinal property.

Similarly, decimals are placed with the integers and maintains the ordinal property.

Outline

The outline of material to learn "decimals" is as follows.

Note: goto detailed outline of Decimals

• Decimals - Introduction

→ Decimals as Standard form of Fractions

→ Expanded form of Decimals

• Decimals - Conversion

→ Conversion between decimals and fractions

→ Repeating decimals

→ Irrational Numbers

• Decimals - Arithmetics

→ Comparing decimals

→ Addition & Subtraction

→ Multiplication

→ Division

• Decimals - Expressions

→ Expression Simplification

→ PEMA / BOMA