## Key Concepts

• Fraction

• Decimal number

• Repeating decimal

• Non-repeating/terminating decimal

**Real Numbers: Rational Numbers as Decimals**

### Introduction to Decimal Numbers:

A decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point.

The dot in a decimal number is called a decimal point.

The digits following the decimal point show a value smaller than one.

**Understanding the parts of a decimal number:**

**Representation of decimal numbers on a scale:**

**Decimal numbers in everyday life:**

**Writing decimals as fractions:**

To convert a decimal to a fraction, we write the decimal number as a numerator and its place value as the denominator.

**Example 1:** 0.07

**1.1.1 Writing repeating decimals as fractions**

**Steps to writing a repeating decimal as a fraction: **

Begin by writing x = the repeating number.

Multiply both sides of the equation by a power of 10, which will move the decimal to the right of the repeating number.

Subtract the equation from step 1 from the equation in step 2.

Solve the resulting equation.

**Example:**

Write

0._{7}^{−}

as a fraction.

**Step1:** Begin by writing x = the repeating number.

x=0.7−

**Step 2:** Multiply both sides of the equation by a power of 10, which will move the decimal to the right of the repeating number.

The repeating number is seven. To move the decimal to the right of the 7, you need to multiply by 10. This gives you the following:

10x= 7.7−

**Step 3:** Subtract the equation from step 1 from the equation in step 2.

10x= 7.7−

− x=0.7−9x=7

**Step 4:** Solve the resulting equation.

9x=7

x=79

**1.1.2 Repeating decimals with non-repeating digits as fractions**

A mixed repeating decimal is a decimal that does not repeat until after the tenths place. The value

712

=0.583− is an example of this. To write a mixed repeating decimal, you will use the same steps as before.

Begin by writing x = the repeating number.

Multiply both sides of the equation by a power of 10, which will move the decimal to the right of the repeating number.

Subtract the equation from step 1 from the equation in step 2.

Solve the resulting equation.

**Example**

Write

0.86−

as a fraction.

**Step 1:** Begin by writing x = the repeating number.

x= 0.86−

**Step 2:** Multiply both sides of the equation by a power of 10, which will move the decimal to the right of the repeating number.

The repeating number is six. To move the decimal to the right of the 6, you need to multiply by 100, which gives you the following:

100x= 86.6−

**Step 3:** Subtract the equation from step 1 from the equation in step 2.

100x= 86.6−

− x=0.86− 99x=85.8− x=0.86- 99x=85.8

**Step 4:** Solve the resulting equation.

99x= 85.8 99x= 85.8

x=85.8 99 x=85.8 99

Since your answer has a decimal in the fraction, you must multiply the numerator and denominator by a power of ten, producing an equivalent fraction with no decimals. Multiplying the numerator and denominator by 10 gives you your answer:

x=858990 x=858990

We can reduce this fraction to

429/495=143/165=13/15

**1.1.3 Decimals with multiple repeating digits as fractions **

Let us see how to simplify a repeating decimal like 0.63636363…

**Step-1:** Let x = recurring number

X = 0.63

**Step-2:** Two digits (63) are repeating.

So, multiply both sides by 10^{2}, i.e., 100.

100x = 100

××

0.63…

100x = 63.63

**Step-3:** Subtract *x* from left side and 0.63… from right side

100x – x = 63.63 – 0.63

99x = 63

**Step-4:** Solve for *x*

99x = 63

X = 63/99

X = 7/11

## Exercise:

# Exercise:

- Rewrite as a simplified fraction

0.2 ̅= ? - Rewrite as a simplified fraction

2.6 ̅=? - Rewrite as a simplified fraction

1.83 ̅=? - Convert the fraction 4/3 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 7/6 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 10/9 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 8/7 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 5/3 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 7/3 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 11/6 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 8/3 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 13/6 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.
- Convert the fraction 11/3 into decimals. If necessary, use a bar to indicate which digit or group of digits repeat.

### What we have learned:

1.1 About decimal numbers, parts of a decimal number, representation of decimal numbers on a scale, application of decimal numbers in everyday life, writing decimals as fractions and How to write repeating decimals as fractions.

1.2 How to write repeating decimals with non-repeating digits as fractions.

1.3 How to write decimals with multiple repeating digits as fractions.

#### Concept Map:

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: