Real Numbers – Concept & Explanation

Key Concepts

• Fraction

• Rational number

• Decimal number

• Irrational number

• Number line

• Perfect squares

Compare and order Real Numbers

Representation of a fraction: 

Charles’s mother made a pizza at home on a Sunday. She wants to serve it to Charles, Lisa (Charles’s elder sister) and Michelle (Charles’s younger sister). 

Can you guess how much of pizza each one will get? 

1313

right? 

1313

is called a fraction. 

Rational number: 

Rational numbers are those numbers that are integers and can be expressed in the form of x/y, where both numerator and denominator are integers. 

1/3

1/3 is an example of rational number. 

Decimal number: 

Similarly, if we convert the fraction 1/2

we get 0.5 which is a terminating decimal number. 

Irrational Numbers: 

An irrational number is a type of real number that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. When irrational numbers are expressed in the decimal form, they go on forever, even after the decimal point, without repeating numbers. Thus, they are also known as non-terminating, non-repeating numbers. 

Let’s see an example: 

If we find the value of √5, 

The answer will be 2.23606797749979 

If we observe the numbers after decimal point, the numbers are non-terminating non-repeating. 

The most common irrational number is: 

Pi (π) = 22/7

= 3.14159265358979… 

Number Line: 

Let’s think about where 4.5, 1.838383… and π should be placed on a number line. 

• 1.838383… is placed closer to the 2 because as a rounded number, it would be rounded to 2. 
• π is placed closer to the 3 because π is approximately 3.1416. 
• 4.5 is halfway between 4 and 5. 

Perfect Squares: 

Table showing perfect squares and their square roots: 

1.3.1 Approximation of Irrational Numbers 

Irrational Numbers: 

An irrational number is a type of real number that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. When irrational numbers are expressed in the decimal form, they go on forever, even after the decimal point, without repeating numbers. Thus, they are also known as non-terminating, non-repeating numbers. 

If N is irrational, then N is not equal to

p/q

, where p and q are integers and q is not equal to 0. 

Let us see an example: 

If we find the value of √5, 

The answer will be 2.23606797749979 

Estimating irrational numbers: 

1. Count up until you hit a square root that works. 

2. Count down until you hit a square root that works. 

3. Square root the high and low number, then graph their points on a number line. 

4. Your estimate should be somewhere between those two numbers. 

Example: 

Estimate √5 using perfect squares, and then graph your estimate on the number line 

Solution: 

Step1: Count up until you hit a square root that works. 

Step 2: Count down until you hit a square root that works. 

Step 3: Square root the high and low number, then graph their points on a number line. 

Step 4: Your estimate should be somewhere between those two numbers. 

Step 5: For better approximation, square decimals between 2 and 3. 

1.3.2 Comparing Two Irrational Numbers 

Let us compare the two irrational numbers √7 and 2.513461 

 Step1: Approximate √7 using perfect squares. 

√4 < √7 < √9 

2 < √7 < 3 

Find a better approximation by using decimals 

2.6 x 2.6 = 6.76 

2.7 x 2.7 = 7.29 

2.6 < √7 <2.7 

Step 2: Approximate 2.513461 as a rational number by rounding to the nearest tenth. 

2.513461 = 2.5 

Step 3: Plot each approximation on a number line to compare. 

With the above approximation, we can say 

2.513461 < √7  

1.3.3 Comparing and Ordering Rational and Irrational Numbers 

Let us write the following set of rational and irrational numbers in order from least to greatest. 

8

15

, 8.22, 8

19, 8.35235246… 

Step 1: To compare numbers, you must first make them all into decimals. 

8

1515

= 8.2 

8.22 

8

1919

= 8.1111111….  

8.35235246… 

Step 2: Approximate 8.35235246… and 8.1111111…. as rational numbers by rounding them to the nearest tenth. 

8.35235246… = 8.3 

8.1111111… = 8.11 

Step 3: Plot each approximation on a number line to compare. 

Exercise:

1.     Estimate √24 and graph your estimation on a number line

• Put the following sets of numbers in order on the number line below.

• Put the following sets of numbers in order on the number line below

• Compare the following rational numbers using the symbols < or >

• Which of the following rational or irrational numbers belongs between the 5 and the 6 on the number line below?

• Compare the following numbers using < or >

√32 ⬜ 5.1 √38 ⬜ √42 √17 ⬜ 5/2 √49 ⬜ 7.1

• Compare the following numbers using < or >

√99 ⬜ 28/3 √17 ⬜ 4.5 43/5 ⬜ √65 √12 ⬜ √21

• List the following numbers in order from least to greatest.

• List the following numbers in order from least to greatest.

• List the following numbers in order from least to greatest.

What we have learnt

• 1.3.1 About fractions, rational numbers, decimal numbers, irrational numbers, number line, perfect squares and Approximation of Irrational Numbers

■ 1.3.2 Comparing Two Irrational numbers

• 1.3.3 Comparing and Ordering Rational and Irrational Numbers

Concept Map: