8th Grade - Irrational Numbers


  • The real number system can be divided into two categories:
    1. Rational Numbers
    2. Irrational Numbers

Irrational Numbers

  • Real numbers that can't be expressed as a ratio of integers (such as pq where p and q are integers, q0) are called irrational numbers
  • In simple terms, real numbers that are not rational are called irrational numbers.
  • Some examples of irrational numbers are π2e (Euler's number), etc.

Properties of Irrational Numbers

  • Irrational numbers are real numbers.
  • The decimal expansion of an irrational number is always non-terminating & non-recurring.
    • Example: the value of π=3.141592......
  • If x is rational and y is irrational, then x+y and x-y are irrational numbers. Also, xy and xy are irrational numbers, y0.

Approximating Values of Irrational Numbers

  • Let us understand the process of evaluating the approximate value of an irrational number by the example below.

Question: Approximate the value of 50.


    • Identify the perfect squares less than and greater than 50, and locate them on a number line.

49<50<64 or 7<50<8

    • 50 lies between the whole numbers 7 and 8. 50 is closer to 49. Hence, the whole number approximation of 50 is 7.
    • To find the decimal component of the approximation, use the hit & trial method.

Try 7.1


  • Another method of approximating the value of an irrational number is by using the shortcut formula xx+y2y. This is used to compute the approximate square root of a non-perfect square number. The number y is the nearest perfect square number to the number x.
  • To calculate the approximate cube root of a non-perfect cube number, we can use the shortcut formula b32b+a2a+ba3 where a is the nearest perfect cube number to the number b.

Solved Examples

Question 1: Order the given numbers from least to greatest.

0.25-10-132, and 50

Solution: The whole number approximation of 10=3, and 50=7. Also, 132=6.5.

Let us now arrange the given numbers based on the information above.



Question 2: Fill in the blank using an appropriate symbol (<>=).

26          2π-1

Solution: The whole number approximation of 26=5. Also, 2π-1=2×3.14-1=6.28-1=5.28

Thus, 26     <     2π-1

Cheat Sheet

  • Irrational numbers are real numbers. 
  • Irrational numbers can't be expressed as a ratio of two integers.
  • Irrational numbers are non-terminating and non-repeating (recurring) decimals. 
  • The square root of a non-perfect square is an irrational number. 
  • π (Pi) is a famous irrational number. The decimal approximation of π is 3.14. 
  • Use approximation to estimate the value of the square root of a non-perfect square. 

Blunder Areas

  • The product of π×π is π2, which is an irrational number.
  • The product of 2×2 is 2, which is a rational number.