Rational and Irrational Numbers - Terminating and Non Terminating Decimals

Terminating and Non Terminating Decimals 

1. Terminating Decimals - A decimal that ends after a finite number of digits.

Eg: 1/4 = 0.25

 2. Non-Terminating Repeating Decimal - A decimal that goes on forever but has a repeating pattern.

Eg: 1/3 = 0.333333  ( Repeating 3)

 3. Non-Terminating Non-Repeating Decimal - A decimal that goes on forever with no repeating pattern.

Eg: π = 3.14159..., and √2 = 1.4142...

 Determine whether a rational number has a terminating or non-terminating repeating decimal representation without performing actual division:

Steps to Check Decimal Representation:

  Given a rational number in the form:

                p/q ( in simple form)

Step 1:

Ensure that p/q is in lowest terms (i.e., p and q are coprime).

Step 2: Factor the denominator q

Check the prime factors of the denominator.

If the only prime factors of q are 2 and/or 5, then the decimal is terminating.

If q has any other prime factors (e.g., 3, 7, 11, etc.), then the decimal is non-terminating and repeating.

 Decimal representation is terminating if and only if the denominator (after simplifying) divides a power of 10 — i.e., is of the form 2^m⋅5ⁿ.

 Examples:

i) 3/40

Given Number is 3/40 

Since, 40 = 2³ x 51  = 40 = 2³ x 5¹

I.e. 40 can be expressed as 2^m x 5ⁿ

Therefore, 3/40 is a terminating decimal.

ii) 7/30

Given Number is 7/30

Since, 30 = 2 x 3 x 5 = 2¹ x 3¹ x 5¹

I.e. 30 cannot be expressed as 2^m x 5ⁿ

Therefore, 7/30 is a non-terminating decimal.