World Of Numbers Class 9

Key ConceptsThis chapter explains how to classify numbers as rational or irrational. Exams commonly test decimal patterns, square roots, and the position of numbers on the number line.

A rational number can be written as p/q, where p and q are integers and q ≠ 0.

Rational decimals either:

terminate, such as 0.75; or

repeat, such as 0.333...

An irrational number cannot be written as p/q.

Irrational decimals are non-terminating and non-repeating, such as 1.414213...

√n is irrational when n is not a perfect square. For example, √7 is irrational.

√49 = 7, so √49/7 = 1, which is rational.

Rational and irrational numbers together form the real numbers. Both can be represented on the number line.

Between any two different real numbers, infinitely many rational and irrational numbers exist.

Before you continue: How can you classify 0.121212... and √12?

Key TermsRational number - A number expressible as p/q, where p and q are integers and q ≠ 0.

Irrational number - A number that cannot be expressed as p/q.

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

Recurring decimal - A decimal in which one or more digits repeat forever.

Non-terminating non-recurring decimal - A decimal that continues forever without a repeating pattern.

Perfect square - A number such as 1, 4, 9, or 16 whose square root is an integer.

Active RecallRational number__________

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Irrational number__________

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Terminating decimal__________

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Recurring decimal__________

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Non-terminating non-recurring decimal__________

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Perfect square__________

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Worked Examples1. Classifying a decimalClassify 0.625.

The decimal ends after three places.

Therefore, it is terminating.

Every terminating decimal is rational.

Answer: 0.625 is rational.Now you try: Classify 0.454545...

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2. Classifying a square rootClassify √8.

8 is not a perfect square.

Therefore, √8 cannot be written as an integer.

√8 is irrational.

Answer: √8 is irrational.Now you try: Classify √81/9.

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3. Simplifying before classifyingClassify √12/√3.

√12/√3 = √(12/3).

√4 = 2.

Since 2 is an integer, it is rational.

Answer: √12/√3 is rational.Now you try: Classify √50/√2.

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Practice Questions(Easy) Classify 0.777... as rational or irrational.

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(Easy) Classify √11 as rational or irrational.

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(Medium) Explain why 3.141592... is irrational if its digits do not repeat.

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(Medium) Is √49/7 rational or irrational? Give a reason.

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(Medium) Find one rational number between 2 and 3.

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(Hard) Prove that √7 cannot be written as p/q, where p and q are integers and q ≠ 0.

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Answer KeyRational

Irrational

It is non-terminating and non-recurring.

Rational; √49/7 = 7/7 = 1.

For example, 2.5.

Assume √7 = p/q in lowest form. Squaring gives p^2 = 7q^2, so 7 divides p. This makes 7 divide q as well, contradicting lowest form. Therefore, √7 is irrational.

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