Number System

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A number 'r' is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0. Examples: 1/2, -3/4, 5 (which is 5/1), 0 (which is 0/1).

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A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0. Examples: √2, √3, √15, π, 0.10110111011110...

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True. Every integer m can be expressed in the form m/1, where both m and 1 are integers and 1 ≠ 0. Therefore, every integer is a rational number.

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Rational numbers have either: 1) Terminating decimal expansions (like 7/8 = 0.875), or 2) Non-terminating recurring (repeating) decimal expansions (like 1/3 = 0.333... = 0.3̄).

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Irrational numbers have non-terminating non-recurring (non-repeating) decimal expansions. For example, √2 = 1.4142135... and π = 3.14159265...

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Let x = 0.6̄ = 0.666... 10x = 6.666... 10x = 6 + 0.666... = 6 + x 10x - x = 6 9x = 6 x = 6/9 = 2/3…

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The collection of real numbers is denoted by R. It includes all rational numbers and all irrational numbers together. Every point on the number line represents a unique real number.

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The sum of a rational number and an irrational number is always irrational. For example, 2 + √3 is irrational because √3 has a non-terminating non-recurring decimal expansion.