The World of Numbers

A rational number can be written as a fraction p/q of two integers with q not zero, and its decimal either ends or repeats. An irrational number cannot be written as such a fraction, and its decimal goes on forever without repeating, like √2 or π.

Real numbers are all the rational and irrational numbers taken together. They include natural numbers, whole numbers, integers, fractions, and non-repeating decimals. Every real number can be marked as a point on the number line.

A decimal represents a rational number if it either terminates (ends) or repeats a pattern of digits forever. For example, 0.75 and 0.333... are rational. A decimal that neither ends nor repeats is irrational.

√2 is an irrational number. It cannot be written as a fraction of two integers, and its decimal expansion 1.41421356... continues without ending or repeating. Roots like this that stay irrational are called surds.

Irrational numbers like √2 can be located using geometric construction, often with the Pythagoras theorem to build a length equal to the root. A right triangle with legs of length 1 produces a hypotenuse of √2, which is then transferred onto the number line with a compass.

The laws of exponents let us simplify powers: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), and (a^m)^n = a^(mn). In Class 9 these rules are extended so they hold for real-number bases and rational exponents.