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A rational number is one that can be written as a fraction a/b of integers with b ≠ 0. In decimal form, every rational number either terminates (like 0.75) or repeats forever (like 0.333...). An irrational number is a real number that cannot be written as such a fraction, so its decimal never ends and never settles into a repeating block. Examples include √2, √3 and π. Rational and irrational numbers together make up the real numbers, which fill the whole number line.
Start by converting fractions to decimals with long division and noticing that the remainders must eventually repeat. Then approximate √2 by squaring guesses (1.4² = 1.96, 1.5² = 2.25) to show the digits keep going without a pattern. Place rational and irrational numbers on a number line together, and practice classifying numbers such as √16, √15, 0.121212... and 22/7.
You already know many kinds of numbers: whole numbers, integers, fractions and decimals. All of these can be written as one integer divided by another nonzero integer. For example, 5 = 5/1, -3 = -3/1 and 0.75 = 3/4. Numbers that can be written this way are called rational, from the word "ratio."
Rational numbers have a recognizable decimal form. When you divide a by b with long division, each remainder is smaller than b, so there are only finitely many possible remainders. Either a remainder of 0 appears and the decimal stops, or some remainder comes back and the digits start repeating. That is why 3/8 = 0.375 terminates and 2/11 = 0.181818... repeats. The process also runs backward: a repeating decimal can be turned into a fraction. If x = 0.181818..., then 100x = 18.181818..., and subtracting gives 99x = 18, so x = 18/99 = 2/11.
Some numbers do not fit this pattern. Try to find √2. Since 1² = 1 and 2² = 4, it lies between 1 and 2. Since 1.4² = 1.96 and 1.5² = 2.25, it lies between 1.4 and 1.5. You can keep narrowing forever, getting 1.41421356..., and the digits never settle into a repeating block. The ancient Greeks proved that no fraction a/b, squared, gives exactly 2. Numbers like this are called irrational. Other examples are √3, √5, any square root of a whole number that is not a perfect square, and the circle constant π.
Rational and irrational numbers together form the real numbers. Every point on the number line is one or the other, never both. To place an irrational number, use rational approximations: √10 is a little more than 3 because 3² = 9, and π sits just to the right of 3.14.
Be careful with arithmetic rules. The sum of a rational number and an irrational number is always irrational, so 1 + √2 is irrational. But two irrational numbers can combine to give a rational one: √2 × √2 = 2, and (1 + √2) + (1 - √2) = 2. Classify each result by checking it, not by assuming.