Summary and Vocabulary of Rational and Irrational Numbers

By Michael Avidon, math editor

Vocabulary

• irrational number: a number that cannot be written as the quotient of two integers
• lower bound: or under-estimate, is a number that is less than the value of a given expression
• radical: a square root, cube root, or other root of a number, written with the symbol
• radicand: the number inside a radical symbol √
• repeating decimal: a decimal with an infinite number of digits that at some point follows a repeating pattern
• terminating decimal: a decimal with a finite number of digits
• upper bound: or over-estimate, is a number that is greater than the value of a given expression

More in this series
At-home math lessons: Rational and irrational numbers Decimal expansions of rational numbers At-home math lesson: irrationality of the square root of 2 Lesson 2: Approximation of Irrational Numbers At-home math lesson: Approximation of pi At-home lessons: the golden ratio At-home math lesson: doubling and dividing

Facts and Theorems

• Any rational number can be represented by a terminating or repeating decimal.
• Conversely, any terminating or repeating decimal represents a rational number.
• A fraction written in reduced form will have a terminating decimal if and only if the prime factors of the denominator are 2 and 5.
• An irrational number has a non-terminating, non-repeating decimal representation.

Other

• An overbar on some digits in a decimal indicates that those digits repeat indefinitely.
• To convert a number x that has a repeating decimal to fraction form, do the following:
  • Multiply by a power of 10 so that the first repeating sequence is just to the left of the decimal point.
  • Multiply by a power of 10 (if necessary) so that the first repeating sequence is just to the right of the decimal point.
  • Subtract the second number from the first to cancel the repetitions.
  • Solve for x.
• If k² < n < (k + 1)², then k < √n < k + 1. You can guess and check (numbers between k and k + 1) to get an approximation of √n to the nearest tenth.
• Estimating cube roots can be done in a similar manner to square roots, by first finding the two perfect cubes that are closest to the radicand.
• Using an approximation to the nearest tenth for part of an expression does not necessarily lead you to an approximation to the nearest tenth for the value of the whole expression.

MORE: Print this out for reference | Next lesson: Lesson 2: Approximation of Irrational Numbers