Rational Numbers Formula

Rational numbers are numbers that can be expressed as a ratio of two integers (a/b where b ≠ 0).

The Formula

Q={ab∣a,b∈Z,  b≠0}

When to use: Any number you can write as a fraction, including decimals that end or repeat.

Quick Example

12,−34,0.75,0.333…

Notation

Q denotes the set of rational numbers; ab denotes the ratio of integers a and b

What This Formula Means

Numbers that can be expressed as a ratio of two integers (ab where b≠0).

Any number you can write as a fraction, including decimals that end or repeat.

Formal View

Q={pq:p∈Z,  q∈Z,  q≠0} with equivalence pq=rs  ⟺  ps=qr

Worked Examples

Example 1

easy
Place the following numbers in order from least to greatest: 34, 0.6, 710.

Answer

0.6<710<34

First step

1
Convert all to decimals: 34=0.75, 0.6=0.6, 710=0.7.

Full solution

  1. 2
    Order the decimals: 0.6<0.7<0.75.
  2. 3
    In original form: 0.6<710<34.
To compare rational numbers in different forms, convert them all to the same representation—usually decimals—then order them. Every rational number can be expressed as a terminating or repeating decimal.

Example 2

medium
Express 0.36‾ as a fraction in simplest form.

Example 3

easy
Plot 12, −14, and 34 on a number line. List them from least to greatest.

Common Mistakes

  • Calling every decimal rational - only terminating or repeating decimals are; non-repeating infinite decimals are irrational.
  • Allowing zero in the denominator - b must be nonzero for a/b to be a rational number.
  • Thinking a number must look like a fraction to be rational - integers like 7 are rational too (7/1).

Why This Formula Matters

Rational numbers complete the number system for everyday arithmetic — every measurement, price, and fraction lives here, and they are exactly the decimals that terminate or repeat. Knowing the boundary sets up the dramatic contrast with irrationals like 2 and π. Recognizing it by "Can this number be written as one integer divided by another (with the decimal ending or repeating)?" — rather than by familiar numbers — is what lets a student tell it apart from irrational numbers and integers and fractions in a mixed problem set.

Frequently Asked Questions

What is the Rational Numbers formula?

Numbers that can be expressed as a ratio of two integers (ab where b≠0).

How do you use the Rational Numbers formula?

Any number you can write as a fraction, including decimals that end or repeat.

What do the symbols mean in the Rational Numbers formula?

Q denotes the set of rational numbers; ab denotes the ratio of integers a and b

Why is the Rational Numbers formula important in Math?

Rational numbers complete the number system for everyday arithmetic — every measurement, price, and fraction lives here, and they are exactly the decimals that terminate or repeat. Knowing the boundary sets up the dramatic contrast with irrationals like 2 and π. Recognizing it by "Can this number be written as one integer divided by another (with the decimal ending or repeating)?" — rather than by familiar numbers — is what lets a student tell it apart from irrational numbers and integers and fractions in a mixed problem set.

What do students get wrong about Rational Numbers?

The procedure for rational numbers is the easy part; the trap is calling every decimal rational. Asking "Can this number be written as one integer divided by another (with the decimal ending or repeating)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Rational Numbers formula?

Before studying the Rational Numbers formula, you should understand: fractions, decimals, integers.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Cube Roots, Square Roots, and Irrational Numbers →
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