Rational Numbers

Q: What is the difference between rational and irrational numbers?

A rational number can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. An irrational number cannot be written this way — its decimal expansion goes on forever without repeating. Examples of irrational numbers include $\sqrt{2}$, $\pi$, and $e$.

Q: Are all integers rational numbers?

Yes. Every integer $n$ can be written as $\frac{n}{1}$, which is a ratio of two integers with a nonzero denominator. So integers like $-5$, $0$, and $7$ are all rational numbers. The integers are a subset of the rationals.

Rational Numbers

All positive and negative fractions, including integers and so-called improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator. Rational numbers are indicated by the symbol .

Note: Real numbers that aren't rational are called irrational.

Nested diagram of number sets: natural, whole, integers ℤ, rationals ℚ, algebraic, reals ℝ, and complex ℂ with pure imaginary...

Key Formula

\mathbb{Q} = \left\{\,\frac{a}{b} \;\middle|\; a, b \in \mathbb{Z},\; b \neq 0\,\right\}

Where:

$\mathbb{Q}$ = The set of all rational numbers
$a$ = Any integer (the numerator)
$b$ = Any nonzero integer (the denominator)
$\mathbb{Z}$ = The set of all integers: ..., −2, −1, 0, 1, 2, ...

Worked Example

Problem: Determine whether 0.75 is a rational number. If so, express it as a ratio of two integers.

Step 1: Recognize that 0.75 is a terminating decimal. Every terminating decimal can be written as a fraction with a power of 10 in the denominator.

0.75 = \frac{75}{100}

Step 2: Simplify the fraction by finding the greatest common factor of 75 and 100, which is 25.

\frac{75 \div 25}{100 \div 25} = \frac{3}{4}

Step 3: Verify the result fits the definition: 3 is an integer, 4 is a nonzero integer, so 3/4 is a ratio of integers.

a = 3 \in \mathbb{Z},\quad b = 4 \in \mathbb{Z},\quad b \neq 0

Answer: Yes, 0.75 is a rational number because it equals

\frac{3}{4}

Another Example

This example shows a repeating (non-terminating) decimal, which is harder to recognize as rational than the terminating decimal in the first example. It also demonstrates the algebraic technique for converting any repeating decimal to a fraction.

Problem: Show that the repeating decimal

0.\overline{36}

(i.e., 0.363636...) is a rational number by converting it to a fraction.

Step 1: Let x equal the repeating decimal.

x = 0.363636\ldots

Step 2: Multiply both sides by 100 (since the repeating block has 2 digits) to shift the decimal point past one full cycle.

100x = 36.363636\ldots

Step 3: Subtract the original equation from this new equation to eliminate the repeating part.

100x - x = 36.3636\ldots - 0.3636\ldots \implies 99x = 36

Step 4: Solve for x and simplify. The GCF of 36 and 99 is 9.

x = \frac{36}{99} = \frac{4}{11}

Answer:

0.\overline{36} = \frac{4}{11}

, confirming it is rational.

Frequently Asked Questions

Is 0 a rational number?

Yes. Zero is a rational number because it can be written as

\frac{0}{1}

(or

\frac{0}{b}

for any nonzero integer

b

). The numerator is an integer and the denominator is a nonzero integer, so it satisfies the definition.

What is the difference between rational and irrational numbers?

A rational number can be expressed as a fraction

\frac{a}{b}

where

a

and

b

are integers and

b \neq 0

. An irrational number cannot be written this way — its decimal expansion goes on forever without repeating. Examples of irrational numbers include

\sqrt{2}

\pi

, and

e

Are all integers rational numbers?

Yes. Every integer

n

can be written as

\frac{n}{1}

, which is a ratio of two integers with a nonzero denominator. So integers like

-5

0

, and

7

are all rational numbers. The integers are a subset of the rationals.

Rational Numbers vs. Irrational Numbers

	Rational Numbers	Irrational Numbers
Definition	Can be written as a/b where a, b are integers and b ≠ 0	Cannot be expressed as a ratio of two integers
Decimal form	Terminates or repeats (e.g., 0.5, 0.333...)	Non-terminating and non-repeating (e.g., 3.14159...)
Examples	1/2, −7, 0, 0.125, 2.¯3	√2, π, e, √5
Symbol	ℚ	Often written as ℝ \ ℚ (reals minus rationals)
Countability	Countably infinite	Uncountably infinite

Why It Matters

Rational numbers appear constantly in algebra, from solving equations to working with proportions and rates. Understanding which numbers are rational helps you determine whether an answer can be expressed exactly as a fraction or must remain in a form like

\sqrt{3}

. In more advanced courses, the distinction between rational and irrational numbers underpins topics such as limits, number theory, and proofs by contradiction.

Common Mistakes

Mistake: Thinking that all decimals are rational because they 'look like' numbers you can write down.

Correction: Only terminating and repeating decimals are rational. A decimal like 0.101001000100001... (with a pattern that never repeats on a fixed cycle) is irrational. The key test is whether the decimal eventually settles into an endlessly repeating block.

Mistake: Believing that a fraction like

\frac{\sqrt{2}}{3}

is rational because it is written as a fraction.

Correction: The definition requires both the numerator and the denominator to be integers. Since

\sqrt{2}

is not an integer,

\frac{\sqrt{2}}{3}

is irrational. The word 'fraction' in the definition specifically means a ratio of integers.

Related Terms

Irrational Numbers — Real numbers that are not rational
Integers — A subset of the rational numbers
Fraction — The form used to express rational numbers
Real Numbers — The union of rational and irrational numbers
Natural Numbers — Counting numbers, a subset of rationals
Whole Numbers — Non-negative integers, a subset of rationals
Ratio — Rational numbers are defined as ratios of integers
Improper Fraction — A fraction where numerator ≥ denominator, still rational