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√ IRRATIONAL

Understand
Irrational Numbers

Discover the numbers that never repeat and never terminate: learn how to tell them apart from rationals and convert repeating decimals to fractions.

8th Grade Math

What Is a Rational Number?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). When you divide, you always get a decimal that either terminates or repeats.

Terminates
\(\frac{3}{8} = 0.375\) — the digits stop.
Repeats
\(\frac{1}{3} = 0.\overline{3}\) — the digit 3 repeats forever.

What Is an Irrational Number?

An irrational number is a number that cannot be written as \(\frac{a}{b}\) with integers \(a\) and \(b\). Its decimal expansion goes on forever without ever settling into a repeating pattern.

\(\sqrt{2}\)
\(1.41421356...\) — never repeats, never ends.
\(\pi\)
\(3.14159265...\) — infinite non-repeating digits.

The Number Line: Where Do They Live?

0 1 2 3 4
\(\frac{1}{2}\)
\(\frac{7}{3}\)
\(\sqrt{2}\)
 ≈1.414
\(\pi\)
 ≈3.14 Rational Irrational

Both rational and irrational numbers live on the same number line. Together they fill every point: the rationals and the irrationals combine to form the real numbers.

The Decimal Expansion Rule

1
Every real number has a decimal expansion: a sequence of digits after the decimal point.
2
If the expansion terminates or eventually repeats, the number is rational.
3
If the expansion never terminates and never repeats, the number is irrational.

Repeating vs Non-Repeating

Rational
\(\frac{1}{3} = 0.\overline{3}\)
 3 3 3 3 3 3 3 3 3 ...
\(\frac{2}{7} = 0.\overline{285714}\)
 285714 285714 285714 ...
\(\frac{5}{4} = 1.25\)
 Terminates (stops) Irrational
\(\sqrt{2} = 1.41421356...\)
 No pattern emerges
\(\pi = 3.14159265...\)
 No pattern emerges
\(\sqrt{3} = 1.73205080...\)
 No pattern emerges

Rational vs Irrational

Property Rational Irrational
Fraction form \(\frac{a}{b}\) Yes No
Decimal terminates? Sometimes Never
Decimal repeats? Sometimes Never
Examples \(\frac{1}{2},\; 0.\overline{6},\; -4\) \(\sqrt{2},\; \pi,\; \sqrt{5}\)

How to Identify the Type

1
Can you write it as \(\frac{a}{b}\)? If yes, it's rational. Done.
2
Is it a square root? Check if the number under the radical is a perfect square. \(\sqrt{9} = 3\) is rational. \(\sqrt{7}\) is irrational.
3
Look at the decimal. If it terminates or has a repeating block, it's rational. Otherwise, irrational.
4
Known irrationals: memorize that \(\pi\), \(e\), and \(\sqrt{p}\) (for any non-perfect-square \(p\)) are irrational.

Converting Repeating Decimals to Fractions

Since repeating decimals are rational, we can always convert them back to \(\frac{a}{b}\). The method uses algebra:

1
Set \(x\) equal to the decimal. Example: \(x = 0.\overline{36}\).
2
Multiply by \(10^n\) where \(n\) is the number of repeating digits. Here \(n = 2\): \(100x = 36.\overline{36}\).
3
Subtract the original: \(100x - x = 36\), so \(99x = 36\).
4
Solve: \(x = \frac{36}{99} = \frac{4}{11}\).

Example 1: Convert \(0.\overline{3}\)

1
Let \(x = 0.333...\)
2
One repeating digit, so multiply by \(10\): \(10x = 3.333...\)
3
Subtract: \(10x - x = 3.333... - 0.333... = 3\)
4
\(9x = 3\), so \(x = \frac{3}{9} = \frac{1}{3}\)

Example 2: Convert \(0.\overline{18}\)

1
Let \(x = 0.181818...\)
2
Two repeating digits, so multiply by \(100\): \(100x = 18.181818...\)
3
Subtract: \(100x - x = 18\), so \(99x = 18\)
4
\(x = \frac{18}{99} = \frac{2}{11}\)

Example 3: Convert \(0.8\overline{3}\)

This one has a non-repeating part (the 8) before the repeating part (the 3). We need two multiplications:

1
Let \(x = 0.8333...\)
2
\(10x = 8.333...\) and \(100x = 83.333...\)
3
Subtract: \(100x - 10x = 83.333... - 8.333... = 75\)
4
\(90x = 75\), so \(x = \frac{75}{90} = \frac{5}{6}\)

Why Is \(\sqrt{2}\) Irrational?

1 1
\(\sqrt{2}\)
 Pythagorean Theorem:
\(1^2 + 1^2 = d^2\)
\(2 = d^2\)
\(d = \sqrt{2}\)
Cannot be written as \(\frac{a}{b}\)

The diagonal of a unit square has length \(\sqrt{2}\). Ancient Greeks proved (by contradiction) that no fraction equals \(\sqrt{2}\). Its decimal, \(1.41421356...\), never repeats.

Famous Irrational Numbers

\(\sqrt{2}\)
\(1.41421356...\) Diagonal of a unit square.
\(\pi\)
\(3.14159265...\) Ratio of circumference to diameter.
\(e\)
\(2.71828182...\) Base of natural logarithms.

Common Mistakes

✕ "Long decimal = irrational"
\(0.142857142857...\) looks long but it repeats: it equals \(\frac{1}{7}\), which is rational.
✕ "All square roots are irrational"
\(\sqrt{25} = 5\), \(\sqrt{0.04} = 0.2\): perfect squares give rational results.
✕ "\(\pi = \frac{22}{7}\)"
\(\frac{22}{7}\) is only an approximation. The true value of \(\pi\) is irrational and cannot be written as any fraction.

Summary: The Full Picture

Number Decimal Rational? Reason
\(\frac{3}{4}\) 0.75 Yes Terminates
\(\frac{2}{3}\) \(0.\overline{6}\) Yes Repeats
\(\sqrt{10}\) 3.16227... No 10 not a perfect square
\(\pi\) 3.14159... No Proven irrational
\(-7\) \(-7.0\) Yes \(\frac{-7}{1}\)

Question 1 of 5

Which of the following numbers is irrational?

Question 2 of 5

Convert the repeating decimal \(0.\overline{45}\) to a fraction in simplest form.

Question 3 of 5

A number has the decimal expansion \(0.101001000100001...\) where the number of zeros between each 1 keeps increasing. Is this number rational or irrational?

Question 4 of 5

Which statement is true about the number \(\sqrt{64}\)?

Question 5 of 5

Convert \(0.5\overline{8}\) to a fraction. Which of the following is correct?