A rational number can be written as a fraction of two integers, so its decimal stops or repeats. An irrational number, such as the square root of 2 or pi, has a decimal that never stops and never repeats. Estimate irrational numbers by squaring decimals, and write repeating decimals as fractions using subtraction.
Key facts:
- Standards 8.NS.A.1 and 8.NS.A.2: tell rational from irrational numbers and approximate irrational numbers.
- Every repeating decimal equals a fraction, such as \(0.\overline{3}\) = 1/3.
- The square root of a whole number that is not a perfect square is irrational.
- Bracket a square root between two decimals whose squares surround the number.
- To compare two numbers, use decimals or square both positive numbers.
What are rational and irrational numbers?
Every number on the number line is either rational or irrational. A rational number can be written as a fraction of two integers, so its decimal either stops or repeats a block of digits forever. An irrational number cannot be written that way. Its decimal never stops and never repeats.
Terminating decimal: a decimal that ends, such as \(0.375\).
Repeating decimal: a decimal with a block of digits that repeats forever, such as \(0.\overline{27}\).
Perfect square: the square of an integer, such as 25 or 49. Its square root is an integer, so it is rational.
The square root of any whole number that is not a perfect square is irrational. So \(\sqrt{2}\), \(\sqrt{10}\) and \(\pi\) are irrational, while \(\sqrt{49}=7\) is rational.
How do you write a repeating decimal as a fraction?
- Let x equal the repeating decimal.
- Multiply both sides by 10, 100 or 1,000 so one full block moves left of the decimal point.
- Subtract the original equation to cancel the repeating part.
- Divide to solve for x and simplify.
Worked example. Write \(0.\overline{27}\) as a fraction. Two digits repeat, so multiply by 100: \(100x=27.2727\). Subtract \(x=0.2727\) to get \(99x=27\), so \(x=\frac{27}{99}=\frac{3}{11}\).
How do you approximate a square root?
Find the two perfect squares that surround the number, then test decimals. Since \(1^{2}=1\) and \(2^{2}=4\), the number \(\sqrt{2}\) is between 1 and 2. Testing tenths shows \(1.4^{2}=1.96\) and \(1.5^{2}=2.25\), so it is between 1.4 and 1.5. One more digit gives \(1.41^{2}=1.9881\) and \(1.42^{2}=2.0164\).
| Test | Square | Compared with 2 |
|---|---|---|
| \(1.4\) | \(1.96\) | too small |
| \(1.41\) | \(1.9881\) | too small |
| \(1.42\) | \(2.0164\) | too big |
| \(1.5\) | \(2.25\) | too big |
How do you compare irrational numbers?
Use decimal approximations, or square both numbers when both are positive. To compare \(\sqrt{7}\) and \(2.6\), note \(2.6^{2}=6.76\), which is less than 7. So \(\sqrt{7}>2.6\). On the number line, the number farther right is greater.
Rational and Irrational Numbers: frequently asked questions
Is zero a rational number?
Yes. Zero can be written as \(\frac{0}{1}\), a fraction of two integers, so it is rational. Every integer is rational because it can be written with a denominator of 1.
Is pi a rational number?
No. Pi is irrational because its decimal never ends and never repeats. Fractions such as \(\frac{22}{7}\) and decimals such as 3.14 are only close approximations.
How can I tell if a square root is irrational?
Check whether the number under the radical is a perfect square. If it is, the root is an integer. If it is not, such as \(\sqrt{20}\), the root is irrational.
Why does a repeating decimal equal a fraction?
Multiplying by a power of 10 shifts the repeating block, and subtracting cancels the endless part. That leaves a whole number equal to a multiple of x, so x is a fraction.
Does a decimal with a pattern have to be rational?
No. A decimal is rational only when one block of digits repeats forever. A pattern such as \(0.101001000100001\) with a growing gap never repeats a block, so it is irrational.
See every chapter on the Grade 8 math page.