8th Grade · 8.NS.A.1

Rational and irrational numbers in 8th grade

Rational and irrational numbers are the two kinds of numbers on the number line. A rational number can be written as a fraction of two integers (whole numbers and their negatives), and an irrational number such as $\sqrt{2}$ or $\pi$ cannot. In 8th grade, your child learns to tell the two apart by their decimals and to change repeating decimals into fractions.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

What it means, in plain words

Rational and irrational numbers look different as decimals. The decimal of a rational number either stops or repeats. $\frac{1}{4} = 0.25$ stops, so it is called a terminating decimal. $\frac{1}{3} = 0.333...$ repeats one digit forever, and $\frac{2}{7} = 0.285714285714...$ repeats a block of six digits. A bar over the digits shows the repeat: $0.\overline{3}$ .

Irrational numbers are the numbers that are not rational. Their decimals never stop and never settle into a repeating pattern. $\sqrt{2} = 1.41421356...$ and $\pi = 3.14159265...$ are the best-known examples. The square root of a whole number that is not a perfect square is irrational, so $\sqrt{7}$ is irrational but $\sqrt{9} = 3$ is rational.

Children also see why a fraction has to stop or repeat. When 2 is divided by 7, the only possible remainders are 1 to 6, so a remainder soon comes back and the digits start over. The reverse job, changing repeating decimals to fractions, uses a short piece of algebra, shown in the first example.

Your child should be able to:

Sort numbers such as $\sqrt{49}$ , $\sqrt{50}$ , $0.\overline{6}$ and $-3$ into rational and irrational
Tell a terminating decimal from a repeating one, and both from a decimal that never repeats
Explain why the decimal of a fraction has to stop or repeat
Change a repeating decimal, such as $0.\overline{36}$ , into a fraction

Example

Write $0.\overline{36}$ (0.363636...) as a fraction in simplest form.

Call the number $x$ : $x = 0.363636...$
Two digits repeat, so multiply by 100: $100x = 36.363636...$
Subtract the first line from the second. The repeating parts cancel: $100x - x = 36$ , so $99x = 36$ .
$x = \frac{36}{99} = \frac{4}{11}$ , so 0.363636... equals four elevenths.

Example

Which of $\sqrt{49}$ , $\sqrt{50}$ and $\frac{2}{3}$ is irrational?

$\sqrt{49} = 7 = \frac{7}{1}$ , so it is rational.
$\frac{2}{3} = 0.666...$ repeats, so it is rational.
50 is not a perfect square, because $7^2 = 49$ and $8^2 = 64$ .
So $\sqrt{50}$ is the only irrational number of the three.

Where children usually get stuck

Your child calls $0.666...$ irrational because the digits never stop.
The child remembers that irrational decimals go on forever and misses the second half of the rule: they also never repeat.
What helps. Ask which fraction equals $0.666...$ . Once your child finds $\frac{2}{3}$ , the number has a fraction, so it is rational.
Your child puts $\sqrt{64}$ with the irrational numbers.
Any square root symbol looks irrational. The child judges the number by its symbol, not by its value.
What helps. Ask your child to work out the root first. $\sqrt{64} = 8$ , a whole number, so it is rational. Only the root of a number that is not a perfect square stays irrational.
To change $0.\overline{36}$ into a fraction, your child multiplies by 10, gets $10x = 3.\overline{63}$ and cannot make the repeating parts cancel.
The child multiplies by 10 every time. The power of 10 has to match the length of the repeat, so a two-digit repeat needs 100.
What helps. Ask your child to count the digits under the bar before starting. One digit means multiply by 10, two digits mean 100 and three digits mean 1,000.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The fraction $\frac{1}{4}$ is given. Aiden filled in the decimal and the type. The decimal must equal the fraction exactly. Which statement is true?
Fraction Decimal Type
$\displaystyle \frac{1}{4}$
0.25 Terminating
A
Both are correct
B
The decimal is correct but the type is wrong
C
The type is correct but the decimal is wrong
D
Both are wrong
Show the answer and how to solve it
Both are correct because $\frac{1}{4} = 0.25$ and 0.25 is a terminating decimal.
1. 1Look at the row. It shows the fraction $\frac{1}{4}$ , the decimal 0.25, and the type Terminating.
2. 2 $\frac{1}{4}$ is already in simplest form. Check the decimal value.
3. 3The decimal 0.25 matches exactly, and since it ends, the type Terminating is correct. Both parts are right, so the answer is 'Both are correct.'
Common mistake. A common mistake is labeling 0.25 as Repeating. Because $\frac{1}{4}$ has a denominator of 4, which has only the prime factor 2, its decimal ends and the type is Terminating.
The idea underneath. Equivalent fractions have the same decimal expansion, so simplifying first can make the terminating-or-repeating type easier to see.
2On level
Look at the decimal below. Which type of decimal is it?
$\displaystyle 2.4500$
A
Terminating
B
Repeating
C
Non-terminating, non-repeating
Show the answer and how to solve it
The answer is Terminating because 2.4500 stops exactly as written; the trailing zeros are place-fillers, not a repeating block.
1. 1Look at the decimal 2.4500. There is no overline, no parentheses, and no ellipsis.
2. 2The trailing zeros do not extend the decimal — $2.4500 = 2.45$ , and the decimal stops there.
3. 3A decimal is terminating if it ends with no continuation symbol. The correct label is Terminating.
Common mistake. A common mistake is assuming the repeated zeros mean a digit is cycling. Without a repeat marker such as a bar or parentheses, the zeros are simply trailing and the decimal terminates.
The idea underneath. A decimal is repeating if some digit block repeats forever, and that cycle can begin after a nonrepeating prefix.

Fraction	Decimal	Type
$\displaystyle \frac{1}{4}$	0.25	Terminating

3On level

Fill in the missing numbers. Write the final fraction in simplest form.

	$\displaystyle x$	$\displaystyle =$	$\displaystyle 0.1\overline{6}$
Shift 1	$\displaystyle x$	$\displaystyle =$	$\displaystyle 1.\overline{6}$
Shift 2	$\displaystyle x$	$\displaystyle =$	$\displaystyle 16.\overline{6}$
Subtract	$\displaystyle 90x$	$\displaystyle =$	$\displaystyle 16.\overline{6} - 1.\overline{6}$
Result	$\displaystyle 90x$	$\displaystyle =$
Fraction	$\displaystyle x$	$\displaystyle =$	$\displaystyle /$

Show the answer and how to solve it

The answer is $\frac{1}{6}$ because shifting to 10x and 100x makes the repeating part cancel, leaving $\frac{15}{90}$ , which simplifies to $\frac{1}{6}$ .

1Let $x = 0.1\overline{6}$ . Multiply by 10 and by 100 so the repeating 6s line up on the right side of both equations.
2Subtract the two shifted equations. The repeating parts cancel because both right sides end with the same repeating 6.
3Divide both sides by 90 to solve for $x$ .
4Simplify $\frac{15}{90}$ by dividing numerator and denominator by $\gcd(15, 90) = 15$ .

Common mistake. A common mistake is writing the denominator as 99, as if the whole decimal repeated from the start. Because 1 is a non-repeating digit, the aligned shifts are 10x and 100x, giving denominator 90, not 99. Also, $\frac{15}{90}$ must still be simplified by dividing by 15.

The idea underneath. Pick powers of 10 that move the decimal point to the start and end of one full repeating block; subtraction then removes the infinite repeating tail.

4Stretch
Alex already sorted these numbers into Rational and Irrational. Move only the misplaced numbers so the sort is correct.
Drag each item into the group where it belongs, or tap the item, then tap the group.
Rational
$\displaystyle {-5}$
$\displaystyle 0.45$
Irrational
$\displaystyle 0.\overline{45}$
$\displaystyle \sqrt{82}$
$\displaystyle \sqrt{81}$
Keyboard instructions: focus an item and press Enter or Space to pick it up. Move focus to a bin or the bank and press Enter or Space to drop it. Press Escape to cancel.
Show the answer and how to solve it
Move $\sqrt{81}$ and $0.\overline{45}$ to Rational. Then Rational has $-5$ , $0.45$ , $0.\overline{45}$ , and $\sqrt{81}$ , and Irrational has only $\sqrt{82}$ .
1. 1Look at the Irrational bin. It contains $\sqrt{81}$ , $\sqrt{82}$ , and $0.\overline{45}$ .
2. 2 $81$ is a perfect square because $81 = 9^2$ , so $\sqrt{81} = 9$ . Since $9$ is an integer, $\sqrt{81}$ is rational and does not belong in the Irrational bin. $82$ is not a perfect square, so $\sqrt{82}$ stays irrational.
3. 3 $0.\overline{45}$ has a repeating decimal expansion, so it is rational and does not belong in the Irrational bin. The negative sign on $-5$ does not change rationality, so $-5$ is rational and was already correctly placed.
4. 4Move $\sqrt{81}$ and $0.\overline{45}$ to the Rational bin. The corrected sort is:
Common mistake. A common mistake is treating every square root as irrational and every repeating decimal as irrational. But $\sqrt{81} = 9$ is an integer and therefore rational, and $0.\overline{45}$ repeats so it is also rational.
The idea underneath. The $\sqrt{\ }$ symbol and the overline notation do not decide the category alone; what matters is whether the value is rational or irrational. Perfect-square roots equal integers (rational), and repeating decimals always equal fractions (rational).