8.NS.1: Rational and Irrational Numbers
Also written as 8.NS.A.1 and CCSS.MATH.CONTENT.8.NS.A.1 · Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers.
The Official Wording
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.
Common Core State Standards, Grade 8, The Number System · Read it at the source · Checked October 14, 2026
What 8.NS.1 Means in Plain English
Eighth graders sort numbers into rational and irrational. A fraction’s decimal either ends, like 5/8 = 0.625, or repeats, like 2/11 = 0.181818..., because long division runs out of new remainders. Numbers like √2 and π never end or repeat, so no fraction equals them: they are irrational.
Students also run it backward: x = 0.181818... gives 100x - x = 18, so x = 18/99 = 2/11. Together the two kinds make the real numbers, which 8.NS.2 places on a number line. High school proves that a rational plus an irrational is irrational (HSN-RN.3).
8.NS.1 I Can Statements
- I can tell whether a number is rational or irrational.
- I can show by long division that a fraction’s decimal repeats.
- I can rewrite a repeating decimal like 0.181818... as a fraction.
8.NS.1 Worked Example
A repeating decimal turned back into a fraction, with one subtraction that makes the endless tail disappear.
Write 0.181818..., where the block 18 repeats forever, as a fraction in lowest terms. Is the number rational?
- Call the number x. The repeating block, 18, is two digits long, and that length decides the next move.x = 0.181818...
- Multiply both sides by 100, one factor of 10 per digit in the block, so the decimal point jumps one full block.100x = 18.181818...
- Subtract the first equation from the second. Both tails are 0.181818..., so they cancel exactly and leave whole numbers.100x - x = 18.181818... - 0.181818..., so 99x = 18
- Divide both sides by 99, then simplify: 18 and 99 share a factor of 9.x = 18/99 = 2/11
Check it: Long-divide 2 by 11: 20 ÷ 11 = 1 remainder 9, then 90 ÷ 11 = 8 remainder 2. The remainder 2 is back where it started, so the digits 1, 8 repeat forever.
Common Mistakes with 8.NS.1
Calling every long decimal irrational
“3.14159 is irrational because it has so many digits.”
Try this: Ask whether the decimal stops or repeats. 3.14159 stops after five places, so it is 314159/100000, a rational approximation of π.
Multiplying by the wrong power of 10
x = 0.181818..., 10x = 1.818181..., so 9x = 1.636363...
Try this: Use one factor of 10 per digit in the repeating block: a two-digit block like 18 needs 100x, so the tails cancel exactly.
Treating every square root as irrational
“√49 is irrational because it has a root sign.”
Try this: Evaluate the root first. √49 = 7, a whole number, so it is rational; roots of non-perfect squares like √50 are irrational.
Where 8.NS.1 Fits
Practice 8.NS.1
Quick Check
- Is 0.375 rational? Write it as a fraction.
- Write 0.777..., with the 7 repeating forever, as a fraction.
- Which of these are irrational: √16, √17, 22/7, π?