8.NS.2: Approximate Irrational Numbers
Also written as 8.NS.A.2 and CCSS.MATH.CONTENT.8.NS.A.2 · Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers.
The Official Wording
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Common Core State Standards, Grade 8, The Number System · Read it at the source · Checked October 14, 2026
What 8.NS.2 Means in Plain English
Eighth graders trap an irrational number between decimals. Since 3² = 9 and 4² = 16, √10 is between 3 and 4; since 3.1² = 9.61 and 3.2² = 10.24, it is between 3.1 and 3.2. Each extra decimal place pins it tighter.
With those bounds students compare irrationals, mark them on a number line and estimate expressions like π², just under 10. The habit pays off in 8.G.7, where Pythagorean answers like √52 need a sensible size, and in high school radical work (HSN-RN.2).
8.NS.2 I Can Statements
- I can find the two whole numbers a square root lies between.
- I can narrow a square root to tenths and hundredths by squaring.
- I can compare irrational numbers and place them on a number line.
8.NS.2 Worked Example
Two irrational numbers that agree to the first decimal place, so the comparison forces one more place.
Without a square root key, decide which is larger, √10 or π, and show where each one sits on a number line.
- Bracket √10 between whole numbers using the perfect squares on either side of 10.3² = 9 and 4² = 16, so 3 < √10 < 4
- Square tenths until one passes 10.3.1² = 9.61 and 3.2² = 10.24, so 3.1 < √10 < 3.2
- π is also between 3.1 and 3.2, so tenths cannot settle it. Go one decimal place further.3.16² = 9.9856 and 3.17² = 10.0489, so 3.16 < √10 < 3.17
- Truncate π to hundredths and line the intervals up.π = 3.14159..., so 3.14 < π < 3.15 < 3.16 < √10
- Mark both on a number line from 3.1 to 3.2 in hundredths.π just past 3.14; √10 just past 3.16
Check it: Square π’s bounds instead: 3.14² = 9.8596 and 3.15² = 9.9225 are both under 10, so π² < 10, and π must be smaller than √10.
Common Mistakes with 8.NS.2
Halving instead of taking the root
√10 ≈ 5
Try this: Ask which number times itself makes 10. Since 3² = 9 and 4² = 16, √10 sits between 3 and 4.
Assuming the root sits halfway
“10 is between 9 and 16, so √10 is about 3.5.”
Try this: Square the guess to test it: 3.5² = 12.25, well past 10, so √10 sits much closer to 3, near 3.16.
Treating 3.14 as π exactly
“π = 3.14, so π and 3.14 are the same point.”
Try this: Write π = 3.14159... with the dots, and say “between 3.14 and 3.15” so the approximation stays an approximation.
Where 8.NS.2 Fits
Practice 8.NS.2
Quick Check
- Between which two whole numbers is √40?
- Use tenths to bracket √5.
- Estimate π² without a calculator.