Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 8th grade irrational numbers? Then this Grade 8 lesson on Rational and Irrational Numbers Explained explains each idea step by step, with worked examples.
Quick answer: Rational numbers are quotients of integers, so their decimals end or loop. Irrational numbers such as √2 and π have endless decimals with no loop. Put the two families together and you get every real number, with no overlap between them: each number belongs to just one family.
Standards: 8.NS.A.1, 8.NS.A.2 · Level: Grade 8
What makes a number rational or irrational?
A rational number is a quotient ab of two integers with b ≠ 0; written as a decimal, it ends or loops through one block. An irrational number has no such quotient, and its digits wander on without a loop.
Square roots of perfect squares, such as √169 = 13, are rational. Square roots of other whole numbers, such as √35, are irrational.
How can a repeating decimal be rewritten as a fraction?
- Name the number x and mark the repeating block.
- Shift the decimal point past one full block: multiply by 10, 100 or 1,000 for blocks of one, two or three digits.
- Subtract x from the shifted equation so the loop disappears.
- Solve for x and simplify.
The block 318 has three digits, so multiply by 1,000: 1,000x = 318.318318…
Subtract: 1,000x − x = 318, so 999x = 318.
Then x = 318999 = 106333.
How can you approximate a root that is not a whole number?
Trap the root between two perfect squares, then refine with tenths. Since 49 < 54 < 64, we know 7 < √54 < 8. Testing 7.3² = 53.29 and 7.4² = 54.76 gives 7.3 < √54 < 7.4.
| Number | Decimal form | Type |
|---|---|---|
| 516 | 0.3125 (stops) | Rational |
| 415 | 0.2666… (repeats) | Rational |
| √26 | 5.0990195… (no pattern) | Irrational |
Which of two irrational numbers is larger?
Write each number as a decimal, or square both numbers when roots are involved. Compare π ≈ 3.1416 with √10 ≈ 3.1623: the first digits agree until the hundredths place, and 3.14 < 3.16, so π < √10.
A rounded decimal such as 3.14 is rational, but π itself is not. An approximation never changes the type of the original number.
Beep! To place √n, find the perfect squares on each side of n. The root lives between their roots, and it sits closer to the nearer square.
- Rational: a fraction of integers; decimal stops or repeats.
- Irrational: decimal never stops and never repeats (π, √2, √35).
- The square root of a non-perfect square is irrational.
- Multiply a repeating decimal by 10, 100 or 1,000, then subtract to find its fraction.
- Estimate √n by trapping it between two perfect squares.
Rational and Irrational Numbers Explained: frequently asked questions
Is zero a rational number?
Yes. Zero equals 0/1, a fraction of two integers, so it is rational. Every integer is rational because it can be written over 1.
Is a repeating decimal rational?
Yes. Every repeating decimal equals a fraction. For example, 0.8888… equals 8/9, which you can prove by multiplying by 10 and subtracting.
Is √2 rational or irrational?
It is irrational. Its decimal begins 1.41421356… and never ends or repeats, and no fraction of integers equals it.
Can an irrational number be written exactly as a decimal?
No. A decimal for an irrational number never ends, so any written decimal is only an approximation. The symbol, such as √7 or π, is the exact value.
How do you know which of two square roots is larger?
Compare the numbers under the radical signs. Since 29 is greater than 23, √29 is greater than √23. For a root versus a decimal, square the decimal and compare.
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