Irrational Numbers and Square Roots
Grade 8 · Math · Study Guide
Explore the real number system, perfect squares, cube roots, and irrational numbers.
The real number system includes rational and irrational numbers. Rational numbers can be written as a fraction p/q (like 1/2, 0.75, or -3). Irrational numbers cannot — their decimals go on forever without repeating. Famous examples: pi = 3.14159... and the square root of 2 = 1.41421...
The square root of n is the value that when multiplied by itself gives n. Perfect squares have whole-number square roots: sqrt(4)=2, sqrt(25)=5, sqrt(100)=10. Non-perfect squares like sqrt(3) or sqrt(7) are irrational — we can only approximate them.
Cube roots work similarly: cbrt(8)=2 because 2 cubed = 8. On the number line, irrational numbers fit between rational numbers — sqrt(2) sits between 1 and 2, closer to 1.4. Knowing perfect squares from 1 to 144 makes estimating irrational roots much faster.
| n | n squared | sqrt(n squared) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
Key terms
- Rational Number
- Any number expressible as p/q where p and q are integers and q is not zero.
- Irrational Number
- A real number that cannot be written as a fraction; decimal is non-terminating and non-repeating.
- Square Root
- sqrt(n) = the value that when squared gives n. sqrt(25) = 5.
- Perfect Square
- A whole number with a whole-number square root: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- Cube Root
- cbrt(n) = value that when cubed gives n. cbrt(27) = 3.
- Real Number System
- All rational and irrational numbers — every point on the number line.
Flashcards
- Irrational number
- Cannot be written as a fraction; decimal is non-terminating and non-repeating. Example: sqrt(2), pi.
- Rational number
- Can be written as p/q. Terminating or repeating decimals are rational.
- Perfect square
- Whole number whose square root is also whole: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- Estimating sqrt(n)
- Find the two perfect squares n falls between; sqrt(n) is between those integers.
- Cube root
- cbrt(n) = number that when cubed gives n. cbrt(8) = 2 because 2^3 = 8.
- Real number system
- Rational numbers plus irrational numbers = all real numbers on the number line.
Practice questions
Which number is irrational?
- 0.75
- sqrt(9)
- sqrt(5)
- -4
Show answer
Answer: sqrt(5)
sqrt(9)=3 and 0.75=3/4 are rational; -4 is rational. sqrt(5) cannot be expressed as a fraction — irrational.
All square roots are irrational numbers.
Show answer
Answer: False
Square roots of perfect squares are rational: sqrt(4)=2, sqrt(25)=5.
The cube root of 64 is ___.
Show answer
Answer: 4
4 cubed = 4x4x4 = 64. So cbrt(64) = 4.
Between which two consecutive integers does sqrt(50) fall?
- 5 and 6
- 6 and 7
- 7 and 8
- 8 and 9
Show answer
Answer: 7 and 8
7^2=49 and 8^2=64. Since 49 < 50 < 64, sqrt(50) is between 7 and 8.
Which of these is a rational number?
- pi
- sqrt(2)
- sqrt(16)
- sqrt(3)
Show answer
Answer: sqrt(16)
sqrt(16) = 4 — a whole number, which is rational. Pi, sqrt(2), and sqrt(3) are all irrational.
Fun facts
- Ancient Greeks were shocked when they proved sqrt(2) is irrational — it shattered their belief that all numbers were rational fractions.
- Pi has been calculated to over 100 trillion decimal places and no repeating pattern has ever been found.
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