Irrational Numbers
Grade 8 · Math · Study Guide
Discover what makes a number irrational, how to estimate square roots, and where irrational numbers fit on the number line.
Real numbers are divided into two groups: rational numbers and irrational numbers. A rational number can be written as a fraction p/q where p and q are integers and q ≠ 0. Their decimal expansions either terminate (like 0.5) or repeat (like 0.333…).
An irrational number cannot be written as a fraction of two integers. Its decimal expansion goes on forever without repeating. Famous examples include π (pi) ≈ 3.14159… and the square root of 2 (√2 ≈ 1.41421…).
Square roots of perfect squares are rational: √4 = 2, √9 = 3, √25 = 5. But square roots of non-perfect squares are irrational: √2, √3, √5, √7, √10, and so on.
To estimate an irrational square root, find the two perfect squares it falls between. For example, √50 is between √49 = 7 and √64 = 8, so √50 is between 7 and 8, closer to 7 (since 50 is closer to 49). A better estimate is about 7.07.
Irrational numbers have a place on the number line, just like rational numbers. The set of all rational and irrational numbers together forms the real numbers.
Key terms
- Irrational number
- A number that cannot be written as a fraction p/q; its decimal is non-terminating and non-repeating
- Rational number
- Any number expressible as p/q where p and q are integers and q ≠ 0
- Perfect square
- An integer that is the square of another integer (e.g., 1, 4, 9, 16, 25)
- Square root
- A number that, when multiplied by itself, gives the original number; √a × √a = a
- Real numbers
- All rational and irrational numbers together; every point on the number line
- Non-terminating, non-repeating decimal
- A decimal that goes on forever without a repeating pattern; the hallmark of irrational numbers
Flashcards
- Rational number
- Can be written as p/q (fraction); decimal terminates or repeats
- Irrational number
- Cannot be written as a fraction; decimal is non-terminating and non-repeating
- √2
- ≈ 1.41421…; irrational — not a perfect square
- π (pi)
- ≈ 3.14159…; irrational; ratio of circumference to diameter
- Perfect squares
- 1, 4, 9, 16, 25, 36, 49, 64, 81, 100; their square roots are rational integers
- Estimating √n
- Find perfect squares on either side; e.g., √50 is between 7 (√49) and 8 (√64)
- Real number line
- Contains both rational and irrational numbers; every point on it is a real number
- Non-repeating decimal
- A decimal that never develops a repeating block of digits; characteristic of irrationals
Questions
Which of the following is an irrational number?
- √16
- 0.75
- √7
- 2/3
Show answer
Answer: √7
7 is not a perfect square, so √7 cannot be expressed as a fraction and is irrational. √16 = 4 (rational), 0.75 and 2/3 are fractions (rational).
The number π (pi) is a rational number because we often use 3.14 to approximate it.
Show answer
Answer: False
3.14 is just an approximation of π. The actual value of π never terminates or repeats, making it irrational.
The square root of a non-perfect square is always _____.
Show answer
Answer: irrational
If a whole number is not a perfect square, its square root is irrational (never-ending, non-repeating decimal).
Between which two integers does √30 lie?
- 4 and 5
- 5 and 6
- 6 and 7
- 7 and 8
Show answer
Answer: 5 and 6
√25 = 5 and √36 = 6. Since 25 < 30 < 36, we have 5 < √30 < 6.
All irrational numbers are real numbers.
Show answer
Answer: True
The real numbers consist of ALL rational and irrational numbers. Irrational numbers are a subset of real numbers.
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