Number Systems and Real Numbers: Class 9 Study Guide
Real numbers include every number on the number line, both rational and irrational.
A rational number can be written as p/q with q not zero. Its decimal expansion either ends (0.25) or repeats (1/7 = 0.142857142857...). An irrational number cannot be written as p/q. Its decimal never ends and never repeats. โ2, โ3 and ฯ are irrational. Together, rational and irrational numbers make up the real numbers, and every real number has a point on the number line.
The square root of a non-perfect-square natural number, such as โ5, is irrational. The sum of a rational and an irrational number is irrational. The square root of a perfect square, such as โ49 = 7, is rational.
Rational or irrational?
| Number | Type | Reason |
|---|---|---|
| 3/8 = 0.375 | Rational | Decimal ends |
| 2/3 = 0.666... | Rational | Decimal repeats |
| โ2 = 1.41421... | Irrational | Never ends, never repeats |
| โ16 | Rational | Equals 4 |
| ฯ | Irrational | Never ends, never repeats |
Laws of exponents for real numbers
For a positive real number a: a^m ร a^n = a^(m+n); (a^m)^n = a^(mn); a^m รท a^n = a^(mโn); and a^m ร b^m = (ab)^m. Fractional exponents mean roots: a^(1/n) is the nth root of a. So 8^(1/3) = 2, and 8^(2/3) = (8^(1/3))ยฒ = 4. In the same way 16^(3/4) = (16^(1/4))ยณ = 2ยณ = 8.
a^(1/n) is the nth root of a.
It keeps numbers small.
Square, cube and so on.
Add exponents when multiplying.
Rationalising the denominator
To remove a root from the denominator, multiply the top and bottom by a suitable number. 1/โ2 = โ2/2. For a denominator with two terms, multiply by its conjugate. 1/(3 + โ2) = (3 โ โ2)/((3 + โ2)(3 โ โ2)) = (3 โ โ2)/(9 โ 2) = (3 โ โ2)/7, using (a + b)(a โ b) = aยฒ โ bยฒ.
Worked examples and common mistakes
Example: Insert a rational number between 1/3 and 1/2. Their average is (1/3 + 1/2) รท 2 = 5/12. Example: Simplify 2^3 ร 2^4 = 2^7 = 128. Common mistakes: thinking that 22/7 equals ฯ (it is only a close fraction), calling a repeating decimal irrational, and multiplying only the denominator when rationalising. Multiply top and bottom by the same number.
Check your understanding: model answers
1. Irrational: the decimal never ends and has no repeating block. 2. 27^(1/3) = 3, so the answer is 9. 3. (โ5 โ 2)/((โ5 + 2)(โ5 โ 2)) = (โ5 โ 2)/(5 โ 4) = โ5 โ 2.
โจ Key Takeaways
- Rational numbers have ending or repeating decimals.
- Irrational numbers have non-ending, non-repeating decimals.
- Rational and irrational numbers together are the real numbers.
- Rationalise a denominator by multiplying by its conjugate.
๐ฌ Questions for Discussion & Reflection
- Is 0.2020020002... rational or irrational?
- Simplify 27^(2/3).
- Rationalise 1/(โ5 + 2).
Print negative numbers worksheets โ