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| \(\blacksquare\) Rational numbers are the numbers that can be expressed in fractional form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\). |
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| \(\blacksquare\) Irrational numbers are the numbers that cannot be expressed in fractional form. |
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| Below are some examples: |
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| Rational number |
Irrational number |
| \(-3=-\dfrac{3}{1}\) |
\(\pi=3.14159265...\) |
| \(\dfrac{1}{3}\) |
\(e=2.71828182...\)(Euler number) |
| \(1.75=\dfrac{7}{4}\) (terminating decimal) |
\(\varphi=1.61803398\) (golden ratio) |
| \(0.555...=\dfrac{111}{200}\)(recurring decimal) |
\(\sqrt{3}=1.732050808...\) |
| \(\sqrt{25}=5\) |
\(\sqrt[3]{9}=2.080083823\) |
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From the examples, root of a number can be either rational or irrational.
An irrational number in the form of root is called surd.
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| \(\sqrt[3]{9}\) is read as "surd \(9\) order \(3\)". |
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Laws of surds:
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\(\begin{aligned} \bullet \quad\sqrt{a} \times \sqrt{b} &=\sqrt{ab} \\\\ \bullet \quad \sqrt{a} \div \sqrt{b}&=\sqrt{\dfrac{a}{b}} \end{aligned}\)
for \(a \gt 0\) and \(b \gt 0\).
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Conjugate surd of \(a+\sqrt{b}\) is \(a-\sqrt{b}\), similarly \(a-\sqrt{b}\) is \(a+\sqrt{b}\).
A rational number is obtained when the conjugate pair is multiplied.
\((a+\sqrt{b})(a-\sqrt{b})=a^2-b\).
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To simplify an expression involving surd as denominator, rationalising the denominator by multiplying the numerator and denominator with conjugate surd.
For example,
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\(\begin{aligned} &\bullet \quad\dfrac{1}{m\sqrt{a}} \times\dfrac{m\sqrt{a}}{m\sqrt{a}} \\\\ &\bullet \dfrac{1}{m\sqrt{a}+n\sqrt{b}}\times\dfrac{m\sqrt{a}-n\sqrt{b}}{m\sqrt{a}-n\sqrt{b}} \\\\ &\bullet \dfrac{1}{m\sqrt{a}-n\sqrt{b}}\times\dfrac{m\sqrt{a}+n\sqrt{b}}{m\sqrt{a}+n\sqrt{b}} \end{aligned}\)
