In our lesson on the number theory, we learnt that the set of real numbers (\(\mathbb{R}\)) has two main subsets:
1. Rational numbers and
2. Irrational numbers
Rational Numbers
Rational numbers are numbers that can be expressed as fractions or in the form \(\frac{a}{b}\), where \(b \ne 0\). We say \(b \ne 0\), this is because it is not possible to divide any number by zero (0).
Recurring and terminating decimals are also said to be rational numbers, as they can be expressed as fractions.
Examples of rational numbers are:
\(*\) Terminating decimal fractions \(\Rightarrow\) \(\frac{3}{8} = 0.375\), \(\frac{4}{5} =0.8\), \(\frac{9}{20} = 0.45\) etc.
\(*\) Repeated decimal fractions \(\Rightarrow\) \(\frac{5}{9} = 0.555 555 ...\), \(\frac{2}{11} = 0.18181818 ...\), \(\frac{7}{15} = 0.466666666 ...\), etc.
Click here to learn how to convert between decimals and fractions.
Irrational Numbers
Irrational numbers are numbers with no terminating or recurring decimals. As a result, they cannot be expressed as fractions \(\frac{a}{b}\) or ratios.
Examples of irrational numbers are as follows:
\(* \ \sqrt{2} = 1.414 \ 213 \ 562 \ ...\)
\(*\ \sqrt{3} = 1.732\ 050 \ 808 \ ...\)
\(*\ \sqrt{5} = 2.236 \ 067 \ 977\ ...\)
\(*\ \pi \)(pi) \(= 3.141 \ 592\ 654\ ...\)
Note:
The square root of perfect squares are rational numbers. Consider the examples below:
1. \(\sqrt{4} = 2\)
Since 2 is an integer, it implies that \(\sqrt{4}\) is not an irrational number.
2. \(\sqrt{9} = 3\)
Since 3 is an integer, it implies that \(\sqrt{9}\) is not an irrational number.
3. \(\sqrt{16} = 4\)
Since 4 is an integer, it implies that \(\sqrt{16}\) is not an irrational number.
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. In other words, it is the product of an integer multiplied by itself.
Examples of perfect squares are:
1. \(4 = 2 \times 2 = 2^2 \Rightarrow 4\) is a perfect square.
2. \(25 = 5 \times 5 = 5^2 \Rightarrow 25\) is a perfect square.
3. \(36 = 6 \times 6 = 6^2 \Rightarrow 36\) is a perfect square.
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
Surds belong to the set of irrational numbers.
A surd is an irrational number that cannot be simplified to remove the square root \((\sqrt{})\), or other roots, eg. \((\sqrt[3]{})\), etc.
Surds are typically represented using the radical symbol \((\sqrt{})\). They are irrational numbers left in root form to ensure accuracy in mathematical calculations.
The most common surds are the square roots of non-perfect squares, eg. \(\sqrt{2}\), cube root of non-perfect cubes, eg. \(\sqrt[3]{5}\), etc.
You say:
1. \(\sqrt{a}\) \(\Rightarrow\) root \(a\) or square root of \(a\).
2. \(\sqrt[3]{a}\) \(\Rightarrow\) cubic root of \(a\) or third root of \(a\).
3. \(\sqrt[4]{a}\) \(\Rightarrow\) fourth root of \(a\).
4. \(\sqrt[5]{a}\) \(\Rightarrow\) fifth root of \(a\).
And so on.
Note:
In indices,
1. \(\sqrt{a} = a^\frac{1}{2}\)
2. \(\sqrt[3]{a} = a^\frac{1}{3}\)
3. \(\sqrt[4]{a} = a^\frac{1}{4}\)
4. \(\sqrt[5]{a} =a^\frac{1}{5}\)
And so on.
Surds are used when exact values are needed in calculations.
There are basically two (2) kinds of surds:
1. Simple surds
2. Compound surds
Simple surds
Simple surds are surds that have only one term under the radical symbol.
Eg. \(\sqrt{2}\), \(\sqrt[3]{5}\), \(\sqrt{3}\), \(\sqrt[4]{10}\), etc.
Compound surds
A compound surd is an expression that contains two or more surds connected by addition or subtraction.
Eg. \(\sqrt{2} + \sqrt{3}\), \(\sqrt{5} - \sqrt{3}\), etc.
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
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