Surds: Real Number and Square Root Form
Rational Number
Any number that can be written as a fraction is called a rational number. The natural numbers and integers are all rational numbers. A terminating or recurring decimal can always be written as a fraction and as such these are both subsets of rational numbers.
Irrational Numbers
Numbers that cannot be written as a fraction are called irrational. Example √2, √5, √7, Π. These numbers cannot be written as a fraction so they are irrational.
Surds
A surd is any number that looks like this √21, √15, √17. A surd is a root that cannot be written exactly as a decimal or a fraction. They are irrational.
Simplifying Surds
A surd can be simplified by rewriting the number inside the root as a product of two other numbers, one of which is its largest square factor. The square root of this factor is then written outside of the root sign.
Rules of Surds
Surds are numbers left in "square root form" (or "cube root form" etc).
Addition and subtraction of surds a√b + c√b = (a + c)√b a√b - c√b = (a - c)√b
Examples
4√7 - 2√7 = 2√7.
5√2 + 8√2 = 13√2
NB1: 5√2 + 3√3 cannot be manipulated because the surds are different (one is √2 and one is √3).
NB2: √a + √b is not the same as √(a + b) .
Multiplication and Division
√a x √a = a
√ab = √a × √b
√(a/b) = √a/√b
Examples
√5 × √15 = √75
= √25 × √3
= 5√3.
Double Brackets and Surds
We must remember that
√9 x √9 = 3 x 3 = 9
So when we multiply a surd by itself we obtain the number under the root sign.
Example
(√5)2 = √5 x √5 = 5
(√8) 2 = √8 x √8 = 8
(√17) 2 = √17 x√17 = 17
We use this fact to help us when we expand double brackets and when we rationalise the denominator of fractions which we will see next
(1 + √3)(2 - √3) [The brackets are expanded as usual]
=
Rationalising the Denominator
To rationalise the denominator means that we want to get rid of the surd sign in the denominator. We do this by multiplying top and bottom by the surd in the denominator.
Any number that can be written as a fraction is called a rational number. The natural numbers and integers are all rational numbers. A terminating or recurring decimal can always be written as a fraction and as such these are both subsets of rational numbers.
Irrational Numbers
Numbers that cannot be written as a fraction are called irrational. Example √2, √5, √7, Π. These numbers cannot be written as a fraction so they are irrational.
Surds
A surd is any number that looks like this √21, √15, √17. A surd is a root that cannot be written exactly as a decimal or a fraction. They are irrational.
Simplifying Surds
A surd can be simplified by rewriting the number inside the root as a product of two other numbers, one of which is its largest square factor. The square root of this factor is then written outside of the root sign.
Rules of Surds
Surds are numbers left in "square root form" (or "cube root form" etc).
Addition and subtraction of surds a√b + c√b = (a + c)√b a√b - c√b = (a - c)√b
Examples
4√7 - 2√7 = 2√7.
5√2 + 8√2 = 13√2
NB1: 5√2 + 3√3 cannot be manipulated because the surds are different (one is √2 and one is √3).
NB2: √a + √b is not the same as √(a + b) .
Multiplication and Division
√a x √a = a
√ab = √a × √b
√(a/b) = √a/√b
Examples
√5 × √15 = √75
= √25 × √3
= 5√3.
Double Brackets and Surds
We must remember that
√9 x √9 = 3 x 3 = 9
So when we multiply a surd by itself we obtain the number under the root sign.
Example
(√5)2 = √5 x √5 = 5
(√8) 2 = √8 x √8 = 8
(√17) 2 = √17 x√17 = 17
We use this fact to help us when we expand double brackets and when we rationalise the denominator of fractions which we will see next
(1 + √3)(2 - √3) [The brackets are expanded as usual]
=
Rationalising the Denominator
To rationalise the denominator means that we want to get rid of the surd sign in the denominator. We do this by multiplying top and bottom by the surd in the denominator.