Surds are special types of irrational number, such as and . Some surds give rational values, such as , however others do not have an exact decimal value.
Think: What is the number between and . Is that number ? Why not?
Basic Properties of Surds
Addition and Subtraction
Treat surds in the same way as pronumerals in algebra. Only add or subtract 'like surds' the same way 'like terms' are calculated.
Multiplication and Division
The same rules for expanding brackets and binomial products that you use in algebra also applies to surds. Go to Group Symbols and Binomials to find out more.
Properties of Surds as Binomials
Rationalising the Denominator
Rationalising the denominator of a fractional surd means writing it with a rational number (not a surd) in the denominator. For example, after rationalising the denominator, becomes .
Back when calculators were available, rationalising denominator is used to make it easier to divide a fraction by a rational number than an irrational one.
When there is a binomial denominator, we use the difference of two squares to rationalise it, as the result is always a rational number.
The sum and difference of two simple quadratic surds are said to be conjugate surds to each other.
Conjugate surds are also known as complementary surds.
Two surds and are conjugate to each other.
To rationalise the denominator of , multiply by .