Surds

1. Surds The types of numbers we use and where surds appear Square numbers and how to use them to simplify surds. What is so special about surds? Surds are a special type of number that you need to understand and do calculations with. The are examples of exact valuesand are often appear in Pythagoras and trigonometry questions Simplifying sums and differences of surds. Rationalising the denominator. Simplifying products and quotients of surds.

2. Our sets of Numbers This is a short tour of the sets of numbers that we use in National 5 maths. At each stage every new set contains all of the previous sets and new types of number. Some of the irrationals in  can’t be formed by taking roots. These include numbers like  which is a type of number called a transcendental number      The first set of number we encounter are the Natural numbers. These are the counting numbers 1, 2, 3, 4, 5, …. The set of natural numbers is called .Using set notation,  = {1, 2, 3, 4, …} The final set of numbers we look at in Nat 5 maths is the set formed when we take roots of the numbers in . The roots are usually square roots and cube roots. This set is the real numbers and is called . The new numbers that are in  are called irrational numbers and include surds like 2 which can’t be written as fraction. Adding the negative whole numbers to  creates a new set called the integers. The set of integers is called . The set of integers  = { … -3, -2, -1, 0, 1, 2, 3, …} When we add 0 (zero) to this set we get the Whole numbers. The set of whole numbers is called  and  = {0, 1, 2, 3, …} Forming fractions using the members of  creates the set of rational numbers . The set of rationalsis  =

3. What is so special about surds? When we take square roots we sometimes end up with whole numbers or fractions (members of  or ). Here are some examples you should understand: For other numbers, we can only find an approximation to the answer and it can only be written to a few decimal places e.g. The decimal values that calculators produce are not exact. They are only approximations because the actual numbers would need an infinite number of decimal places to be written down and there is no pattern to the digits. Surds let you write down the exact values of numbers like this. Can you spot which of these are really surds and which are not? Not a surd surd surd surd = 2 = 2 = 5

4. Square numbers and how they simplify surds To use this technique you need to be able to use this surds rule and you need to know that the square root of a perfect square like 9 or 16 is a whole number. Key skill: Know your list of squares up to 202 = 400. Key skill: Find square factors of other numbers It will save you a LOT of time if you know the first 20 perfect squares by heart... Do you? To simplify a surd, rewrite the number as a product where one of the factors is a square: Example: The square root of 5 is a surd and cannot be simplified so we can’t go any further.Now try to simplify these:

5. Simplifying sums and differences Once surds are written in simplified form, they can be added and subtracted by collecting like terms: The idea is that each different surd can be treated as a like term Usually, you will have to simplify the surds before you can collect like terms Here are two examples for you to try:

6. Products, quotients & the rules of surds The surd rules you will find most useful are: For example: You need to gain an understanding of these rules so that you which is the best rule to apply when you are working with simplification questions: The next slide contains a selection of surds questions for you to try: As with simplifying fractions, always try to simplify as far as possible:

7. Simplify these surd expressions as far as possible

8. Rationalising the denominator In the days before computers, finding the approximate value of a number like Would be extremely time-consuming and difficult. The technique of rationalising the denominator produces an equivalent fraction where thesurd terms are in the numerator and the denominator becomes a rational number (for us this will mean an integer). The simplest fractions to rewrite are those with a lone surd in the denominator like: Rationalising the denominator is also an example of a really useful algebraic trickcalled ‘multiplying by one’. (If you look back at completing the square, this uses an equally useful trick called ‘adding zero’. Neither of these operations can change the value of a number but if you are clever about it, you can produce an expressionthat is equal but in a more useful form . If we multiply this fraction by one then it will have an identical value. Also if we multiply 2 by itself it will equal 2 (a rational number). So how can these two ideas be combined? Make a fraction equal to one which includes 2. Now when you do the multiplication it becomes: This achieves the goal: You have an equivalent fraction which no longer has a surd in the denominator

9. Examples for you to try Rationalise the denominator: As always, do this on paper before checking the answer. At Nat 5 you’ll be expected to simplify your rationalised fraction if possible

10. Harder examples When the denominator includes a surd and a whole number, there is anotheralgebraic trick (based on the difference of 2 squares factorisation) that you need to learn: For these examples you need to create a fraction like this: Here multiplying by b doesn’t do the job, try it! The surd stays in the denominator where it is not wanted. For example

11. Two for you to try: