2. Aims: To be to be able to classify types of numbers To be able to write a surd in its simplest form To be able to add, subtract and multiply surds SURDS Lesson 1

3. Types of number We can classify numbers into the following sets: The set of natural numbers, : Ν Positive whole numbers {0, 1, 2, 3, 4 …} The set of integers, : Positive and negative whole numbers {0, ±1, ±2, ±3 …} The set of rational numbers, : Numbers that can be expressed in the form, where n and m are integers. All fractions and all terminating and recurring decimals are rational numbers; for example, ¾, –0.63, 0.2. The set of real numbers, : All numbers including irrational numbers; that is, numbers that cannot be expressed in the form, where n and m are integers. For example,  and. Numbers written in this form are called surds. When the square root of a number, for example √2, √3 or √5,is irrational, it is often preferable to write it with the root sign.

4. Manipulating surds When working with surds it is important to remember the following two rules: You should also remember that, by definition, √ a means the positive square root of a. and Also:

5. Simplifying surds Start by finding the largest square number that divides into 50. We can do this using the fact that For example: We are often required to simplify surds by writing them in the form Simplify by writing it in the form

6. Simplifying surds Simplify the following surds by writing them in the form a √ b.

7. Simplifying surds

8. Adding and subtracting surds Surds can be added or subtracted if the number under the square root sign is the same. For example: Start by writing and in their simplest forms.

9. Basic multiplying and dividing surds

10. Expanding brackets containing surds Simplify the following: Problem 2) demonstrates the fact that ( a – b )( a + b ) = a 2 – b 2. In general: Do exercise 2A page 30 (Do a, c, e questions from each number)