22 Proof Proof questions. This chapter will show you how to:


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1 22 Ch qxd 23/9/05 12:21 Page a 2 b This chapter will show you how to: tell the difference between 'verify' and 'proof' prove results using simple, stepbystep chains of reasoning use a counter example to disprove a statement prove simple geometrical statements 22.1 questions questions test the Using and Applying part of what you learn in Mathematics. You need to: 1 Find examples that match a general statement, or give counter examples which disprove a statement. 2 Show how one statement follows from another. 3 Explain your reasoning or criticise a piece of faulty reasoning. 4 Know the difference between a practical demonstration of something and a proof. Many proof questions are based on the properties of numbers. You need to know about: odd and even numbers prime numbers factors and multiples square and cube numbers. Certain words occur in proof questions: consecutive (one after the other) integer (whole number) product (multiply). You need to know the rules for addition and multiplication of whole numbers: Addition Multiplication For the general statement Dogs have 2 ears. You can find examples of dogs with 2 ears to match the statement. For All dogs are black a brown dog is a counter example that disproves the statement. Chapter 14 covers most of these. Look back if you need help. odd odd even odd even odd even odd odd even even even odd odd odd odd even even even odd even even even even
2 22 Ch qxd 23/9/05 12:21 Page v. verify You need to know the difference between proof and verify. Key words: verify proof Verify means to check something is true by substituting numbers into an expression or formula. means to show something is true using logical reasoning. Example 1 illustrates the difference. Example 1 n is any integer. Explain why 2n 1 must be an odd number. integer whole number Answer 1 Try n 1 2n Try n 2 2n Try n 3 2n The answer is always an odd number. Is it always? The answer only shows it is for n 1, 2, 3. This is not a proof because it does not explain why 2n 1 must be an odd number. This answer verifies that 2n 1 is an odd number when n 1, 2 and 3. Will 2n 1 be an odd number for n 4, 5, 6, 7,? You cannot test it for all numbers it would take forever! Look at Answer 2. n is any integer. Explain why 2n 1 must be an odd number. Answer 2 2n 2 n Multiplying any whole number by 2 gives an even number, so 2n is an even number. So 2n 1 even number 1 odd number 2n 1 must always be an odd number.
3 22 Ch qxd 23/9/05 12:21 Page Answer 2 is a proof. It shows whatever number you start with (n could stand for any whole number), 2n 1 will always be an odd number. This example shows that there is a huge difference between verifying a result (simply checking with numbers) and proving a result. Verify is a practical demonstration of the result. shows how one statement follows from another using simple chains of reasoning. Example 2 Explain why the sum of any three consecutive integers is always a multiple of 3. Verify Try 1, 2 and Try 5, 6 and Try 10, 11 and The result works for these examples. Consecutive means one after the other. Multiple of 3 a number in the 3 times table. If x is one of the numbers then the others are (x 1) and (x 2) x (x 1) (x 2) 3x 3 3(x 1) 3(x 1) means 3 (x 1) So the answer is always a multiple of 3. For the consecutive numbers you could choose (x 1), x and (x 1). questions often have the word explain in the question. You only get full marks if you prove the result. You may get some marks for verifying (giving numerical examples), but answers which just check the result using numbers often score no marks at all! usually involves some algebra skills. In example 1 you needed to know that 2n means 2 n, a very basic algebra fact. In example 2 you had to collect like terms and take out a common factor. For help with these, look back at Chapter 4.
4 22 Ch qxd 23/9/05 12:21 Page Exercise 22A 1 p is an odd number and q is an even number. (a) Explain why p q 1 is always an even number, (b) Explain why pq 1 is always an odd number. 2 n is a positive integer. Explain why n(n 1) must be an even number. 3 x is an odd number. Explain why x 2 1 is always an even number. 4 If b is an even number, prove that (b 1)(b 1) is an odd number. 5 x is an odd number and y is an even number. Explain why (x y)(x y) is an odd number. 6 Explain why the sum of 4 consecutive numbers is always an even number by counter example Key words: counter example by counter example asks you to show that a statement is incorrect by finding one example where the stated result does not work. You can substitute numbers into an expression or formula until you find a case where the result is not true. Example 3 Tony says that when n is an even number, 1 2 n 3 is always even. Give an example to show that he is wrong. Even numbers are 2, 4, 6, Try n n (2) even Try n = n (4) odd When n 4 the result is not true, so Tony is wrong. The case where n 4 is a counter example. Sometimes you need to try several values before you find a counter example.
5 22 Ch qxd 23/9/05 12:21 Page Example 4 Simon says that when you square a number, the answer is always bigger than the original number. Give a counter example to show that Simon is wrong. Try the same Try bigger Try bigger Trying 4, 5, will be even bigger answers than these. Try 2 ( 2) 2 ( 2) ( 2) 4 bigger Try a decimal number smaller than 1, such as 0.5 (0.5) which is smaller (0.5) 2 gives a smaller number, so Simon is wrong. Work systematically, trying different values. Squaring numbers bigger than 3 will give bigger numbers. negative negative positive (0.5) 2 is a counter example. Exercise 22B 1 Sam says that when k is an even number, k k is always odd. Give an example to show that he is wrong. 2 Heather says that m 3 2 is never a multiple of 3. Give a counter example to show that she is wrong. 3 p is an odd number and q is an even number. Andrew says that p q 1 cannot be a prime number. Explain why he is wrong. 4 a and b are both prime numbers. Give an example to show that a b is not always an even number. 5 Ian says that n 2 3n 1 is a prime number for all values of n. Give a counter example to show that he is wrong. 6 Give a counter example to each of these statements: (a) the square root of any number is always smaller than the original number (b) the cube of any number is always bigger than the square of the same number.
6 22 Ch qxd 23/9/05 12:21 Page in geometry To prove a result in geometry you use stepbystep reasoning, showing clearly how one statement follows from another. Don t just verify by finding examples that work. For geometry proofs you need to know: angle properties of parallel lines angle properties at a point and on straight lines. Never use a protractor to check angles. The diagrams are not usually accurately drawn. Example 5 Prove that the sum of the angles of a triangle is 180 This result was proved in Chapter 5. b B D Start by drawing a diagram. A y a c x C E Draw a triangle ABC. Extend side AC to E. Draw line CD parallel to AB. Label the angles as shown on the diagram. x a (corresponding angles) y b (alternate angles) x y c 180 (sum of angles on a straight line at point C) So a b c 180. The sum of the angles of ABC is 180. Look back at angle properties and parallel line properties in Chapter 5 to remind yourself of these facts. To set out a geometry proof: State each step clearly. give a reason for each step. Steps need to follow on from each other.
7 22 Ch qxd 23/9/05 12:21 Page Exercise 22C 1 Prove that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. 2 Prove that the sum of the interior angles of a quadrilateral is Prove that the interior angles of a regular pentagon are 108. Summary of key points You need to know the difference between proof and verify. Verify means to check something is true by substituting numbers into an expression or a formula. It is a practical demonstration of a result. means to show something is true using logical reasoning. questions often have the word 'explain' in the question. by counter example asks you to show that a statement is incorrect by finding one example where the stated result does not work. You can substitute numbers into an expression or formula until you find a case where the result is not true. To prove a result in Geometry you use stepbystep reasoning, showing clearly how one statement follows from another. Questions on proof can be set at various grade levels depending on the content of the material. They will typically be at grades E, D or C.
8 22 Ch qxd 23/9/05 12:21 Page Examination Questions 1 TO FOLLOW!!!
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