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Year 9 Proof

Year 9 Proof. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Objectives: Understand what is meant by a proof, and examples of what does and what doesn’t constitute a proof. Last modified: 7 th March 2014. How Many Examples Needed?.

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Year 9 Proof

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  1. Year 9 Proof Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: Understand what is meant by a proof, and examples of what does and what doesn’t constitute a proof. Last modified: 7th March 2014

  2. How Many Examples Needed? In 1772 Euler noticed that the following equation gives prime numbers for many positive integers n: We might wonder if the statement “For all positive integers n, n2– n + 41 is prime” is true. Try this for a few . How many examples of n would we need to show this statement is: True False One   A few Infinitely many   Infinitely many One  A few  ? The value of for which this statement is false is: 41

  3. Counterexamples A counterexample is an example used to disprove a statement. Discussing in pairs, find counterexamples for the following statements: 1 ? 2 ? 3 ? 4 ?

  4. Find a Rule! 5th 1st 2nd 3rd 4th 5 4 9 13 22 35 57 2 6 8 14 22 36 58 Each of the numbers is the sum of the two numbers to the left. You get to set the first 2 numbers. Try a few starting numbers, can you discover a rule that allows you to quickly work out the last (5th) number given the first two, without having to work out the ones in between? In this example we had 5 squares after the first two. Can you generalise your rule to work for any number of squares?

  5. Solution a b a+b a+2b 2a+3b 3a+5b 5a + 8b Our rule could be this: The last number is 5 times the first one plus 8 times the second one. We could let our first few terms be algebraic. This allows us to represent any starting numbers we choose. It also helps us more easily spot a pattern. Generify! For the nth square, we multiply the first number by the nth Fibonacci number and the second number by the (n+1)th Fibonacci number, and add the results. ? ?

  6. Exercise 1 Find a rule which gives you the orange number given the black numbers(s). Q1 191 6 a 5 a 2a+1 11 5 b a-b 1 23 4a+3 2b-a 47 4 8a+7 16a+15 -3 2a-3b 95 5b-3a 7 32a+31 -10 5a-8b ? Q2 ? By looking at your expressions in Q2 above, can you come up with a formula for the nth term of the sequence based on the first term a? (where the black box is the 1st) Un = a 2n-1 – 2n-1 + 1 ? QN 5 2 11 23 254 ? a b ab+1 ab2+b+1 a2b3 + 2ab2 + ab + b + 2

  7. Reasoning about sides and angles. Introducing variables for unknowns. Geometric Algebraic Some types of proofs Games Even/odd If I have a ‘perfect Connect 4’ strategy for winning, how can I show it’s perfect? e.g. “Show that n2 + n is always even.”

  8. Divisibility Proofs Show that the sum of any three consecutive integers is a multiple of 3. QueProblemo? He hasn’t shown it’s true for all possible integers. ? Kyle’s proof: “5 + 6 + 7 = 18, which is divisible by 3”. We could represent 3 consecutive integers as: Then: which is divisible by 3. ? ?

  9. Divisibility Proofs How could I algebraically represent: ! ? ? ? ? ? ?

  10. Exercise 2 Prove algebraically that the sum of two consecutive odd numbers is divisible by 4. which is divisible by 4. I think of two consecutive integers. Prove that the difference of the squares of these integers is equal to the sum of the two integers. (This was a recent GCSE question) Two numbers are: x and x + 1 Difference of squares: (x + 1)2 – x2 = x2 + 2x + 1 – x2 = 2x + 1 Sum of numbers: x + (x + 1) = 2x + 1 These are equal. Prove that the difference between the squares of two odd numbers is a multiple of 8. which is divisible by 8. Q1 ? Q2 ? Q3 ?

  11. Further Algebraic Proofs (This is perhaps the most important thing you’ll see in maths at school!) “Show that if X, then Y” 1. “Assume that X is true”. 2. Model X using algebra. 3. Do some manipulationto get towards Y. 4. “Therefore Y”

  12. Further Algebraic Proofs “I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9” 71 – 17 = 54 Have a few minutes to try and prove this. You can use the breakdown on the next page to help you...

  13. Further Algebraic Proofs “I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9” 1. “Assume that X is true”. Let “ab” and “ba” be two numbers. ? 2. Model X using algebra. Then the numbers have values 10a + b and 10b + a. ? 3. Do some manipulationto get towards Y. The difference is 10a + b – (10b + a) = 9a – 9b = 9(a-b) ? 4. “Therefore Y” This is divisible by 9. ?

  14. Rubbish Proofs Darth Kitty’s solution: “Pick the number 31 for example. The reverse of this is 13. And 31 – 13 = 18, which is divisible by 9. Hence the statement is true.” What is wrong with their proof? ? Showing something is true for one example is not sufficient. We need to show the statement is true for ANY POSSIBLE starting number we choose.

  15. Rubbish Proofs Photoshop Kitty’s solution: “Here’s all possible two-digit numbers: 11 – 11 = 0, 12 – 12 = 0, ... 21 – 12 = 9, 22 – 22 = 0, 32 – 23 = 9, 42 – 24 = 18, ... 98 – 89 = 9, 99 – 99 = 0 These are all divisible by 9. What is wrong with their proof? ? Technically it’s a valid proof, because we’ve shown the statement is true for every possible two-digit number. But it’s bad maths having to list out every case, and would become difficult if for example we tried to generalise the statement to 3-digit numbers.

  16. Exercise 3 – Digit Problems I think of a 3-digit number and then reverse its digits. Prove that the difference between these two numbers is a multiple of 11. 100a + 10b + c – (100c + 10b + a) = 99a – 99c = 11(9a – 9c) A divisibility test for 7 is to subtract twice the last digit from the remaining numbers, and see if the result is divisible by 7. e.g. 392 -> 39 – 4 = 35, which is clearly divisible by 7, so 392 is. Prove this works for all three-digit numbers. Let our number be n with the digits “abc”. n = 100a + 10b + c. If we subtract twice the last digit from the remaining numbers, we get 10a + b – 2c. Now if this is divisible by 7, 10a + b – 2c = 7k for some integer k. Thus 100a + 10b – 20c = 70k Then n = 100a + 10b + c = 70k + 21c = 7(10k + 3c) which is divisible by 7. Q1 ? QN ?

  17. Another Example of Good/Bad Proofs “An arithmetic sequence is one in which the difference between successive terms remains constant (for example, 4, 7, 10, 13, …). Suppose that a right-angled triangle has the property that the lengths of its sides form an arithmetic sequence. Prove that the sides of the triangle are in the ratio 3:4:5.” Example of a bad proof: What would a good proof look like? ? “If the sides of the triangle are in the ratio 3:4:5, we could have any multiple of this. e.g. 6, 8, 10. These are still in an arithmetic sequence and satisfy Pythagoras Theorem.” • Use our general structure for an algebraic proof! • “Assume that the sides form an arithmetic sequence. • Model this assumption, using algebraic expressions for side lengths that represent all possible arithmetic sequences, e.g. a-b, a, a+b. • Do some manipulation using Pythagoras... • Therefore, the sides have ratio 3:4:5.” Why is it bad? ? We’re supposed to be showing that IF the sides form an arithmetic sequence THEN the sides have ratio 3:4:5. They’ve just shown the reverse!

  18. Sprucing up our Proof I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9. “Let the number have the digits ‘ab’. Then the number is 10a + b. If we reverse the digits to get ‘ba’, which has the value 10b + a. Subtracting to find the difference: 10a + b – (10b + a) = 9a – 9b = 9(a-b). 9(a-b) is divisible by 9.” Our proof so far…

  19. Sprucing up our Proof I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9. “Let the number have the digits ‘ab’. Then the number is 10a + b. If we reverse the digits to get ‘ba’, which has the value 10b + a. Subtracting to find the difference: 10a + b – (10b + a) = 9a – 9b = 9(a-b). 9(a-b) is divisible by 9.” QED QED is short for the latin “Quad erat demonstrandum”. It translates to “Which had to be demonstrated”. It signals that we’ve finished our proof. ? ?

  20. Sprucing up our Proof I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9. Let the number have the digits ‘ab’. Then the number is 10a + b. If we reverse the digits to get ‘ba’, which has the value 10b + a. Subtracting to find the difference: 10a + b – (10b + a) = 9a – 9b = 9(a-b). 9(a-b) is divisible by 9. QED When we’re finding a difference, we usually subtract the smaller number from the larger one, so that our difference is positive. • Questions for discussion: • When will this be a problem in our proof? • Could we restrict our original number? • If so, does that mean our proof no longer covers all possible 2 digit numbers?

  21. Our super mathematically mature proof! I think of a two-digit number. I then reverse the digits. Prove that the difference between the two numbers is a multiple of 9. Let the number have the digits ‘ab’, and without loss of generality, assume that a≥b. Then the number is 10a + b. If we reverse the digits to get ‘ba’, which has the value 10b + a. Subtracting to find the difference: 10a + b – (10b + a) = 9a – 9b = 9(a-b). 9(a-b) is divisible by 9. QED “Without loss of generality” or “w.l.o.g.” means that we’ve made some restriction to ensure some kind of correctness, but it doesn’t make our proof any less ‘general’ (i.e. it still covers all cases)

  22. Games Other Types of Proof • [Source: JMO] “X and Y play a game in which X starts by choosing a number, which must be either 1 or 2. • Y then adds either 1 or 2 and states the total of the two numbers chosen so far. X does likewise, adding either 1 or 2 and stating the total, and so on. The winner is first player to make the total reach (or exceed) 20. • i) Explain how X can always win. • … 1 2 For (i), what do we actually have to show? ? That for any way in which Y plays, we can always find a resulting move that guarantees us to win.

  23. Even/Odd Other Types of Proof You will do this kind of proof next year... “Show that for any integer n, n2 + n is always even.” ? All integers are either odd or even. If n is odd: n2 is odd because odd x odd = odd. Then odd + odd = even. If n is even: n2 is even because even x even = even. Then even + even = even. What we’ve done here is split up all numbers according to some property (i.e. odd or even). This made proving our assertion for all integers easier.

  24. Summary So what makes a good (and valid!) proof? • Use of algebra: • When appropriate, use variables to represent something that could take any value (e.g. the two digits!) This keeps your argument as general as possible. • Then use algebra to manipulate these variables based on the question. • Keep things general!Avoid making assertions that would ignore certain possibilities. • Don’t make leaps of logicTry not to ‘assume’ too much. You need to clearly show your steps without making assumptions, which may turn out to be wrong! • Avoid circular argumentsIf the statement to prove is something like “If X, then show that Y”, then your proof needs to be something like: “Assume that X, then [manipulation involving X]... And thus Y is true.” You can’t assume Y is true at any point, because that’s what you’re trying to prove! • Consider edge casesConsider unusual cases. e.g. What if the first digit is 0? What if the digits are the same? Does our proof still work? • Avoid listing out lots of casesIf you’re having to list out lots and lots of cases, then you’re possibly missing ways to narrow down your search. Could you use algebra?

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