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Download A. Counter examples 1. Brian says all prime numbers are odd. Prove
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Transcript
Proof To prove a statement we need to give an accurate mathematical argument. This could involve many mathematical techniques, usually algebra. To prove a statement is false, finding one counter example is sufficient A. Counter examples 1. Brian says all prime numbers are odd. Prove he is incorrect. 2 is a prime number and 2 is even 2 is a counter example 2. Helen says all odd numbers are prime. Prove Helen is incorrect. 9 is an odd number and 9 is not prime because it has more than 2 factors. The factors of 9 are 1, 3, 9 9 is a counter example 3. Colin says if x and y are prime numbers will x2 + y2 will be an even number. Is he correct? If x = 3 and y = 5 32 + 52 = 9 + 25 = 34 34 is even 2 2 If x = 11 and y = 7 11 + 7 = 121 + 49 = 170 170 is even If x = 2 and y = 5 22 + 52 = 4 + 25 = 29 29 is odd Try some examples to see if you can find a counter example It just takes one counter example to prove he is wrong He is incorrect 4. Sandra says the cube of a number is always larger than its square. Is she correct? If x = 3 If x = 1 If x = -2 32 = 9 2 1 =1 -22 = 4 33 = 27 true This counter example shows the cube is the same as the square 3 1 = 1 false -23 = -8 false This counter example shows the cube is the smaller than the square She is incorrect B. Algebraic proof Quite often the easiest way to prove a statement is to use algebra *To choose 2 consecutive numbers using algebra we could use n and n + 1, however we would not know which one was odd and which was even. *To choose an even number we could use 2n, as doubling any number will make it even. *Any two even numbers could be 2n and 2m. *As 2n will be even using 2n + 1 or 2n – 1 will give us an odd number *2n is a multiple of 2, 3n is a multiple of 3, 4n is a multiple of 4, etc. 1. Prove the sum of three consecutive numbers will be a multiple of 3 Let the numbers be n, n + 1, n + 2 Start by showing the numbers you are going to use n + n + 1 + n + 2 = 3n + 3 = 3(n + 1) As we could divide by 3 the sum has to be a multiple of 3 2. Prove the sum of two consecutive numbers is odd Let the numbers be n, n + 1 Start by showing the numbers you are going to use n + n + 1 = 2n + 1 2n + 1 can’t be divided exactly by 2 so the sum must be odd © www.teachitmaths.co.uk 2016 28266 Page 1 of 2 Proof 3. Prove the product of two consecutive numbers is even Let the numbers be n, n + 1 Show the numbers you are going to use Product = n(n + 1) = n2 + n We cannot show that we can divide by 2 Choosing a different start may help! The two numbers must be of the form 2n, 2n+1 or 2n-1, 2n depending on whether the smaller number is even or odd Choose new starting values Let the numbers be 2n, 2n + 1 We have now shown we can divide by 2, so the product must be even Product = 2n(2n + 1) = 4n2 + 2n = 2(2n2 + n) It is easy to show that the same approach will work with 2n-1, 2n. Check this. 4. Show that the sum of three consecutive even numbers will be a multiple of 6 Let the numbers be 2n, 2n + 2, 2n + 4 Show the numbers you are going to use Sum = 2n + 2n + 2 + 2n + 4 = 6n + 6 As we could divide by 6 the sum has to be a = 6(n + 1) multiple of 6 5. Show that the difference of two consecutive square numbers is odd Let the numbers be n, n + 1 Start by showing the numbers you are going to 2 2 use, then square them n and (n + 1) = (n + 1)(n + 1) 2 =n +n+n+1 = n2 + 2n + 1 2 Difference = n + 2n + 1 - n2 = 2n + 1 2n has to be even, so 2n + 1 must be odd 6. Prove that (n + 3)2 ≡ (n + 2)(n + 4) + 1 Show the LHS = RHS (n + 3)2 ≡ (n + 3)(n + 3) ≡ n2 + 3n + 3n + 9 ≡ n2 + 6n + 8 + 1 ≡ (n+2)(n+4) + 1 Both sides are the same 7. Prove that (2n + 1)2 – (2n - 1)2 will be a multiple of 8 Square each bracket = (2n + 1)(2n + 1) – (2n - 1)(2n – 1) 2 2 = (4n + 2n + 2n + 1) – (4n - 2n – 2n + 1) = (4n2 + 4n + 1) – (4n2 - 4n + 1) Be careful with the signs when doing the subtraction = 4n2 + 4n + 1 – 4n2 + 4n – 1 = 8n and 8n is a multiple of 8 as it can be divided by 8 8. Prove that a = 3 and b = -2 when 3(ax + 2) – 2(x - b) ≡ 7x + 4 3ax + 6 – 2x + b ≡ 7x + 4 x(3a – 2) + 6 + b ≡ 7x + 4 3a – 2 = 7 6+b=4 3a = 9 b=4–6 𝟗 a=𝟑 a=3 © www.teachitmaths.co.uk 2016 Multiply out the brackets then equate the coefficients b = -2 28266 Page 2 of 2