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PROOFS
Algebra: prove or disprove
PROOFS
- Mathematical arguments to establish the universal validity of the statement.
- UNIVERSAL: under all conditions, no exceptions are allowed. 2 โThe sun rises in the east and sets in the west.โ (^) Always: UNIVERSAL โEvery person sees a rainbow on a rainy day.โ May be or may not be: NOT UNIVERSAL
PROVE AND DISPROVE
- For any statement
- Proof: general
- Disproof: only one case (counterexample) 3 For all positive integers ๐, ๐๐^ โ ๐ + ๐๐ gives a prime number. To prove the statement, you will have infinite cases. To disprove the statement, you only need one value of ๐ Disproof: Put ๐ = 11 11 เฌถ^ โ 11 + 11 = 11เฌถ Which is not prime.
DISPROVE: COUNTEREXAMPLE
Statement Counterexample Prime numbers are always odd 2 The square root of a number is always smaller 0.25 = 0. than the number itself. 7, 9 19, If ๐ is prime then ๐ + 2 is prime ๐ = 11 2 11 เฌถ^ + 11 = 11 22 + 1 = 23 ร 11 2๐เฌถ^ + 11 is prime for all integer values of ๐ 4
PROVE
- Sum of two odd integers is always even.
- Prove: Consider two numbers,
- ๐ด = 2๐ + 1, ๐ โ โค
- ๐ต = 2๐ + 1, ๐ โ โค ๐ด + ๐ต = 2๐ + 1 + 2๐ + 1 ๐ด + ๐ต = 2๐ + 2๐ + 2 ๐ด + ๐ต = 2(๐ + ๐ + 1) 7
ALGEBRA ADDITION RULES
Number-1 Number-2 Addition Even Even Even Even Odd Odd Odd Even Odd Odd Odd Even 8
ALGEBRA MULTIPLICATION RULES
Number-1 Number-2 Multiplication Even Even Even Even Odd Even Odd Even Even Odd Odd Odd 9
REPRESENTING NUMBERS
10 Mathematical Representation Description Two consecutive integers ๐, (๐ + 1) Three consecutive integers ๐ โ 1 , ๐, (๐ + 1) Two consecutive odd integers 2๐ โ 1 , (2๐ + 1) One less than multiple of 7 7๐ โ 1 A number divided by 5 leaves (5๐ + 3) a remainder of 3 Three consecutive even 2๐ โ 2 , 2๐, (2๐ + 2 ) integers
PRACTICE
- Show that square of an odd number is also odd.
- Show that 5 เฌผเฌฝ^ โ 401 is not a prime number. 13
PRACTICE
- If 5 3๐ฅ โ 5 โ 2 2๐ฅ + 9 โก ๐๐ฅ + ๐ is an identity, find the values of ๐ and ๐
- Prove that ๐ + 3 เฌถ^ โ 3๐ + 5 โก ๐ + 1 ๐ + 2 + 2 14
PRACTICE
- Tom says that 7๐ฅ โ (2๐ฅ + 3)(๐ฅ + 2) is always negative. Is he correct? Explain your answer
- Prove that ๐ + 3 เฌถ^ + ๐ 3 โ ๐ โ 3(๐ + 4) is a multiple of 3 for all integer values of ๐ 15
PRACTICE
- Prove algebraically that the sum of the squares of two consecutive multiples of 5 is not a multiple of 10. 16
PRACTICE
- The diagram shows a cross placed on a number grid.
- L is the product of the left and right numbers of the cross.
- T is the product of the top and bottom numbers of the cross.
- M is the middle number of the cross.
- Show that when M = 35, L โ T =
- Prove that, for any position of the cross on the number grid above, L โ T = 99. 19
