Number proof and structure
Objective. Prove results involving factors, irrationality and recurring decimals.
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- 2. Worked example
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Learn: the key idea
Number proof and structure uses algebraic representation of integers, such as 2n for any even number or n and n+1 for consecutive integers, to prove a general statement is always true rather than checking examples. Proofs of irrationality often use contradiction: assume a number is rational, write it as a fraction in lowest terms, and derive a contradiction. Recurring decimal proofs multiply the decimal by a power of 10 to align the repeating block, then subtract to eliminate the recurring part algebraically. For example, to prove the sum of two consecutive integers is odd, write n+(n+1) = 2n+1, which is always one more than an even number.
Worked example
Prove algebraically that the sum of any two odd numbers is always even.
- 1Let the two odd numbers be 2a+1 and 2b+1, for integers a and b
- 2Add them: (2a+1) + (2b+1) = 2a+2b+2
- 3Factorise: 2a+2b+2 = 2(a+b+1)
- 4Since a+b+1 is an integer, 2(a+b+1) is a multiple of 2, so the sum is always even
Answer: Sum = 2(a+b+1), always even
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Start practisingCommon mistake. Pupils often 'prove' a statement using only one or two numerical examples, such as checking 3+4=7 and 5+6=11 and concluding all consecutive sums are odd, instead of using algebraic expressions like n and n+1 that cover every case.
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