PROOF
Algebraic Proof
If I said that any two odd numbers add to make an even number, you could try a range of examples and eventually agree that two odd numbers do add to make an even number. However, the only way to know this is certainly true for all values is by proving it algebraically. By finding examples, we are only proving that the statement is true for those particular values, rather than all values. In this section, we look at algebraic proof by first looking at how numbers can be represented algebraically.
Prior knowledge
Key Terms
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Proof: Proving a statement to be true for all values.
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Even number: A number in the two times table, of the form 2k.
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Odd number: A number that is not in the two times table, of the form 2k+1.
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Any two integers: Two whole numbers, k and j.
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Consecutive numbers: Two integers that follow each other, of the form n and n+1.
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Multiple: A number in a particular times table.
Exam Questions
Identities
An identity is similar to an equation, however, it is true for all values of x, rather than just some particular values of x. Identities are represented using an equals sign with three lines (≡). For example, the equation 2x+1=5 is true only when x=2, and this is what makes it an equation. 2x+3x≡5x is true regardless of the value of x and hence is an identity. Equations can be solved to find the particular values of x, whereas with identities there is no need. However, if I knew that 2x+ax≡5x, I could deduce that the constant a has value 3 because I know that the left hand side is identical to the right hand side. In this section, we look at how identities are used.
Key Terms
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Equation: Two expressions that are equal for a particular value(s) of x.
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Identity: Two expressions that are equal for all values of x.
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​Comparing coefficients: Comparing coefficients of terms to find unknown coefficients.