1.3.5

Proof by Exhaustion

Test yourself

What is Proof by Exhaustion?

Proof by exhaustion is a method of proving that a mathematical statement is always true by working it out and showing it is true for every possible case.

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How do you prove by exhaustion?

  • To prove a conjecture by exhaustion, split it up into different cases.
  • The different cases come from looking at all of the possible values within the range that the conjecture is true for.
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Example

  • Prove that no square number ends in a 7.
  • It's impossible to write out all the square numbers to show that they don't end in a 7.
  • We have to think of another way to cover all of the possible cases.
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What are the different cases?

  • All numbers ending in the same digits have square numbers that also end in the same digits.
  • For example. as 42 = 16, all numbers ending in 4 will have square numbers that end in 6.
    • This is a known theorem, so we can begin our proof with this statement.
  • So the cases to consider are the squares of the 10 different digits that a number can end in.
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Square numbers

  • Writing out the square numbers, we have:
    • 02 = 0, 12 = 1, 22 = 4
    • 32 = 9, 42 = 16, 52 = 25
    • 62 = 36, 72 = 49, 82 = 64 and 92 = 81
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Inspection

  • We see the possible digits they can end in are:
    • 0,1,4,5,6,90, 1, 4, 5, 6, 9
  • The number 7 is missing from this list.
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Final statement

  • No square number can end in a 7.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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