1.3.5
Proof by Exhaustion
What is Proof by Exhaustion?
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.
How do you prove by exhaustion?
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.
Example
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.
What are the different cases?
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.
Square numbers
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
Inspection
Inspection
- We see the possible digits they can end in are:
- The number 7 is missing from this list.
Final statement
Final statement
- No square number can end in a 7.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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