Proving Theorems Using Number Properties

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will explain proving theorems using number properties. We will go over multiple examples of using various number properties to prove different theorems, and the steps involved in doing this.

Proving Even Multiples

It is said that mathematicians make great lawyers. That's probably because they are so great at proving things!

Suppose Earl Integral is a math lawyer, and he is hard at work on a case. His client, the defendant, has claimed that if two even numbers are multiplied together, then the result is also an even number. The plaintiff is claiming that since there are so many even numbers, it is not possible for the defendant to test them all and make this claim.

Earl to the rescue! On the day of the trial, Earl tells the jury that he will prove, beyond a shadow of doubt, that what his client says is true.

Earl explains that by definition, an even number is a number of the form 2k, where k is an integer. He tells the jury to consider any two even numbers. By definition, they would have the form 2m and 2n, where m and n are integers. If we multiply these two numbers together, we get the following:

  • (2m)(2n)

Remove parentheses using the associative property

  • 2 × m × 2 × n

Rearrange using the commutative property

  • 2 × 2 × m × n

Add in parentheses using the associative property

  • 2(2mn)|

We end up with 2(2mn). Since m and n are integers, it follows that 2mn is also an integer, because integers are closed under multiplication. Therefore, we have that the product of any two even numbers is equal to 2 times an integer (the definition of an even number). Thus, the product of any two even numbers is also an even number.

Earl just won the case for his client.

Proving Theorems

Earl's argument is called proving a theorem in mathematics. Proving a theorem uses properties, other proven theorems, and rules to prove mathematical theorems. In Earl's proof, he used quite a few different number properties, properties about sets of numbers that are always true.


numbpropthm1


In general, number properties are a great tool to help us to prove theorems. Let's take a look at a couple of examples.

Example 1: Odd Number Subtraction

Suppose we are given that a + b is odd, where a and b are integers, and we want to prove that a - b is also odd. This is another example of when number properties come in really handy! For this proof, we are going to use the following number properties:

  1. An odd number is a number that can be written as 2k + 1, where k is some integer.
  2. If a = b, then a ± c = b ± c.
  3. If a = b, then we can substitute a for b or b for a in any expression.
  4. a(b ± c) = ab ± ac
  5. The integers are closed under subtraction.

We are given that a + b is odd, so by property 1, a + b = 2k + 1, where k is some integer. We want to show that a - b is also odd, so we want to show that a - b = 2m + 1 for some integer m.

a + b = 2k + 1 By property 1
a = 2k - b + 1 By property 2
a - b = (2k - b + 1) - b = 2k - 2b + 1 By property 3 and simplifying
a - b = 2k - 2b + 1 = 2(k - b) + 1 By property 4

We have that a - b = 2(k - b) + 1, and by property 5, since k and b are integers, k - b is an integer, call it m. Therefore, a - b is of the form 2m + 1, where m is an integer, so by property 1, a - b is odd.

So neat! Let's consider one more example!

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