Copyright

Proof of Theorems Using Number Properties

Instructor: Michael Gundlach
In school mathematics, we often learn rules, such as the product of two even numbers is even. In this lesson, you'll learn how to prove this and many similar statements students often learn in grade school.

A Need for Proof

You probably learned in grade school that an even number multiplied by another even number equals an even number. Some easy examples make this statement believable.

2 × 2 = 4

4 × 12 = 48

8 × 16 = 128

But, how do we know that there aren't two weird even numbers that multiply together to give an odd number? This is where a proof comes in handy. When we're able to prove from basic number properties that such statements are true, we'll know that no such evil numbers exist.

By the end of the lesson, we'll have proved that the product of two even numbers is even and the product of two negative numbers is positive.

Definitions and Number Properties

In order for us to do these proofs, we're going to need some definitions and number properties. We'll start with some definitions.

An even number is an integer that is divisible by 2. In other words, if an integer x is even, then x = 2k, where k is some other integer.

An odd number is an integer that is not divisible by 2. Therefore, if y is odd, we can write it as y = 2l +1, to show that y has a remainder of 1 when divided by 2.

A negative number is any real number, a, that can be added to one positive number, b, resulting in a sum of zero. This can be expressed as a = (-1)b or a + b = 0. In other words, a = -b, and b > 0.

These definitions may seem a little silly, but they're very important in proving properties about numbers. We'll also need a few properties of integers and real numbers to help us with the proofs we're going to do.

According to the closure property, we know that the integers are closed under multiplication and addition. In other words, adding and multiplying integers will always give us integers.

The positive real numbers are also closed under multiplication and addition.

The commutative property of multiplication says that for all real numbers (x and y), xy = yx

The associative property of addition tells us that if we are adding 3 or more numbers, it doesn't matter which pair of numbers we add together first. In symbols, this means that if a, b, and c are real numbers, then (a + b) + c = a + (b + c).

The associative property of multiplication tells us that if we are multiplying 3 or more numbers, it doesn't matter which pair of numbers we multiply together first. In symbols, this means that if a, b, and c are real numbers, then (ab)c = a(bc).

The distributive property tells us that for real numbers a, b, and c that a(b + c) = ab + ac.

The identity property of multiplication tells us that any number multiplied by 1 equals itself.

With these definitions and properties in hand, we're now able to prove some of our basic number properties.

Even-Odd Product Properties

Let's start out by proving that the product of two even numbers is even.

Let a and b both be even numbers. We're using letters to represent them to show that a and b are arbitrary even numbers. A variable is arbitrary when it could represent any member of a particular set; we assign no special properties to the variables other than being part of a set.

In this case, we're saying a and b are even, and no more. Since they are even, we can write a = 2k and b = 2l by our definition above, where k and l are integers. The use of two different letters, k and l, for this step is important. It reminds us that a and b could be two different even numbers.

Now, let's see how we can use this to show that the product of two even numbers is even. We're going to multiply a and b together and see what we can do to make it plain that their product is even.

ab = (2k)(2l) by substitution.

(2k)(2l) = 2(k2l) based on the associative property.

So ab = 2(k2l).

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support