### Topics

## Introduction to :

Understanding **equality**, or sameness, is a universal theme in all areas of mathematics. When mathematicians say, "2 + 2 = 4," they mean that the two things on either side of the equals sign are literally the same thing: they are two ways of describing the same object. They are equal.

In mathematics, the objects we work with are usually numbers. The thing is, we can't usually see them (talk about imaginary numbers, huh?), and we're forced to make do with looking at their descriptions instead.

Somebody with too much free time might choose to write "(1 + 1 + 1 + 1) – (1 + 1 + 1) + 1 + (1 – 1 – 1) – (1 – 1) + (1 + 1)" instead of the significantly easier to read "3," even though they describe the exact same number. Way to mess with us.

#### Example 1

Identify the property exemplified here. Turkey Sandwich = Turkey Sandwich. | |

A turkey sandwich is a turkey sandwich no matter how you look at it. Just like contortionists. They can be quite flexible (or *re*flexible), but they're still the same thing. This is an example of the reflexive property. | |

#### Example 2

We're given that *A* = *B* and *C* = *D*. As a result, we state that *A* – *C* = *B* – *D*. This equation is an example of which property or properties? | |

Well let's take a look-see. If we take *A* = *B* and we subtract the same number from it, that's the subtraction property. That gives us *A* – *C* = *B* – *C*, right? The problem is that we're subtracting *C* from *A* and *D* from *B*. For that property to be true, we need *C* to be the same as *D*. Luckily, we're told that right off the bat. That way, we can make use of the substitution property and replace a *C* with a *D*. That'll give us *A* – *C* = *B* – *D*. | |

#### Example 3

If *A* = *X*, *B* = *Y*, and *C* = *Z*, then can we say that *A*(*B* + *C*) = *Z* × *Y* + *X* × *Z*? | |

Nope. If we look carefully at the left side of the equation and apply the substitution property, we get *X*(*Y* + *Z*). Then, the distributive property gives us *X* × *Y* + *X* × *Z*. We can't be sure that *Z* × *Y* is the same as *X* × *Y*, so that isn't right. | |

#### Example 4

If *A* + *B* = *C*, then is it possible to claim that *X*(*A* + *B*) = *C* × *X*? Which properties allow us or forbid us from doing so? | |

It's not only possible. It's possimpible. First off, the multiplication property allows us to multiply both sides of an equation by the same number. In this case, that number is *X*. Depending on how we multiply them, we may end up with our answer or *A* × *X* + *B* × *X* = *C* × *X*. If that's how we want it, we can use the distributive property to factor out the *X* and end up with our answer. Hooray. | |

#### Exercise 1

Give an example of the symmetrical property.

Hint

Think of the symmetrical property like a line of symmetry that you can flip things across.

Answer

There are many options but it essentially states that if *A* = *B*, then *B* = *A*.

#### Exercise 2

If we're given a puppy, what can we say about the puppy and which property would we be using?

Hint

Even though the puppy isn't a contortionist, the same property applies.

Answer

The reflexive property: puppy = puppy.

#### Exercise 3

^{A}⁄_{B} = ^{C}⁄_{D}. Which statements would make this true?

Hint

You should use the division property as long as you promise not to divide by zero.

Answer

*A* = *C* and *B* = *D* ≠ 0.

#### Exercise 4

*X* = *Y* and *Y* = 3. Which properties allow or forbid us to say that 3 = *X*?

Hint

We aren't adding, subtracting, multiplying, or dividing.

Answer

The transitive and the symmetrical properties.

#### Exercise 5

*A* = *B* and *X* = *Y*. Which properties allow or forbid us to say that ?

Hint

Assume that *C* ≠ 0. We don't want to go down that road.

Answer

The addition property, the division property, the distributive property, and the substitution property.

#### Exercise 6

*A* = *B*, *B* = *C*, *C* = *D*, and *D* = *E*, *E* = *F*, and *F* = *G*. Which properties allow or forbid us to say that *A*(*B* + *C*) = *D* × *E* + *F* × *G*?

Hint

The statement is true. Which properties allow it?

Answer

The distributive property, the transitive property many times over, and the substitution property.