Equality

Background 

As we start to evaluate expressions, terms that will be used more and more is "equals" or "equal to", for instance, 2 + 3 equals what? Or 2 + 3 = ?

Question

What does "equals" mean?

Answer

It allows us, with math symbols, to say that the term on the left of the equals sign has the same value (or values) as the term on the right hand side.

Analysis 

Before I dive into this, let me be clear that we're talking about Equality in the math sense of the word and not in terms of human relations (and while I may be accused of some sort of bias against numbers, I do believe that all people are equal to each other but not all numbers are equal to each other. And just to be clear, that was supposed to be funny...)

In math, we run into situations where we want to say that This Thing is the same value as That Thing. When we have a situation where This Thing is the same value as That Thing, we say that they are equal. We can write it like this:

This Thing = That Thing

with those two little lines, known as an "equals sign" in the middle meaning "is equal to".

So how do we use it?

We can say something really obvious like this (I'll colour the 4s in different colours because I'll start playing with them in a minute):

4 = 4

or 4 equals 4. That's almost so obvious that you're looking for the trick or the twist. No trick, no twist - the 4 on the left side and the 4 on the right side are the same thing.

Let's play with this idea a little more. I can write the value 4 in a couple of different ways:

2 + 2 = 4
3 + 1 = 4

I can even combine those two statements:

2 + 2 = 4 = 3 + 1

The 2 + 2 = 4 on the left side is the same value as the 3 + 1 = 4 on the right side.

The Equal Sign as a Scale

Sometimes people will think of the equals sign as a scale - that the two sides of the scale are of equal weight.

So what happens if I add 1 more to the left side? For the two sides to continue to be equal, I need to add 1 to the right side. Let's work with that 4 = 4 again:

4 = 4

and now I add 1 to both sides:

4 + 1 = 4 + 1

5 = 5

(notice I simplified the left and right side 4 + 1 and put it on the line below). Does 5 = 5? yes.

The same thing works with subtraction:

4 - 1 = 4 - 1

3 = 3

Notice that the 3 and the 5 I got by subtracting and adding 1 from 4 make the values different - I'm not trying to say that 3 and 5 also equal 4. Nothing of the sort - we're talking about keeping the balance equal between the left and right sides.

Asking if the right side and the left side are equal:

One more thing before closing this topic - sometimes we aren't sure if the left side and the right side are equal. We can ask the question with a question mark over the equals sign:



We could ask if 2 + 2 equals 3 + 1:

2 + 2  3 + 1

We add 2 + 2 on the left side and we add 3 + 1 on the right side and get:

4  4

It does, so we can finish with

4 = 4  ✅(we can add the checkmark or a smiley face just so we know we've verified the equality).

If it doesn't however, we can use a "not equals" sign, which is an equal sign with a slash through it: 

Let's try working with it:

2 + 3  1 + 3

5  4

And we finish by saying 5 and 4 are not equal:



Vocabulary used:

For more information check out these links (comment to add your favourite link):


Where might you have come from?

Fact-orials Index

Operations:

• Addition
• Subtraction

Where might we go?

Relations:

• Equality and Rational Numbers
• Inequality