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# Two Plus Two Equals Two Times Two

November 2, 2016

You know that 2 + 2 = 4 right? A six year old child probably knows this too.

You also know that 2 x 2 = 4. An 8 year old child would know that.

Some children find it interesting that by adding 2 to itself you get the same result as multiplying it by itself. Maybe trivial, but still interesting... Does this hold for every number?

Obviously not.

1 + 1 = 2 and this isn't 1 x 1. If you try 3, 4, 17.8 or most other numbers then you will also see that they don't have this property that 2 does. Even if you try negative numbers, e.g, -2:

(-2) + (-2) = -4  and this doesn't equal (-2) x (-2) = 4.

I'm not sure if you remember that a negative times a negative yields a positive, but it does (more on this in another post). Hence our conclusion is that most numbers don't possess this property that makes self addition and self multiplication appear as being the same operation.

Now ask your child to find another number (other than 2) that has this property....

.... Well, another number that has this property is obviously 0:

0 x 0 = 0 + 0.

Ok, so you've found out that this exclusive club of numbers has 0 and 2.  Is that it? Any other numbers?

Now if you have a teenager in the house, someone who has already dealt with some basic algebra in high school, ask them to formulate an equation that will describe all the numbers Y, that have this property. Assuming that the teenager agrees to take on the task, you might get something like this,

Y + Y = Y x Y,

or if you rewrite it you can write, 2 Y = Y^2  (where Y^2 is read as Y squared and 2 Y is 2 times Y).

Look at the equation. It makes sense. Right? It specifies the rule that Y added to itself equals its product with itself. This is what equations often do: They define properties.

Then by basic algebra your teenager would get,

2Y - Y^2 = 0  or Y(2-Y) = 0.

When written this way, it is clear that the only Y values that solve this quadratic equation are Y = 0 and Y = 2. Indeed a quadratic equation has at most two distinct solutions.

So an 8 year old child can probably suggest by trial and error that 0 and 2 are the only numbers that have this property. But a 15 year, can prove it. Now you can too!

Got it? Leave us a comment if you have questions.

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