In the Lost Tools of Writing we learn about the five common topics. These topics are 5 categorical questions that we ask in order to invent something. Asking questions is essential for thinking. I tell my children that there is a world of difference between the man who enters a room saying, "this is nice" and the one saying, "how can I make this better?" To journey from accepting everything we're fed to persuading others to the truth, we must learn how to ask great questions. This is true for writing as well as mathematics. Over the next few weeks, let's explore this world of the common topics and how they reveal themselves in the world of mathematics.
Definition : The first common topic is focused on exploring what something is. To be or not to be. We aren't looking at what something is becoming, but what something is. We must find it's essence. Setting aside the world of philosophy, we can explore definition by asking questions like, "what is x?", "what does x do?", and "what does x have?"
In mathematics this is apparent in the mere definition of numbers. We have integers, fractions, irrational numbers, etc. In prealgebra we see the importance of definition in the world of exponents. Exponents come with a bucket full of properties. When we multiply them something happens, when we divide them something else happens. It's difficult to remember all of the properties. How do you keep them all straight? We remember what they are. Every single property of exponents is an extension of the very definition.
An exponent is repeated multiplication. Thus 2 ^ 3 means the product of three 2s: 2 x 2 x 2. This reality helps us see the truth that if I had the product of 3 2s multiplied by the product of 4 2s, then I'd have: 2^3 * 2^4 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 ^7 . A shortened way to complete this operation is to notice that by definition the product of three 2s multiplied by the product of four 2s is going to be the product of seven 2s. Do you see it?
I will spare you the details of how the definition of exponents literally defines all of the properties and instead ask you to explain this one to someone else. Let's see if you really understand the true definition of exponents!