SAT Math and Logic

A large part of understanding mathematics is knowing what words mean. There’s no doubt about it, we even explain mathematics using plain ol’ English. I was reminded the other day that the same word can have different meanings to different people. In fact, while I may have an idea clear in my mind, I can be explaining a concept to a student who “knows” the same words that I do, but we are having two entirely different conversations.

As an example I recently heard somebody tell me that he was paid “biweekly.” But I really knew that he was not paid twice per week because that was not the context used. Interestingly enough the word “biweekly” actually does hold dual meanings. It can mean, “twice a week,” or it can mean, “every two weeks.” There are really two things to say about this.

The first is that this is a classic example of where social-wide ignorance has created ambiguity. Truly it is rare to come across a well-spoken individual: One skilled in oratory and definition. Rare indeed is the person who actually says something (just listen to any politician). I am sure all of you can remember a time when you have had a conversation with somebody for several minutes and walked away wondering what it was you just spent all that time talking about. The world is filled with people who can tell you all about something they have no real knowledge of.

Secondly, don’t assume that just because you have heard a word before or seen it in a book that you own the word in your repertoire. One of the only dangers of reading is that when you see a word repetitiously you can get a false sense of confidence in its meaning. The band 311 has a song entitled, “Reconsider Everything” which really sums up the idea here:

“What if the truth is that there is no truth
The only thing I can prove is there is no proof
Don’t be so sure that your source is correct
People believed it before, before they had checked”

Since the foundation of mathematics is arithmetic, it would be great to have an understanding of the different types of numbers that you will be working with in the first place. For lack of a word that actually exists, let us call this realm of all possible numbers the Numberverse.

For the SAT, the Numberverse is only as large as Real Numbers (you will never be tested on imaginary/complex numbers on the SAT). The following diagram I found at wlonk.com:

I like it because it is simple and does a great job at showing examples within each set. However, you will notice that in Natural Numbers it contains the value zero. I picked this as an example because it makes the assumption that zero is a Natural Number. I have always been taught, and continue to teach, that zero is NOT a Natural Number because I cannot actually possess zero of something. I simply do not have it. I have always called the set containing zero, “Whole Numbers” and showed Natural Numbers as a subset of Whole Numbers.

Even in mathematics there are philosophical debates at every level about simple things like this. All I can tell you is to assume nothing and Reconsider Everything.

– Andrew Turner

www.thinkarchimedes.com