Why Is Math Important?

Date: 01/02/2002 at 15:30:23
From: Amanda Dalton
Subject: Why math is important

Why is math so important? Some of it we don't need to know in the 
real world so why do they teach us things we won't need to know?

Date: 01/02/2002 at 16:57:46
From: Doctor Achilles
Subject: Re: Why math is important

Hi Amanda,

Thanks for writing to Dr. Math.

This is a really hard question to answer well. I can think of a few 
reasons.

The first is that it's often surprising how many places strange math 
ideas pop up. We have an FAQ page dedicated to this, and it links to a 
lot of archives that show how math is used in a lot of different 
professions (from physics and medicine to law and just your everyday 
life):

   Why Study Math?
   http://www.mathforum.org/dr.math/faq/faq.why.math.html   

But there are two really important reasons for learning math that 
don't directly link to how you will use it in your daily life.

The first is basically "the more you know, the more options you have 
available." I'm 21 years old. Almost everyone I know who is my age 
either had no idea what he or she wanted to do 5 years ago, or they 
thought they knew but then changed their minds later. (Actually, a 
fair number of my friends STILL don't know what they want to do.) I 
have one close friend who wanted to be a computer scientist until he 
got to college, and then he changed his mind and he's now studying 
politics. But all the math that he learned for computer science has 
helped him with the statistical analysis and economics he has to study 
as part of his politics work. Fractions, geometry, algebra, and trig 
will all be integrated into just about anything you ever want to 
study, and having practiced those things in high school will allow 
you to not have to worry about relearning them later. Slope-intercept 
form for economic graphs or for population models in biology should be 
second nature, so that you don't have to worry about math later and 
can focus on what you really do want to study.

For example, let's say you want to be an ecologist. Ecologists often 
study population size. Let's say that you're studying a population 
whose size increases like this: size = 2 * time + 100. This is a 
standard slope-intercept equation. A lot of people don't want to learn 
slope-intercept because it doesn't seem to have any point. But if you 
understand it enough that you don't have to think about it, then when 
you see an equation like that, you won't have to worry about the math 
at all and you can just think about the ecology.

So basically, the idea here is that if you learn math now, then when 
you're confronted with math later in life, you won't have to worry 
about it at all and instead you can just pay attention to what you 
want to.

The alternative strategy is to not learn math now. You can wait and 
see what math you end up needing for your job or your future education 
and just learn that. That's okay in principle, but you'll end up 
playing "catch-up" all the time.

When I finished the math requirements at college, I was really happy 
at the time because I didn't like math then and I was glad to be done 
with it. These past couple of years, I've gotten interested in 
computer science as a hobby. However, not having any advanced math I 
can't go very far with it, so if I wanted to do anything with it, I'd 
have to go back and take a lot of math classes that would've been 
easier for me to take a couple of years ago.

Since you don't know for sure what you're going to do with your life, 
it's best to keep your options open. Of course, you can't learn 
everything, and sooner or later you have to decide what you do want to 
study. But, whatever it is, having a good background in math will put 
you ahead.

The second reason to study math is that it gives you a different 
perspective on things. I think that most people hate math because it 
is taught just as an exercise in memorization. You get the impression 
that all there is to math is just a bunch of formulas that you can 
look up in a book. I think of math as something totally different.  
Check out these two links:

   What is Mathematics?
   http://www.mathforum.org/dr.math/problems/erum.09.22.00.html   

   Philosophy of the Truths of Mathematics
   http://www.mathforum.org/dr.math/problems/lauren.02.28.01.html   

The way to be good at math is not to memorize a whole lot of different 
things, it's just to memorize a few small things and then play around 
with them and see what else you get.

For example, what is algebra? Well, you already know about 
multiplication, division, addition, and subtraction. One day (a long, 
long time ago) somebody who knew all of those things was sitting 
around and thinking. Below is my own cartoonish recreation of what 
might have happened.

This person (we'll call him or her "Pat") was sitting around thinking 
about addition.

Pat knew that 3 + 4 = 7

Then Pat thought "what would happen to the equation if I added one to 
both sides?" Pat tried it and got:  3 + 4 + 1 = 7 + 1

Pat realized right away that this new equation was also true. So then 
Pat went back to the original equation: 3 + 4 = 7 and decided to 
subtract 3 from both sides and got: 3 + 4 - 3 = 7 - 3.

Pat then did some arithmetic, and ended up with: 4 = 4.

Right away, Pat realized that this was something that could be applied 
to new, different things. What Pat thought next was "what if I didn't 
know one of the numbers?"

Pat was already familiar with equations like this: 3 + 4 = ?, and knew 
that you could solve those equations.

What Pat decided to try was something a little different: ? - 4 = 7.  
Pat knew from before that you can add or subtract the same number from 
both sides of an equation (see above) and you'll still have a true 
equation. So what Pat did was to add 4 to both sides of this equation 
and got:

  ? - 4 + 4 = 7 + 4

After a little bit more arithmetic, Pat ended up with ? = 11.

So math is not just a bunch of memorization. If it's taught well, math 
is just an extension of a few basic ideas to more and more complicated 
applications. Algebra is not something new to learn, it's just another 
way to do addition and subtraction and multiplication and division, 
only with numbers and letters instead of just with numbers.

For a look at how complicated mathematical formulas are really just a 
new way to look at old formulas you already know, check out:

   Remembering Area Formulas
   http://www.mathforum.org/dr.math/problems/summer.12.23.01.html   

A classic example for me of how math is an exciting exploration of 
simple ideas that leads you to interesting results is the "golden 
ratio."  The golden ratio is a number (about 1.62). If you make a 
rectangle where one side is equal to some number (whatever number you 
like) and the other is equal to that number times the golden ratio, it 
turns out to be a really pretty rectangle (at least as pretty as 
rectangles get). In fact, if I told you to just draw a rectangle on a 
piece of paper, you'd probably draw one that came pretty close to the 
golden ratio.

Also, if you then take that rectangle and cut it up into a square and 
another rectangle, the new rectangle that you make will also have the 
golden ratio. You can then take the new rectangle and do the same 
thing to that, on down forever, and each rectangle you make doing that 
will have the golden ratio. You can then connect the corners of these 
rectangles you've made and it makes a really pretty spiral (called the 
golden spiral) which has other neat mathematical properties.

So one day there was some guy named Fibonacci. And he thought of an 
interesting way to make a sequence of numbers. You start with the 
sequence {1, 1}.  Then to get the next number of the sequence, you 
just add the last two numbers in it together. So you get {1, 1, 2}.  
Then you add the last two number of that sequence together to make a 
new sequence {1, 1, 2, 3}. After a couple of more times, you'll end up 
with {1, 1, 2, 3, 5, 8, 13}.

Now here's the cool part: One day someone asked themself what the 
ratio is between each number in the Fibonacci sequence and the number 
before it. So what that person did was take a number and divide it by 
the one before. At first it didn't look too interesting, but the 
person kept at it for a little while. As he got to bigger and bigger 
numbers, he realized that the ratio of numbers in the Fibonacci 
sequence gets closer and closer to the golden ratio. He then went on 
to prove using calculus that as the numbers go on to infinity, the 
ratio will EXACTLY approach the golden ratio.

Just imagine how exciting that must have been. To start with a simple 
pattern like the Fibonacci sequence that anyone can understand and to 
end up finding a ratio that has really cool mathematical properties 
when you look at rectangles and spirals!

All of math is, as far as I see it, an exploration of what happens 
when we assume certain basic "axioms" about the world and then see 
what we get from that. There have been some very surprising results 
that have opened the door to new questions about what our simple 
assumptions and trivial starting places will take us next.

If you want to learn a bit more about the golden ratio and the 
Fibonacci sequence, check out the Dr. Math FAQ:

   Golden Ratio, Fibonacci Sequence
   http://mathforum.org/dr.math/faq/faq.golden.ratio.html   

So math really is important for your daily life, no matter what you 
want to do. It's true that not everything you learn will be needed 
later, but you never know what you'll want to do, and the more you 
know and have to build on, the better off you'll be. Also, if you look 
at it the right way, math is not a chore of memorization, it's a game 
where you start with a few simple rules and see what more complicated 
rules you can come up with using those simple tools.

If you have other questions, please feel free to write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/