What Is A Number?

Mathematics is a system of two types of things.  These two types of things are objects and rules.  Objects are called numbers or sets.  Rules have many different names.

                To begin with (by the way, I just love starting sentences with prepositions), a number is usually NOT something you count with.  Most numbers are for other things than counting.  No one will use the number i to count with.  (What comes before i, h?  Not h; h is Planck's constant).  The number i is the square root of negative one.  That means that i multiplied times itself is negative one.  No real number multiplied times itself is a negative number.  The number i is called the imaginary number, because it doesn't look at all like a real number.  (Actually, all numbers are imaginary, it's just that we are so familiar with some of them that they seem real to us.)

                You can only count with counting numbers. 

                YOU CAN'T COUNT WITH ANYTHING ELSE!!!  DON'T TRY, OR YOU'LL BE IN THE SOUP!

                I prefer to call counting numbers natural numbers, because there is something outdoorsy about them.  They remind me of camping trips in the mountains.  I love camping trips in the mountains. 

                There is a history of mathematics, which may or may not be true, but we pretend that it is.  According to this history, a long time ago mathematics was used for counting and nothing else.  People would count their cows and consider this the height of mathematical achievement.  It wasn't until people considered it too inconvenient to count everything every time they wanted to know how much they had that they began adding the ones they got today and subtracting the ones they gave away to make a daily total. 

                Eventually people needed a more efficient way to add; addition just took too much time.  Multiplication was invented.  Now, instead of adding three a hundred times, one just multiplied three times one hundred.  Rules for multiplying made it very efficient compared to addition.  Then mathematics started getting complicated.  Scientists started expecting fancy tricks from mathematicians to help them communicate what they were learning.  Exponentiation was welcomed eagerly.  Logarithms became popular.  Abstract concepts of number became necessary.  Functions became common knowledge.  Differentiation and integration became part of every scientist's education.  Physicists require even more complex and abstruse mathematics.  We now have things called numbers which bear little resemblance to anything we can count with. 

                In the beginning, numbers were only for counting.

                A long time ago, there were two types of numbers.  There were cardinal numbers and ordinal numbers.  This is made complicated because we use the same number names in both cases.  Cardinal numbers indicate how many, and ordinal numbers indicate which one.  I have ten fingers, and my number ten finger is a thumb.  (Actually, I have not firmly decided that my thumb is a finger, but I feel like it is right now.)

                Later, people started playing with subtraction.  Subtraction can sometimes be considered as counting backward.  It can also sometimes be considered to be addition of negative numbers.  Negative cardinal numbers were thought of as how many to add to a cardinal number to give zero.  Addition with counting numbers is very limited, so everyone has to be careful when doing it, and subtraction can cause big problems.  This is why we invented a number system with both positive, negative, and zero numbers.  We gave the name INTEGERS to the set of natural numbers, their negatives, and zero.  Negative ordinal numbers simply go before positive ordinal numbers.  Zero was a problem, so we say zero is neither positive nor negative.  How one thinks of a negative number is irrelevant as long as one uses them correctly.  However, counting with positive and negative numbers is ridiculous, and counting with zero makes me scratch my head and wonder.

                Eventually rational numbers happened.  We call a rational number a ratio of two integers.  By the time rational numbers were popular, many mathematicians had become philosophers.  Pythagoras was one such.  Some people say Pythagoras was a nut.  Quite frankly, I agree.  Pythagoras was the first fanatical and influential numerologist.  He thought number was everything.  He thought God was in the numbers, and numbers describe everything.  He had one very big problem though; he thought all numbers could be written as ratios of integers.  There is a story that when someone first showed the Pythagoreans the existence of the square root of two, and how the square root of two can not be written as a ratio of integers, that  someone had to take a long swim.  Let this be a lesson to you: If you are going to propose a controversial idea, make sure you are not on a boat!

                Numbers are now divided into algebraic numbers and transcendental numbers.  Algebraic numbers can be written as a sum of rational numbers with rational number exponents.  Transcendental numbers can not.  Pi is the most famous transcendental number.  The square root of two is only an algebraic irrational number, poor thing.  It is written 2^1/2.

                The square root of a negative number presents problems.  Mathematicians deal with these problems by using the number i.  The number i is the square root of negative one.  It is called the imaginary number (hence the letter i).  This is a very fun number.  One must stretch one's imagination to consider this number.  Numbers which are usual numbers, and don't need any mention of i are called real numbers (even though there aren't any unreal numbers - yet).  Numbers written as a sum of a real number added to a (non-zero) real number multiplied times i are called imaginary numbers.  If you call them anything else, they won't come!  (Notice THE imaginary number is AN imaginary with zero as the real part) Imaginary numbers are used for really fun physics, especially electronics.  My mind almost salivates when I think of these!  Yum!  One very interesting question about imaginary numbers is: Which is greater, i or 2i?  2i or not 2i, that is the question.

                A vector is a number which indicates magnitude and a direction.  I have seen people have difficulty with this.  This is surprising.  You know what direction is:  It's That-A-Way.  And magnitude just means size.  There are numerous numbers which have magnitude and something else.  Five feet is a combination of the magnitude five and the unit of length called a foot.  A foot is not a direction so five feet is not a vector.  Displacement is a vector.  Displacement is when you move something.  When you move something, you describe the movement by saying how far you moved it (magnitude) and which direction you moved it. 

                There is a very interesting type of number called a matrix.  A matrix is a collection of one or more rows and one or more columns of numbers.  Notice, the numbers of matrices may themselves be matrices.  Another interesting type of number is an n-dimensional matrix.  Normal matrices have only two or three dimensions.

                It is very easy to think of matrices, vectors, imaginary numbers and such-like if you stop thinking of numbers as things you count with.  Numbers are not just for counting, numbers are for fun!