Reading and Number

Reading and Number

When a Thing isn’t Itself

I apologize ahead of time because much of the following is rote information.  They are not discoverable facts about math, but rather conventions and language that give us a way to communicate about math.  That may make it less interesting than a lot of things we will discuss later, but it is very important because without it communication is difficult.

Math is all about numbers and their relationships.  The numbers can represent tangible things (applied mathematics) or nothing at all (pure mathematics).  The beauty in mathematics is that when truths are found abstractly there is often an application where the relationship provides great insight, even though the application may not yet have existed when the relationship was discovered. 

An often sited example exists in the Fibonacci sequence, which is 1, 1, 2, 3, 5, 8, 13, 21 ... and the next number is 34 because that is the sum of the previous two entries.   The sequence was invented as a way to practice arithmetic but there are some fascinating things found in it, like the Golden Ratio, which we’ll leave for another time.  However, the sequence of numbers is often how things are found to grow naturally.  An example is how the seeds of a sunflower are laid out, or pedals on a flower.  Most flowers have a number of pedals that is in the Fibonacci sequence.   The neatest thing about this is that the sequence was developed to help people learn arithmetic, not to articulate how flower pedals are formed or to describe how things look beautiful to us (which involves the Golden Ratio).

Without chasing that rabbit any further, let’s begin with what a number is.  The symbol we use for the number five is just a symbol, like any other.  It has no inherent meaning, it’s just a series of lines and curves carefully joined.  Some examples would be 5,aK5.  These are all just symbols.  Some make sense to us, some do not.   The distinction between the symbol 5 and the thing 5 is important to make because math is a written language.  Sometimes 5 is just five things, like five zombies.  Sometimes a 5 means you will perform some operation 5 times, like x⁵ means you’re multiplying x by itself 5 times.

The point is that symbols represent things.  People often complain that math was fine until the alphabet was introduced.  Those people failed to recognize that the alphabet is just a series of symbols that we attach sounds to, but they’re perfectly valid for other uses, like X marks the spot! 

What I will attempt to explain is how the number line is a line, not a discrete collection of points.  It’s continuous, meaning that we can find a number that exists between any two other numbers, no matter how close. 

Let’s start at the beginning, with the number one.  One is called the identity because if you multiply or divide by it, the value of the other number does not change.   It is also strange because it is neither prime or composite (which we will discuss later). 

If we began to build our number line with 1, we would have just one point.

The first set of numbers are the Natural Numbers which start at 1 and continue infinitely by increments of one.    That is, 1, 2, 3, 4, …  They continue in this fashion infinity.  Please note that infinity is not a number, just a concept of something that continues forever.    

Now we can create a new point for each one of the natural numbers, for as long as we desire, making sure to place them in order as numbers respect size.  But there are gaps between each.

Start at zero, then count the natural numbers and you have the Whole Numbers.  The number zero means absence of value, nothing. 

The next set of numbers are called Integers, and it is with these that many people first encounter trouble with math.  Integers are signed numbers, positive or negative.  If no symbol (+  - ) is assigned, the number is assumed to be positive.  (We are just introducing the types of numbers, not discussing each in this section.  There will be more on integers later.) 

The integers are just a set of numbers made of the whole numbers and their negatives, which we call their opposite numbers.  Note, zero has no opposite as it has no value.  It is neither positive of negative.

So now we have a zero in the middle and a string of positive numbers going to the right and a string of negative numbers going to the left.  But still, there are gaps between consecutive integers.

The next set of numbers are the numbers between the integers, the fractions, repeated or terminating decimals, called the Rational Numbers.  A rational number is any number that can be written as a fraction.  So, 11 is rational, because 11/1, which can be treated as 11 divided by 1, is 11.  The number 3.1 is also rational because it is 3 and one tenth.

If we were to list all of the numbers we have so far between any two consecutive integers, like 1 and 2, writing a dot for each, we could zoom in and find gaps.  We do not yet have a continuous string of numbers.   We can zoom in very close, as close as we wanted really, and find gaps.  For this example we will only look at the second decimal place, between 1.41 and 1.42.  Between these two numbers is the square root of two.  That means if you had a box whose area was two, the side length would be between 1.41 and 1.42. 

The number itself cannot be written because it is a non-terminating, non-repeating decimal.   In fact, the following link displays the first 1,000,000 digits, http://apod.nasa.gov/htmltest/gifcity/sqrt2.1mil.  In case you’re wondering, this is still just an approximation of the number times itself that is 2. 

These numbers are called Irrational Numbers, because they cannot be written as a fraction (rational is from ratio, and irrational means not a ratio). 

There are more irrational numbers than rational numbers and there are irrational numbers that are called transcendental numbers which are not solutions to polynomial equations.  It gets weird.  Some of these even got people killed!  (More on that later.)

Now we have a complete number line.  The irrational numbers fill the gaps between consecutive rational numbers.  The list of numbers we have so far comprise what is called the Real Numbers.  It is an unfortunate name causing many people to believe that the next set of numbers, called Imaginary Numbers, are just that, of fiction. 

The imaginary numbers are numbers that involve the square root of a negative.  Since we have not discussed square roots in detail yet, you may just have to take this one on faith!

In summary:

The Real Numbers are:

• ·         Natural Numbers are counting numbers starting with 1.
• ·         Whole Numbers are the natural numbers and zero.
• ·         Integers are the whole numbers and their opposites.  (Zero has no opposite.)
• ·         Rational Numbers include all of the integers and the fractions between them (which include terminating or repeating decimals).
• ·         Irrational Numbers are non-terminating and non-repeating decimals

There are other numbers that involve the square root of negative one which we will not be addressing now.