Numbers – one dimension
Numbers and the things we can do to them (operations) make up maths. Their simplicity and amazing complexity can be used to think up a universe. Or put another way, the beauty and intricacy of the universe can be described by the basic logic of maths. It can be a large thought experiment (meditation). Only a few physical constants need to be measured to define our universe ( eg speed of light).
Numbers are the way we count things and these can be viewed as a number line. In the western world we think of a line starting on the left and continuing (forever?) to the right in a decimal system ( ie using the digits 1-9).
Discrete numbers are called integers and we count to the right in steps of one integer (unit). We can also think of the line as a continuous entity with the spaces between the integers filled with parts of these integers or fractions. These are called rational number as they can be expresssed as a ratio (or fraction). In between these are irrational numbers (these are not a ratio) and transcendental numbers (also irrational).
Here we have transcendental enlightenment very early in the piece! Maths is so much better than numerology in viewing the world and predicting the future.
Another number (or non number) was added to the line in the western world by a guy by the name of Fibonacci in 1202. This number was zero.
While the concept of zero was around for a long time, Fibonacci included it in our “Arabic decimal notation”. Here we use the digits 1 to 9 and include a zero where no number appears. The Arab and eastern world had been using a zero for quite a while before it was adopted in Europe!
The number line so far is all about adding positive numbers. Add 1 to zero to get 1. Add 1 to 1 to get 2. Or if you like 1+1=2. And so we have an equation where the stuff on the left of the equal sign is equivalent to the stuff on the right. We can also count in twos or threes (or any other number) where 2+2=4, 4+2=6 etc.
We can also imagine doing the reverse and taking numbers away, or subtracting eg 4-2=2. Which quickly leads us into negative space.
Now we have a one dimensional line with positive space to the right of zero and negative space to the left. The concepts of addition and subtraction are called operators – these are things that we can do to numbers.
Numbers – two dimensions
We count or measure distance along a single one dimensional line. We can also imagine working in two dimensional space by adding an orthogonal dimension (basis of Euclidean geometry).
Here we have positive numbers generally designated as up and negative down. Zero is the same in all dimensions ( we can get to any number of dimensions later). People learned quickly that this was a useful way to measure area (of land for example). In maths we often call the horizontal axis x and the vertical axis y
It also led to new things we could do to our numbers (new operations). In the example above I have coloured the first 4 rows and 4 columns in positive space. We can add the 4 rows and 4 columns to get to 16 squares. We can also introduce the new concept (or operators) of multiplication where 4×4=16 and division where 16/4 = 4. While these concepts are also useful in one dimension they are better visualised in two. We can also imagine a new operation where the 4 is squared (or raised to the power of 2). In 3 dimensions we would cube the number or raise it to the power of 3.
The other way of viewing this is that 4 is the square root of 16. Which leads to the order of mathematical operations. We keep seeing these puzzles on facebook where an equation is written without the brackets to show which bits to do first. By definition we always do the higher powered operations first, starting with roots and powers, then multiplication and division and finally addition and subtraction. Equal level operations (eg addition and subtraction) are done from left to right.
It was simple, at least until we got to roots. Each number can be squared by muliplying that number with itself. Similarly we can find the square roots by doing the reverese eg 3 squared is 9 (3×3 =9) and the squre root of 9 is 3. This also works for any number including fractions or irrational numbers. For example 1.5 squared can be expressed as 3/2 x 3/2 = 9/4 or 2.25 and the square root of 2.25 is 1.5.
That is, all except for the negative numbers! What number can we multiply with itself to get -1? Two positives multiplied give a positive and 2 negatives give a positive (think of not not having something).
In 1572 a guy by the name of Rafael Bombelli decided that the imaginary number designated might be a useful concept and boy was he right. Today the number is used in all sorts of mathematics, including most things electrical and signal analysis (eg seismic processing). Imaginary space is visualised as another dimension, but more about that another time.
The thing with appreciating maths is to not keep worrying about the concepts or how to use them. If you are watching cricket you do not worry yourself with understanding reverse swing you initially just appreciate that it works. Once you start to enjoy the game you may become more interested in understanding the details.
Numbers – circles
If we go back to our 2 dimensional number lines and imagine putting a pin in at the zero point (origin) and tying a piece of string to the pin with a pencil at the other end we can enscribe a shape we call a circle. All points on the circle are equidistant from the origin. This distance is called a radius. Two radii (radiuses) in a straight line create the diameter. The perimeter of the circle is called the circumference and is always the same ratio to the diameter. If the diameter is one unit then the circumference is called pi (). Pi is a transcendental number. It can not be expressed as a fraction, although 22/7 is used as an approximation. If you try to calculate pi exactly the decimal digits go on forever without repeating. Computers approximate pi (albeit with lots of digits).
So now we have 2 strange numbers, one imaginary and another transcendental. Both appear in all sorts of weird and wild places in mathematics and physics. Both help create our amazingly complex and crazy universe. As beginners in maths appreciation we do not have to understand why they are or why they create amazing intricacy. Just appreciate that they do!
OK, we have had number lines and circles to get some weird numbers, but compound interest. Isn’t that just something bankers invented to take all our money?
It probably did begin with money lenders but the guy that studied it best, Jacob Bernoulli discovered the concept of natural logarithms in 1683 which introduces our next transcendental number e. It is another magic number like pi but totally unrelated (or so you would think).
Bernoulli suggested if you have 100% interest on a loan of say $1 and paid the interest at the end of the year you would have $1(capital) + $1(interest) = $2 at year end. If you compounded the interest every six months you would receive 50% of the interest each time on the compounded amount and have $1+$0.50 =$1.50 at the half year mark and $1.5+$0.75=$2.25 at year end. You may think that shortening the interest period more and more would keep adding to the amount you make. Bernoulli showed that if you keep shortening the period you eventually reach a limit of 2.71828… Like pi the digits go on forever without repeating (or so we think). This is our next transcendental number e. It is involved in things exponential, including growth and decay curves, plagues and vaccinations.
Again don’t worry about why or how limits exist, just accept that they do. They are very important in mathematics and we will return to them a number of times.
You may have heard of Leonhard Euler (pronounced Oiler) in a number of recent movies about numbers (eg Hidden Figures and the Imitation Game). He keeps popping up because he was a pretty good mathematician. Amongst many mathematical discoveries he came up with “the beautiful equation”.
We normally use the decimal system for counting and adding. This was described in the number line I used earlier. It is a great system for people to think in, but it requires 9 symbols and a non symbol 0 (or the off symbol).
It is easier to program computers using binary with just a 0 and 1. One off symbol and one on symbol. It is the system of mathematical logic. The mathematics is all the same but binary is cutting counting back to the basics. Our basic symbols and operators in maths are therefore 0 and 1, the three basic operators, power, multiply and add, the two transcendental numbers e and π (pi) and our imaginary number i.
Euler’s huge insight was that he put all of these together to come up with
Don’t you just love its simplicity and elegance (don’t worry about what it means or how it was derived as yet). It is amazing how things got so complicated but so simple just from looking at a few numbers. The equation incorporates all the basic operators and numbers in elegant beauty.
The universe figured this out a long time ago and incorporated it in basic quantum physics!! The beautiful equation is called Eulers Identity. It is a specific case of Eulers Equation where x = π. Here we substitute each x with π.
(and because cos π = -1 and sin π = 0) this gives = -1 or if we add 1 to each side we get back to Eulers identity. Again, don’t worry about the details (we will look at these new operators cos and sin a bit later).
Who needs numerology, astrology or any other made up stuff. This is the real thing and way easier to understand. Or I probably should admit that it is way easier to show or prove rather than understand. It also does a significantly better job of viewing the present and predicting the future. It is used in complex analysis, signal processing, quantum mechanics and solving differential equations. This may sound abstract but these are the basics of processing satellite signals, climate modelling, seismic processing, electronic circuits, television broadcasts etc.
It has also been suggested that the equation may be the basis for a complete and consistent theory of causality and fundamental interactions, in other words it may be what makes the universe work!
To be able to use all of this information would require great amounts of detailed study, but we can all appreciate its beauty and elegance. We don’t need to know what chords are being played to appreciate music. We don’t have to be able to catch a ball to appreciate sport. If or when we appreciate something, then we start to get interested in the details. To bat well in cricket you will need to practice your shots with great dedication and time. Same with maths. You will only do this if you love the idea of maths (or cricket) and not because someone made you do it!