For our restricted purposes here, a mathematical operation is some way of combining two numbers to get a third number. The third number you get will depend on the type of operation you use. For consideration of our formula, we need to look at addition, multiplication and raising to the power.
Addition is pretty much the easiest thing you can do if you are given two numbers. If you are given three apples (for some reason explanations like this always involve fruit) and then four more apples, then how many apples do you have in total? The process you go through to answer that question is called addition. (The answer is seven apples).
So basically, if you have two or more groups of things, then to work out how many things there are in all the groups combined you add them together. Simples.
We represent the process of adding two numbers by putting the symbol [+] between them.
What is Multiplication?
Multiplication is a special type of addition. If you have two numbers, then if you multiply them together it means that you take the first number and duplicate it so that you end up with the second number of individual instances of the first number. You then add up the results. So 3 multiplied by 4 is 3+3+3+3 - notice there are 4 threes in that list. It doesn't matter which way round you do the operation. So 4+4+4 is the same as 3+3+3+3. That is pretty much all there is to multiplication for our purposes.
The representation of multiplying two numbers is a bit trickier than adding, where the symbol is [+]. We are first taught in school to use [x] to show we are multiplying. So three multiplied by four is 3x4. That's the symbol you will find on your calculator to multiply. This is fine, until we get to algebra. That is a subject for another day, but suffice to say a big part of it is using variables to represent unknown numbers. Now these variables could be represented by a pictogram of a gerbil, or a house, or a dragon or anything you like really. Guess what one of the most common symbols chosen to represent these variables is? Yes, [x]. So if you were multiplying [x] by three you would find yourself writing 3xx. Not ideal. In fact far from ideal. So, there are two other ways to symbolise multiplication. The first is to put a dot between the two numbers (or variables) like this \(3\cdot x\). The second is just to forget about a symbol altogether giving 3x. Obviously the latter option cannot be used for two numbers on their own because we would not be able to tell the difference between the number 34 and three times four.
What is raising to the power?
This is a special type of multiplication, almost in the same way as multiplication is a special type of addition. Again you are given two numbers. This time you gather up a second number of first numbers and then multiply them together rather than adding. So two to the power of two is 2x2 or 2+2. Two to the power of 3 is 2x2x2, or (2+2)+(2+2). The brackets there do not do anything mathematically, they are just there to show the two sets of (2+2) that are being added together. The change from power of two in the last example to power of three in this example just means multiple everything by two again. So given that two to the power of two is (2+2), two to the power of three just means double that, or (2+2)+(2+2). So you have two groups of two groups of twos.
In the same way three to the power of four is 3x3x3x3, or ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3)). I have used double brackets to show that there are three groups of three groups of threes. If that looks a bit complicated, consider this:
three to the power of one = 3 (you only have one number so you cannot do anything with it, it just sits there)
three to the power of two = three copies of the last answer = 3+3+3
three to the power of three = three copies of the last answer = (3+3+3)+(3+3+3)+(3+3+3)
three to the power of four = three copies of the last answer = ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))
An important point to note here is that we are just adding up to get the answer because 'powering' a number is a type of multiplication, which in turn is just a type of addition.
One difference between powering and multiplication is that it DOES matter which number does what. So while 2x3 (3+3) = 3x2 (2+2+2), two to the power of three, 2x2x2=(2+2+2+2)=8, DOES NOT equal three to the power two, 3x3=(3+3+3)=9.
Using a computer keyboard the symbol for powering numbers is [^], so 2^3 is two to the power of three. Normally though we do not use that symbol, or indeed any symbol. Instead we just write the number to be powered first, then the value of the power second. To distinguish this from multiplying (where we also usually do not use a symbol) we write the second number in superscript (which means a bit smaller and higher up). So two to the power of three is 2^3 or \(2^3\). We also call the small number, the power to which we are raising, the exponent. From this we get phrases like "exponential increase".
If you raise the number to the power of two, that is known as 'squaring' the number. This is because to find the area of a square you multiply the length of the sides by itself. If you raise a number to the power of three that is known as "cubing" the number because, similarly, you would need to multiply together three lengths of the side of the cube to get the volume of it. We really stop using special numbers after squares and cubes because our brains do not work in the fourth dimension.
We need to say a quick word about something that is not immediately obvious. One plus zero, is one. If you have something and don't add anything you have the thing you started with. Two multiplied by zero is zero. If you want zero individual instances of two, then when you come to add them up you will get nothing. So what is three to the power of zero? Is it three, or is it zero? Well, it isn't three, because as we have seen that is three to the power of one. Can it be zero? Hmmm.
Lets look at three to the power of three. That is three multiplied together three times or three multiplied by three multiplied by three. What is three to the power of one? Three. And what is three to the power of two? Three multiplied by three. So what is (three to the power of one) MULTIPLIED by (three to the power of two)? Well that is (three) MULTIPLIED by (three multiplied by three). Or just three multiplied by three multiplied by three. So we can say that:
\[3^3=3^2\cdot3^1=3^{2+1}\]
This is going to work for whatever numbers we choose, because raising to the power is just telling us how many of the chosen numbers we are multiplying together. We still end up just doing a multiplication. We have seen that it doesn't matter which order we do our multiplication in - and certainly not if it is just the same number being multiplied together again and again. So if we have a much bigger number like five to the power of six, that is just six fives in a row getting multiplied together. You could also look at it as a group of four fives multiplied together and then multiplied by a group of two fives multiplied together. It really doesn't matter where you draw the lines.
\[5\cdot5\cdot5\cdot5\cdot5\cdot5=(5\cdot5\cdot5)(5\cdot5\cdot5)=(5\cdot5)(5\cdot5\cdot5\cdot5)\]
Good. Makes sense. So what is three to the power of zero? Well if we want our rule to hold, that you just add the exponents, then the answer has to be something that satisfies this formula:
\[3^3=3^{3+0}=3^3\cdot3^0\]
If we make three to the power zero equal to zero then that doesn't work. The bit at the end will break because you are multiplying something BY zero, which makes the whole thing zero. So what we want three to the power of zero to be is a number that DOES NOT change other numbers when it multiplies them. What number is that? One. One times anything is just the anything you started with. So:
\[3^3=3^{3+0}=3^3\cdot3^0=3^3\cdot1=3^3\]
We can safely conclude then that anything to the power of zero is the number one. Well, OK, maybe it wasn't that QUICK a word.
So, in our formula, we have addition when we add one:
\[+1\]
We have multiplication where we multiply \(\pi\) by \(i\) (we will come on to what these symbols mean later):
\[\pi i\]
And we then raise \(e\) to the power of \(\pi i\):
\[e^{\pi i}\]
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