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 \(\pi i\):
\[e^{\pi i}\]
Monday, 28 March 2011
Monday, 21 March 2011
What is Zero?
If you look at Roman Numerals, i.e. I for 1, II for 2, V for 5, VIII for 8, X for 10 and so on, you should spot two things. One thing is easy to spot, and the other is quite hard.
It is easy to spot that the complexity of Roman Numerals is not obviously connected to their size. For instance 37 is XXXVII, while 52 is LII. It is quite hard to spot that this is because their system does not place varying values on the symbols (I, X, L, V etc) depending on where they fall in the sequence. Instead the value is always the same for each symbol, X always is 10 and L is always 50.
The whole number being written is calculated by applying a sequence of rules, which basically says, read from left to right adding each symbol's value unless the symbol to the right of it is greater than it, in which case you deduct its value from that symbol. So, XI is 10+1=11 and IX is 10-1 which is 9. Taken a little further, 1998 is MCMXCVIII which is M(1000)+CM(1000-100)+XC(100-10)+V(5)+I(1)+I(1)+I(1). What is important to note is that the two [C] symbols BOTH mean 100 despite them falling in different positions.
Our system of numerals is different. We also use different symbols to represent the constituent parts of big numbers. So far, so Roman. What is different is that the symbols have no relation to each other but their value depends on where in the sequence they fall. In one sense we could be said to read our sequence of symbols from right to left. The right most symbol (if we are dealing with whole numbers only) tells us how many one's are in the total number. The next rightmost symbol tells us how many ten's, the next how many hundred's and so on. So if we see 123, we quickly add 3 one's, two ten's and one hundred.
If it seems odd that we read the number from right to left, remember it is done subconsciously. If read from left to right, one can not know what the first symbol in the number represents (be it millions, hundreds of thousands or so on) until you have counted how MANY symbols are in the number. So logically you start at the right, where you KNOW the symbol means one's, and then add up moving from right to left. In reality though, it doesn't really matter which direction you read the symbols in, as long as you know what value each symbol position has in advance. This is unlike Roman Numerals, where the end result would be very different if you started at one end rather than the other - see XI and IX for example. Also, note that if you have the number 144, the two [4] symbols mean different things. The middle one means 4x10 or 40, and the right most one means 4x1 or 4. So the symbols increase in relative value the further left you go because they are multiplied by larger and larger sums. So the complexity of our numbers is directly proportionate to their numerical size. 237 is a bigger sequence of symbols than 84 and so on. The Romans were no thickies, built loads of stuff, and conquered the majority of their world (just don't tell the Chinese), so why were they using this cumbersome system where thirty seven takes longer to write than fifty two, and both are longer than one thousand and one?
The answer is rather profound. If we want to write the number one hundred and nine, then we start with the symbol [9] at the right, followed by the symbol [0] to show there are no tens, followed by the symbol [1] to show there is one hundred: 109. The Romans could not do this because they had no symbol which meant nothing. They had I, V, X, L, C, D, and M, each of which represents a specific number, but they had nothing which represented zero, because they did not consider that to BE a number. For us it would be like asking "What is the wavelength of black light?" (I do not mean the purpley ultraviolet stuff that shows up washing powder residue and dandruff with equal aplomb). Black light does not have a wavelength because there are no waves, it is the absence of light. Or it would be like asking "How does the sound of a violin not playing differ from the sound of a trumpet not playing?". It is a meaningless question. The absence of something does not need to be measured, and we do not need a symbol to represent it.
You may say that they could represent nothing by just not writing anything, which makes a sort of sense, but the problem is then distinguishing between 10 and 100, or 11, 101 and 1001. No, for a system of numbers based on symbols having different values depending on their position you need a clear symbol to represent "none of this value needed". A Roman would be puzzled by this; it would be like making up a shopping list with one amphora of wine, three buckets of milk, two handfuls of berries, no apples, and five bags of nuts. Why do you even bother to mention the apples if you want none of them?
So we are forced to have the symbol [0] because of the simpler way we write out numbers. But just the fact that we got [0] makes us think differently about the concept. Once it is a symbol, is it a number? I do not propose to try to answer that, but I will say that we can sometimes use it like a number. We can add it to other numbers (we get the number we started with), we can multiply other numbers by it (we get zero every time), we can even multiply other numbers by themselves [0] times - meaning raise them to the power 0 - (oddly we get one and not zero every time). We cannot divide numbers by zero. That breaks things.
For our present purposes then, [0] is as fundamental as [1].
It is easy to spot that the complexity of Roman Numerals is not obviously connected to their size. For instance 37 is XXXVII, while 52 is LII. It is quite hard to spot that this is because their system does not place varying values on the symbols (I, X, L, V etc) depending on where they fall in the sequence. Instead the value is always the same for each symbol, X always is 10 and L is always 50.
The whole number being written is calculated by applying a sequence of rules, which basically says, read from left to right adding each symbol's value unless the symbol to the right of it is greater than it, in which case you deduct its value from that symbol. So, XI is 10+1=11 and IX is 10-1 which is 9. Taken a little further, 1998 is MCMXCVIII which is M(1000)+CM(1000-100)+XC(100-10)+V(5)+I(1)+I(1)+I(1). What is important to note is that the two [C] symbols BOTH mean 100 despite them falling in different positions.
Our system of numerals is different. We also use different symbols to represent the constituent parts of big numbers. So far, so Roman. What is different is that the symbols have no relation to each other but their value depends on where in the sequence they fall. In one sense we could be said to read our sequence of symbols from right to left. The right most symbol (if we are dealing with whole numbers only) tells us how many one's are in the total number. The next rightmost symbol tells us how many ten's, the next how many hundred's and so on. So if we see 123, we quickly add 3 one's, two ten's and one hundred.
If it seems odd that we read the number from right to left, remember it is done subconsciously. If read from left to right, one can not know what the first symbol in the number represents (be it millions, hundreds of thousands or so on) until you have counted how MANY symbols are in the number. So logically you start at the right, where you KNOW the symbol means one's, and then add up moving from right to left. In reality though, it doesn't really matter which direction you read the symbols in, as long as you know what value each symbol position has in advance. This is unlike Roman Numerals, where the end result would be very different if you started at one end rather than the other - see XI and IX for example. Also, note that if you have the number 144, the two [4] symbols mean different things. The middle one means 4x10 or 40, and the right most one means 4x1 or 4. So the symbols increase in relative value the further left you go because they are multiplied by larger and larger sums. So the complexity of our numbers is directly proportionate to their numerical size. 237 is a bigger sequence of symbols than 84 and so on. The Romans were no thickies, built loads of stuff, and conquered the majority of their world (just don't tell the Chinese), so why were they using this cumbersome system where thirty seven takes longer to write than fifty two, and both are longer than one thousand and one?
The answer is rather profound. If we want to write the number one hundred and nine, then we start with the symbol [9] at the right, followed by the symbol [0] to show there are no tens, followed by the symbol [1] to show there is one hundred: 109. The Romans could not do this because they had no symbol which meant nothing. They had I, V, X, L, C, D, and M, each of which represents a specific number, but they had nothing which represented zero, because they did not consider that to BE a number. For us it would be like asking "What is the wavelength of black light?" (I do not mean the purpley ultraviolet stuff that shows up washing powder residue and dandruff with equal aplomb). Black light does not have a wavelength because there are no waves, it is the absence of light. Or it would be like asking "How does the sound of a violin not playing differ from the sound of a trumpet not playing?". It is a meaningless question. The absence of something does not need to be measured, and we do not need a symbol to represent it.
You may say that they could represent nothing by just not writing anything, which makes a sort of sense, but the problem is then distinguishing between 10 and 100, or 11, 101 and 1001. No, for a system of numbers based on symbols having different values depending on their position you need a clear symbol to represent "none of this value needed". A Roman would be puzzled by this; it would be like making up a shopping list with one amphora of wine, three buckets of milk, two handfuls of berries, no apples, and five bags of nuts. Why do you even bother to mention the apples if you want none of them?
So we are forced to have the symbol [0] because of the simpler way we write out numbers. But just the fact that we got [0] makes us think differently about the concept. Once it is a symbol, is it a number? I do not propose to try to answer that, but I will say that we can sometimes use it like a number. We can add it to other numbers (we get the number we started with), we can multiply other numbers by it (we get zero every time), we can even multiply other numbers by themselves [0] times - meaning raise them to the power 0 - (oddly we get one and not zero every time). We cannot divide numbers by zero. That breaks things.
For our present purposes then, [0] is as fundamental as [1].
Monday, 14 March 2011
What is One?
Let's start at the very beginning. What is '1'? The obvious answer is that it is the first number. But is it? (See '0' next). Or is it even a number? Y'see every other number you can count on your fingers, toes, abacus etc etc, is made up of lots of '1's. So there is a school of thought that '1' isn't actually a number, but is instead the building blocks of all numbers.
One can now wander off down a long and winding path that is a career in Number Theory. Note that that is a sensible academic discipline which attracts seriously minded professional people dedicated to deep study and contemplation. Numerology is NOT to be confused with Number Theory. THAT is a nutty mystical system (in the loosest possible sense) of superstitions which attracts people with serious mental health issues dedicated to deep confusion and picking lucky lottery numbers.
Let us not wander down that path though. For our purposes here, let us instead contemplate 1 as a unit of something. The concept of a unit of something is a tricky one, and one which was never satisfactorily explained to me in school. For length, for instance, you have to decide what "units" you are working in - which is not quite the same as saying [imperial] or [metric]. No, the units will be miles or kilometres, yards or metres, inches or centimetre and so on. You need to get this bit right so that if you measure length 2 and add a length 3, what you get is a total length of 5. If you add 2 metres to 3 miles you do not get 5 of anything, you get a mess, and your probe fails to enter orbit. As I said though, this is not just imperial versus metric. If you add 2 metres to 3 kilometres, again you do not end up with 5 of anything.
So when you are deciding on which units you are using to measure you are really deciding an absolute reference length. ALL other lengths you are going to be measuring are then multiples (or maybe fractions) of that reference length. That means that you can do normal maths on these lengths - addition or multiplication for instance - and you will get sensible answers WHICH ARE themselves multiples or fractions of the SAME reference length.
I remember being very confused in school trying to answer the question "How many square centimetres are in a cubic centimetre?". I kept imagining pieces of paper cut into squares a centimetre on each side and then stacked one on top of the other, and being confused because I could not work out how many would fit to a cubic centimetre. I now know that I was confused about this because I was dealing with two completely different units. A cubic centimetre is a unit in its own right, and is not some multiple of units of square centimetres. There is no exchange rate. A cubic centimetre is a unit of volume and represents a cube (funny that) with sides which are one centimetre in length. A square centimetre is a unit of area and is a square with sides which are one centimetre in length. No one had properly explained the concept of units to me.
So 1 can be seen as the ultimate unit. Even centimetres can be broken down into smaller divisions. For instance a centimetre is actually one hundredth of a metre, which in turn is actually the distance travelled by light in a vacuum in roughly one three hundred thousandth of a second. But 1 does not need to be broken down any more, nor can it be. It is the absolute basic unit, from which everything else is referenced. So when I say I am adding 83 to 32, I actually MEAN I am adding 83 ones to 32 ones.
At a very basic level, and I LIKE very basic levels, 1 is the basic unit which every other thing is measured relative to. It therefore represents the difference between something and nothing. Which takes us neatly on to...
One can now wander off down a long and winding path that is a career in Number Theory. Note that that is a sensible academic discipline which attracts seriously minded professional people dedicated to deep study and contemplation. Numerology is NOT to be confused with Number Theory. THAT is a nutty mystical system (in the loosest possible sense) of superstitions which attracts people with serious mental health issues dedicated to deep confusion and picking lucky lottery numbers.
Let us not wander down that path though. For our purposes here, let us instead contemplate 1 as a unit of something. The concept of a unit of something is a tricky one, and one which was never satisfactorily explained to me in school. For length, for instance, you have to decide what "units" you are working in - which is not quite the same as saying [imperial] or [metric]. No, the units will be miles or kilometres, yards or metres, inches or centimetre and so on. You need to get this bit right so that if you measure length 2 and add a length 3, what you get is a total length of 5. If you add 2 metres to 3 miles you do not get 5 of anything, you get a mess, and your probe fails to enter orbit. As I said though, this is not just imperial versus metric. If you add 2 metres to 3 kilometres, again you do not end up with 5 of anything.
So when you are deciding on which units you are using to measure you are really deciding an absolute reference length. ALL other lengths you are going to be measuring are then multiples (or maybe fractions) of that reference length. That means that you can do normal maths on these lengths - addition or multiplication for instance - and you will get sensible answers WHICH ARE themselves multiples or fractions of the SAME reference length.
I remember being very confused in school trying to answer the question "How many square centimetres are in a cubic centimetre?". I kept imagining pieces of paper cut into squares a centimetre on each side and then stacked one on top of the other, and being confused because I could not work out how many would fit to a cubic centimetre. I now know that I was confused about this because I was dealing with two completely different units. A cubic centimetre is a unit in its own right, and is not some multiple of units of square centimetres. There is no exchange rate. A cubic centimetre is a unit of volume and represents a cube (funny that) with sides which are one centimetre in length. A square centimetre is a unit of area and is a square with sides which are one centimetre in length. No one had properly explained the concept of units to me.
So 1 can be seen as the ultimate unit. Even centimetres can be broken down into smaller divisions. For instance a centimetre is actually one hundredth of a metre, which in turn is actually the distance travelled by light in a vacuum in roughly one three hundred thousandth of a second. But 1 does not need to be broken down any more, nor can it be. It is the absolute basic unit, from which everything else is referenced. So when I say I am adding 83 to 32, I actually MEAN I am adding 83 ones to 32 ones.
At a very basic level, and I LIKE very basic levels, 1 is the basic unit which every other thing is measured relative to. It therefore represents the difference between something and nothing. Which takes us neatly on to...
Monday, 7 March 2011
Interesting Maths Stuff
Pythagoras theorem is probably most famous amongst non-mathematicians. You will in all probability know it. It goes, take a right angled triangle (a triangle in which one of the internal angles is 90 degrees). Measure the length of the shortest side, and multiply it by itself ("squaring" the number). Measure the next longest side and square it as well. (You will notice that the two shortest sides are the ones that meet at the 90 degree angle. This always happens.) Add the two squared numbers together. Now, you have a much bigger number.
You now need to work out what number you need to multiply by itself to get that big number. This is known as the square root of the number. You can either use the \(\surd\) button on your calculator, or you can work it out gradually by guessing ANY number, dividing the big one by it, and seeing if the result is the same as your guess. If it is then a) you already knew the answer, b) you have experienced a massive coincidence, or c) your subconscious knows more about maths than you do. If, more likely, the result is NOT the same as your guess, guess another number between your original guess and the result of the division. Divide the big number by that. Again if the result is not the same as the new guess then guess again between the result and the new guess. Keep doing this - guess new ones between the last guess and the last result of the division into the large number, and the true root will make itself known.
I have made a spreadsheet in Openoffice which demonstrates this, and you can download it here. Without doing anything apart from guessing and dividing, you can force the square root of the number to come out of the shadows. You pop the number you want to find the root of in the blue box, and your first guess in the orange box. The spreadsheet will then run through the operation described in the previous paragraph 25 times, showing you the results each time. Despite the fact this is completely random, it forces the root out. I have also put a cube root finder in as well, which works in exactly the same way. The only difference is that it divides the first number by the guess twice, rather than once.
If that wasn't cool enough already, now measure the longest side of the triangle. Guess what its length is - exactly the number which we just guessed! That is a round about way of saying that the square of the hypotenuse is the sum of the squares on the other two sides. The classic triangle to refer to is one with the longest side length 5, and the other two sides length 3 and 4. (Don't worry about which units we are using, it doesn't matter). 5 fives are twenty five, which is nine (3 threes) plus sixteen (4 fours). That is Pythagoras theorem, and it is justifiably famous. Now, you can do a lot with this equation. You can invent engineering and build the modern world for one thing, but at its core it just predicts lengths of triangles. That isn't very interesting to the Universe.
A possible contender for most interesting formula, and one of great interest the universe is, of course, \(E=mc^2\). This probably vies with Pythagoras for most well known. It says very simply that [E]nergy and [m]ass are exchangeable with an exchange rate of the speed of light (c) multiplied by itself. This one is a bit more profound than Pythagoras, but at a basic level, it lets our engineers build nuclear power stations.
For what I would call a profoundly interesting equation you want this:
\(e^{\pi i}+1=0\)
If there is a formula which demonstrates the profound connections which underlie the structure of our mathematical universe, it is that. It is called Euler's Identity, and it isn't an identity. I'll try and explain that in due course. To try and establish why I think it is interesting, I am going to break it down into its constituent parts, stare in fascination at the fact they all combine, and then work out why it is that they combine.
You now need to work out what number you need to multiply by itself to get that big number. This is known as the square root of the number. You can either use the \(\surd\) button on your calculator, or you can work it out gradually by guessing ANY number, dividing the big one by it, and seeing if the result is the same as your guess. If it is then a) you already knew the answer, b) you have experienced a massive coincidence, or c) your subconscious knows more about maths than you do. If, more likely, the result is NOT the same as your guess, guess another number between your original guess and the result of the division. Divide the big number by that. Again if the result is not the same as the new guess then guess again between the result and the new guess. Keep doing this - guess new ones between the last guess and the last result of the division into the large number, and the true root will make itself known.
I have made a spreadsheet in Openoffice which demonstrates this, and you can download it here. Without doing anything apart from guessing and dividing, you can force the square root of the number to come out of the shadows. You pop the number you want to find the root of in the blue box, and your first guess in the orange box. The spreadsheet will then run through the operation described in the previous paragraph 25 times, showing you the results each time. Despite the fact this is completely random, it forces the root out. I have also put a cube root finder in as well, which works in exactly the same way. The only difference is that it divides the first number by the guess twice, rather than once.
If that wasn't cool enough already, now measure the longest side of the triangle. Guess what its length is - exactly the number which we just guessed! That is a round about way of saying that the square of the hypotenuse is the sum of the squares on the other two sides. The classic triangle to refer to is one with the longest side length 5, and the other two sides length 3 and 4. (Don't worry about which units we are using, it doesn't matter). 5 fives are twenty five, which is nine (3 threes) plus sixteen (4 fours). That is Pythagoras theorem, and it is justifiably famous. Now, you can do a lot with this equation. You can invent engineering and build the modern world for one thing, but at its core it just predicts lengths of triangles. That isn't very interesting to the Universe.
A possible contender for most interesting formula, and one of great interest the universe is, of course, \(E=mc^2\). This probably vies with Pythagoras for most well known. It says very simply that [E]nergy and [m]ass are exchangeable with an exchange rate of the speed of light (c) multiplied by itself. This one is a bit more profound than Pythagoras, but at a basic level, it lets our engineers build nuclear power stations.
For what I would call a profoundly interesting equation you want this:
\(e^{\pi i}+1=0\)
If there is a formula which demonstrates the profound connections which underlie the structure of our mathematical universe, it is that. It is called Euler's Identity, and it isn't an identity. I'll try and explain that in due course. To try and establish why I think it is interesting, I am going to break it down into its constituent parts, stare in fascination at the fact they all combine, and then work out why it is that they combine.
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