If you have a wheel a metre across, it will roll out about\u00a03.14159…metres each revolution. This number, which we call \u03c0\u00a0turns out to be some sort of fundamental property of ‘space’.<\/p>\n
The\u00a0Greeks\u00a0were not very happy about the ‘messiness’ of this number. They preferred numbers that could be expressed as fractions – while 22\/7 was close to\u00a0\u03c0, it was not exact and they lost a lot of sleep trying to find a neat way to write\u00a0\u03c0.<\/p>\n Mathematicians have since grudgingly accepted that it cannot be written as a fraction, and indeed it cannot be written down at all because it has no ‘pattern’ and never ends, those digits just keep coming at (almost) random! Here are the first 100…<\/p>\n 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 ….<\/p>\n The wife of pi…<\/p><\/div>\n Understandably, they decided to call this sort of number\u00a0‘irrational<\/a>‘.<\/em><\/p>\n Part 2: Consider now, ‘powers’<\/strong><\/p>\n Mathematicians may work tirelessly on some very pointless looking things, however, they are still fairly lazy when it comes to writing stuff down. They like shorthand. So rather than writing 3+3+3+3+3 they invented ‘multiplication’, giving them 5×3.<\/p>\n Likewise, rather than writing 3x3x3x3x3 they invented ‘powers’, so they could write 35<\/sup>.<\/p>\n Of course they then\u00a0realized\u00a0these tricks could be extended past ‘whole’ numbers. 2.5×3 is 7.5. But what about 32.5<\/sup>?<\/p>\n It works of course,\u00a0the answer<\/a> turns out to be about\u00a015.59 plus change.<\/p>\n But what<\/em> does it mean?\u00a032.5<\/sup>\u00a0is three, times by itself, 2.5 times! or 3x3x30.5<\/sup>. What on earth is that?<\/p>\n Well it turns out, when you ponder this (maybe I should say, if<\/em> you ponder this), that\u00a030.5<\/sup>\u00a0is the same as\u00a0\u221a3. So ‘root three’ is three times itself half a time…<\/p>\n [pregnant pause]<\/p>\n Ok, let’s look at it another way<\/p>\n Consider, for example 3<\/span>2<\/sup>x3<\/span>3<\/sup>, which is the same as (3×3)x(3x3x3) which is the same as 3x3x3x3x3 which is the same as\u00a0\u00a03<\/span>5<\/sup>\u00a0, so\u00a03<\/span>(2+3)<\/sup>.<\/span><\/p>\n So using that logic…<\/p>\n 3 = 31<\/sup>\u00a0=\u00a03(0.5+0.5)<\/sup>\u00a0=\u00a030.5<\/sup>\u00a0x 30.5<\/sup><\/p>\n And what times\u00a0itself\u00a0is equal to 3? Well\u00a0\u221a3! So\u00a030.5<\/sup>\u00a0is\u00a0\u221a3…<\/p>\n It makes sense now, and we can even get used to saying things like\u00a031.9<\/sup>\u00a0x 30.1<\/sup>\u00a0= 9.<\/p>\n Of course, these fractional powers also commonly yield those ‘messy numbers’, so abhorred by the Greeks. \u221a3\u00a0is, roughly:<\/p>\n 1.73205080756887729352744634150587236694280525381038062805580…<\/p>\n The logic follows through for negative numbers. 3-2<\/sup>\u00a0<\/sup>\u00a0is just 1\/32<\/sup>\u00a0which is \u00a01\/9.<\/p>\n Part 3. Now consider ‘e’<\/strong><\/p>\n y=e^x. The slope is always the same as the value! This has the interesting effect that the tangent to the line always intercepts the y axis precisely 1 unit back…<\/p><\/div>\n y=e^x<\/p><\/div>\n Here is a third sort of messy number, one which the Greeks are probably glad they missed. We have Leonhard Euler<\/a> to thank for discovering this one.<\/p>\n He noted there was a number ‘e<\/em>‘ giving an equation of the form y=ex<\/i><\/sup>\u00a0(see the graphs pictured), where the slope of the curve is the same as the height of the curve at each point<\/em>.<\/p>\n Strange and pointless sounding perhaps but pretty simple. So\u00a0y=2x<\/i><\/sup>\u00a0doesn’t work,\u00a0y=3x<\/i><\/sup>\u00a0doesn’t work, but by trial and error you can find\u00a0a value for a that works, which is, roughly:<\/p>\n 2.71828182845904523536028747135266249775724709369995…<\/p>\n It too has no pattern and no repeats so is also\u00a0‘irrational<\/a>‘. <\/em>This number has a whole book written about it<\/a>, for those who are keen.<\/p>\n Part 4. Now consider ‘i’<\/strong><\/p>\n The last piece of the puzzle now.<\/p>\n Consider the equation 3 + x = 0<\/p>\n Now solve for x. Seems pretty easy, but really you are cheating. There is no number that solves that equation<\/span>. Really, to solve it you had to ‘invent’ the concept of a negative number.<\/p>\n Ok. Now consider the equation x2<\/i><\/sup>\u00a0+ 1 = 0<\/p>\n Ah. Trickier! However it turns out that we can do the same trick; this time we simply invent another sort of number – the ‘imaginary’ number. Now if you’ve never heard about these numbers before, you may think I’m joking. Alas I am not. This is what mathematicians have been up to for the last few hundred years, just making stuff up as they go along.<\/p>\n So we define i as\u00a0\u221a-1, or a number, that when multiplied by itself, yields the more respectable -1.<\/p>\n Aside: Just as -1 is a number which, when multiplied by any negative number renders it decent (i.e. positive) once more.<\/em><\/p>\n So i2<\/i><\/sup>\u00a0+ 1 = 0 and the equation is solved. It turns out mathematicians were suddenly able to solve loads of really <\/em>tedious equations using this trick, which made their entire week.<\/p>\n So we have now got i!\u00a0At last we are ready to put the puzzle together.<\/p>\n Simplicity emerges from the complex…<\/strong><\/p>\n So, now I ask, what happens if you raise e to the power of i?\u00a0What does it even mean? It means, e times itself \u221a-1 times. Ouch. Nonsense surely?<\/p>\n Well it works out at roughly 0.540302306 + 0.841470985 i, which is a right mess, something they call, for fairly self-explanatory reasons, a complex number<\/a>.<\/p>\n So now lets stick our old friend from the circle,\u00a0\u03c0, in there\u00a0and see what happens:<\/p>\n What is\u00a0ei\u03c0<\/sup>?<\/p>\n Surely an even bigger mess? I mean its all these messy irrational<\/a>\u00a0numbers combined with this home-made\u00a0imaginary<\/a>\u00a0number…<\/p>\n![\"\"](\"http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/2a\/Pi-unrolled-720.gif\/200px-Pi-unrolled-720.gif\")
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