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Wednesday, 16 November 2011

2=1?

Hey! Before I start this post, I better introduce myself, I'm Theo, and I'm a new writer on this blog along with Ned. Just a word of warning, this post does get quite technical later, but I do think you guys will like it, so please persevere, and enjoy!



I start with a problem for you guys...
Let a = b
a2 = ab
a2 +a 2 = a2 + ab
By subtracting 2ab from both sides we get:
2a2 - 2ab = a2 - ab
Then we can factorize to get:
2(a2 - ab) = a2 - ab
Which simplifies to:
2=1

Millenia of maths disproved? Not exactly...
Obviously there is a fallacy here, but can you see where and why?

If you wanna figure it out, don't read ahead yet, if you have done or want to cheat read on!







The problem here is the very last stage, from "2(a2 - ab) = a2 - ab" to "2=1." We have already stated that a2=ab, and therefore a2 - ab=0, meaning that simplifying that last stage is actually simplifying 2(0)=1(0) by dividing both sides by 0, which gets you into a bit of a mess...

And that leads me nicely on to 0 and its properties.

The furthest back we can trace the concept of '0' is in the Babylonian sexagesimal (base 60) number system, but rather than being considered a number in its own right, it was simply used as a place holder, a way of differentiating 101 from 11 for example.

0 was first considered as a number with which arithmetic could be done by Brahmagupta in 628 AD. Brahmagupta established some of the basic rules and definitions of 0...

0 + a positive number = that same positive number
0 + a negative number = that same negative number
x + -x = 0
0/x=0
However, he somewhat ducks the issue on  x÷0; he states simply that it equals x/0, and then contradicts the modern position further by saying that 0÷0=0.

If we treat division as the inverse of multiplication, then by simple substitution we can see that it's not as simple as that, as any number multiplied by 0 will equal it (3x0=0, then it seems logical that 0÷0=3 as well), so 0÷0 could be any number between −∞ and +∞, it is completely indeterminate. This is what is meant when it is said that 0÷0 is undefined: it has an uncountably infinite number of solutions and it could be any one of them.

Somebody called Dr James Anderson has recently said that he has come up with a solution to this problem: define the result as "nullity," or Φ, which "lies off the number line," essentially encompassing any number from negative to positive ∞. However, this doesn't do anything except attach a name to this; it is not a constant, and therefore nullity is essentially shorthand for any other number, not much better than the computer's "NaN" response.

He also works on the assumption that a positive number divided by 0 is ∞, and a negative number divided by 0 is -∞, both of which are untrue and both of which can be disproved.

The proof from this arises from the fact that 0 is neither positive nor negative, and again relies on division being the inverse of multiplication. Put simply, it uses the rule that if xy=z, then z/y=x. It is already clear that if y = 0, then x cannot be less than ∞, and 0 multiplied by a finite number is 0. However, as I am about to show, even if it is, we run into problems.

What we are essentially trying to do is, by repeated multiplication, make y closer to z and eventually equal it. For example, if we take y to be 3 and z to be 24, the by multiplying y by any number greater than 1,(let's use 2 for this example to start with), we make it closer to z (3<3×2<24), and then by multiplying by 2 again we make it closer still (3<3×2<3×2×2<24) until eventually we reach a value that is equal or greater than z, in this case, x=8.

However, if y = 0, then we have a problem. 0 × 2 = 0. No matter how many times we multiply 0 by another number, it will not get any closer to the number we are trying to achieve. Since it will never get closer, even multiplying it by 2 an infinite number of times won't help. Written algebraically, this says that y(n)≠z for non-zero z and n>1. Since n=∞, we can re-write that equation as × ∞ ≠ z, so therefore even if x=∞, it still isn't possible.

Thus, we have proved that division by zero gives neither a finite nor infinite result, that is to say, it has no result at all. And if you do dare divide by zero, then you find that 2=1, and maths as we know it collapses!





Thanks for reading guys, I know that was quite heavy at times but I hope you found it interesting, and Ned and I will post more soon!

Theo




Hey guys, it's Ned again. I hope you enjoyed this post. If you did, please take a moment to visit our last two articles:
The Irregularity of Time <1/2> - The first of two posts about why time isn't a constant as we think.
Is Infinity Possible? - What is there not to love about a never-ending number?

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