River Wissey Lovell Fuller

Algebra Cadabra - the magic of maths

March 2009

An amusing use of algebra to test your brain power

A book I read recently includes a mathematical proof that can show that King Arthur was a carrot. Since it uses basic algebra, I thought it would be interesting to set it out here. I'm sure the author won't mind if I use it. I have seen the first part in a few other places, so I suspect it is a well known example of a mathematical paradox. You will be relieved that I have included no exercises for you to do!

Before we start, a few things to bear in mind

* In algebra, we can use the * symbol to mean multiplication, and the / sign to mean division

* The left hand side of an equation must always be exactly equal to the right hand side. As long as we do the same thing to both sides, the two sides will remain equal. For example, if we have x = y, we can also say x + 5 = y + 5. We could also say x * 2 rabbits = y * 2 rabbits. And so on

* In mathematics, if we multiply 0 by anything else, we get 0. For example, if we multiply 2 rabbits by 0, we get 0. If we multiply 6 by 0, we get 0. And so on.

* If two things are each equal to a third thing, then the two things are equal to each other. For example, if x = z and y = z, then it follows that x = y

* x2 means the same as x * x, and xy means x * y

* Multiplication and division are reverse processes, and they can cancel each other out

Part 1

Let x and y be equal to 1. This means x = y.

Since x = y, we can say y2 = xy (equation 1)

Since x equals x, we can say that x2 = x2 (equation 2)

From equation 1, we can subtract y2 from one side of equation 2, and xy from the other to maintain equality. This gives us x2 - y2 = x2 - xy (equation 3)

We now do something called factoring. We can say that x2 - xy equals x(x - y). In other words x * (x - y). This is a perfectly valid mathematical technique. We can also say that x2 - y2 = (x + y)(x - y). In other words (x + y) * (x - y). Again this is perfectly correct, mathematically. We can substitute these new expressions into equation 3 to give (x + y)(x - y) = x(x - y) (equation 4)

We will now divide each side of the equation by (x - y). In the left hand side we will have (x + y) * (x - y) / (x - y). The two (x - y) expressions cancel each other out to leave (x + y). Similarly, in the right hand side we will have x * (x - y) / (x - y). The two (x - y) expressions cancel each other out to leave x. Again, these algebra techniques are perfectly valid. This gives us x + y = x (equation 5)

If we subtract x from both sides of equation 5, we get y = 0 (Equation 6)

Since we set x and y equal to 1 at the start, it means that 1 = 0 (equation 7)

This is a profound result, and it allows us to develop the argument about King Arthur.

Part 2

We know that King Arthur had one head. Equation 7 tells us that 1 = 0 so this means that Arthur had no head, because one head equals no head. Similarly, we know that Arthur had no leafy tops, so, using equation 7, we can say that King Arthur had one leafy top.

If we multiply both sides of equation 7 by 2, we get 2 = 0 (equation 8)

King Arthur had two legs, therefore he had no legs. He had two arms, therefore he had no arms. Now multiply equation 7 by Arthur's waist measurement. We get Arthur's waist measurement = 0 (equation 9)

This means that King Arthur tapered to a point.

If we measure the typical light coming from King Arthur, and multiply it by both sides of equation 7, we can see that the wavelength * 1 equals the wavelength * 0. This means that Arthur's colour wavelength = 0 (equation 10)

Similarly, if we multiply the wavelength of orange, which is about 590 nanometers, by equation 7, we get

590 = 0 (equation 11)

Because two things that are equal to a third are equal to each other, it means that Arthur was orange.

We can sum up by saying King Arthur had no arms or legs, and he tapered to a point. Instead of a head he had a leafy top, and he was orange. Therefore he was a carrot.

In conclusion

This is clearly nonsense, but each step of the argument is correct - except for one. Look at the step from equation 4 to equation 5, where we divide both sides of the equation by (x - y). This looks valid, but remember that x and y are both equal to 1, so (x - y) = 0. In other words, the step between equation 4 and 5 attempts to divide by zero, and in mathematics this is not allowed. The ridiculous conclusion that we came to was made possible by allowing this fundamental rule to be broken.

David Smith

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