Dimensional delusions

Here's a (pretty simple) maths problem for you: a rectangle has an area of 15cm² and a perimeter of 16cm. What are the lengths of its sides? The answer, if you can't work it out (or even if you can) is 3cm and 5cm. Here's how you might work it out:

• let the longer sides be m and the shorter sides be n
• mn = 15 (i.e. m = 15/n)
• 2m + 2n = 16 ( i.e. m + n = 8 )
• therefore 15/n + n = 8
• i.e. 15 + n² = 8n
• i.e. n² - 8n + 15 = 0
• Factorising (n - 3)(n - 5) = 0
• Therefore n = 3cm (the smaller solution)
• And m = 5cm

But hang on a second… Look at the sixth line:

n² - 8n + 15 = 0

Just think about what that's saying (bearing in mind that we now know n = 3cm): that's saying that 9cm² (an two-dimensional area) take away 24cm (a one-dimensional length) and add 15 (a no-dimensional integer) equals nothing.

HOW THE HELL CAN YOU TAKE AWAY A LENGTH FROM AN AREA? THEY'RE TWO ENTIRELY DIFFERENT THINGS. IT'S LIKE TAKING AWAY CREAM CAKES FROM THE COLOUR BLUE!

Maths is crazy.