[split] 0.999... equals 1

[Tiberius]

Every number fits on the number line, which is infinitely long in both directions, with decimal point in the middle.

Each position on the number line can take one of 10 digits (0-9), depending on the value you want to represent. The positions to the left of the decimal point represent the powers of 10 in the positive direction, so every step you take to the left, the value is worth 10 times more than the previous would have been. The positions on the right of the decimal point represent the powers of 10 in the negative direction, so every step you take to the right, the value is worth 10 times less than the previous would have been.

For example:

00001.00000 = 1 (There is a 1 in the first position on the left, so it is worth 1 x 10^0, which is 1. There are no other values in other positions.)

00010.00000 = 10 (There is a 1 in the second position on the left, so it is worth 1 x 10^1, which is 10. There are no other values in other positions.)

00000.10000 = 0.1 (There is a 1 in the first position on the right, so it is worth 1 x 10^-1, which is 0.1. There are no other values in other positions.)

Ok, so let's do a hard one. Combining multiple positions:

00101.00100 = ?

Well, there is a 1 in the third position along to the left, and a 1 in the first position along to the left, and also a 1 in the third position to the right. So:

(1 x 10^2) + (1 x 10^0) + (1 x 10^-3) = 101.001

This is simple base 10 (decimal) arithmetic, but done in it's true form. It's not very good to teach to people just starting out, but once you have to calculate in more bases (like base 2 - otherwise known as binary) it becomes invaluable.

You can see that it works with any number 0 - 9, so:

00823.90000 = (8 x 10^2) + (2 x 10^1) + (3 x 10^0) + (9 x 10^-1) = 823.9

So, the extra zeros are neccessary, since they represent powers of 10 in the decimal system. Another way of writing the above calculation would have been:

(0 x 10^4) + (0 x 10^3) + (8 x 10^2) + (2 x 10^1) + (3 x 10^0) + (9 x 10^-1) + (0 x 10^-2) + (0 x 10^-3) + (0 x 10^-4) + (0 x 10^-5)

However since all of the calculations that have a multiple of 0 in them are equal to 0, there is no point in including them in the calculation.

Some of you are probably wondering about the extra bases, so I'll quickly do an example of binary to decimal conversion. Binary is the base 2 system, which means that for each position on the number line, there can only be 2 values (0-1) rather than 10 (0-9). Not only this, but each step represents a power of 2 (in positive / negative direction).

So the number: 01010.10100 = (1 x 2^3) + (1 x 2^1) + (1 x 2^-1) + (1 x 2^-3) = 10.625 (in decimal).

Easy!