Math question? How do you convert?

Question

How do you convert numbers from base 10 to base 2?
How do you convert numbers from base 2 to base 10?

Answer 1

BASE 10 TO BASE 2:

The places in a base 2 number are descending powers of 2.
128, 64, 32, 16, 8, 4, 2, 1

Let's say you wanted to represent 181 in binary (base 2).

Look at the powers of two in descending order. You need a 128, so put a 1 under it:
128, 64, 32, 16, 8, 4, 2, 1
.. 1

181 - 128 = 53

You have 53 left... you can't use 64, so put a 0 under it.
128, 64, 32, 16, 8, 4, 2, 1
.. 1....0

But you can use 32. Put a 1 under it.

128, 64, 32, 16, 8, 4, 2, 1
.. 1... 0 .. 1

Again subtract 53 - 32 = 21

You can use the 16, so put a 1 under it.
128, 64, 32, 16, 8, 4, 2, 1
.. 1... 0 .. 1 .. 1

Again subtract 21 - 16 = 5

You can't use the 8, so put a 0:
128, 64, 32, 16, 8, 4, 2, 1
.. 1... 0 .. 1 .. 1..0

You can use the 4, so put a 1 under it:
128, 64, 32, 16, 8, 4, 2, 1
.. 1... 0 .. 1 .. 1..0..1

Subtract 5 - 4 = 1

You can't use the 2, so put a 0 under it. Finally, use the 1 and put a 1 under it. Subtracting leaves 0, so you are done (if it doesn't you made a mistake).

128, 64, 32, 16, 8, 4, 2, 1
.. 1... 0 .. 1 .. 1..0..1..0..1

So 181 is equal to:
10110101 (base 2)

Note: Just like in base 10, you can ignore leading zeroes. So rather than having to write 00000101 to mean 5, you can just say 101 base 2. I always start with the first power of 2 that is less than or equal to my number and go from there...

BASE 2 TO BASE 10:

This is even easier. Just write the powers of 2 over each of the places in the binary number. Here it helps to start from the rightmost digit and go *left*, doubling each time.

So if you had 101011 you would write:

32 16 8 4 2 1
1 .. 0 .1 0 1 1

If the number has a 1 under it, add it. So this would be 32 + 8 + 2 + 1 = 43

101011 (base 2)
= 43 (base 10)