Number Systems: The Binary Number System

Binary

Let's look at base-two, or binary, numbers. How would you write, for instance, 12₁₀ ("twelve, base ten") as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000, and so forth. Similarly in base two, you have columns or "places" for 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, and so forth.

The first column in base-two math is the units column. But only "0" or "1" can go in the units column. When you get to "two", you find that there is no single solitary digit that stands for "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the units column, indicating "1 two and 0 ones". The base-ten "two" (2₁₀) is written in binary as 10₂.

A "three" in base two is actually "1 two and 1 one", so it is written as 11₂. "Four" is actually two-times-two, so we zero out the twos column and the units column, and put a "1" in the fours column; 4₁₀ is written in binary form as 100₂. Here is a listing of the first few numbers:

decimal (base 10)	binary (base 2)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000

Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.

• Convert 101100101₂ to the corresponding base-ten number.

I will list the digits in order, and count them off from the RIGHT, starting with zero:

digits:	1  0   1  1  0  0  1  0  1
numbering:	8  7   6  5  4  3  2  1  0

The first row above (labelled "digits") contains the digits from the binary number; the second row (labelled " numbering") contains the power of 2 (the base) corresponding to each digits. I will use this listing to convert each digit to the power of two that it represents:
1×2⁸ + 0×2⁷ + 1×2⁶ + 1×2⁵ + 0×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰     = 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1     = 256 + 64 + 32 + 4 + 1     = 357   Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved
Then 101100101₂ converts to 357₁₀.
- (Purplemath)