Binary Representation

We've looked at how the number of bits used to represent an integer affects how big it can be. Now we turn to the specific way a single-word integer is encoded as bits.

People generally use base ten (decimal) digits to write numbers. Computers use base two (binary).

In base 10, there are ten digits (0-9), and each place is worth ten times as much as the place to its right.

In binary, base 2, there are only two digits (0 and 1), and each place is worth two times the place to its right.

Reading Binary

In base 10 notation, each place value represents a power of ten: the units place (10⁰ = 1), the tens place (10¹ = 10), the hundreds place (10² = 100), the thousands place (10³ = 1000), etc. So, for example:

9827   =   9 × 10³  +  8 × 10²  +  2 × 10¹  +  7 × 10⁰

Base 2 uses the same idea but with powers of two instead of powers of ten. Binary place values represent the units place (2⁰ = 1), the twos place (2¹ = 2), the fours place (2² = 4), the eights place (2³ = 8), the sixteens place (2⁴ = 16), etc. So, for example:

10010₂   =   1 × 2⁴  +  0 × 2³  +  0 × 2²  +  1 × 2¹  +  0 × 2⁰   =   16  +  2   =   18₁₀

To translate from binary (for example, 101101₂) to base 10, first, write the number out on paper. Then write out the binary place values by doubling left from the units place:

101101₂ has only six digits, so we don't need powers of two to the left of that.

Instead of the big blue arrow that nobody understands, let's use five small blue arrows between the cells of the second row.

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

This means that this number is, in base 10, 32 + 8 + 4 + 1 = 45. So, 101101₂ = 45₁₀.

Writing Binary

To translate from base 10 (like 89₁₀) to base 2, first write out the binary place values by doubling left from the units place until you get to a value larger than your number (128 for this example). Then think, "I can take out a 64, so I write a 1 there, and there's 25 left (89 − 64). I have 0 thirty-twos, because I only have 25. But I can take out 16, and there's 9 left. So, 8 and 1 are the last nonzero bits.

Mary, this is a sloppy use of tables when I probably should be using some kind of CSS. Fix later. --MF, 12/8/17 There is a commented out, incorrect sidenote here. -bh

89 25 9 1 0

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

I'm confused about why the arrow is going from right to left. Will that confuse students? Most of the work happens going left to right. Maybe there should be another arrow below the table that goes the other way... --MF, 2/12/18 It may be left over from when we were using the correct algorithm, which does go right to left. -bh

Now, read the number off: 1011001₂ = 89₁₀.

Here's a more precise description of this algorithm to find the base 2 representation of any positive integer:

1. First, find the largest power of two that is less than the number.
2. Then, subtract that power of 2 from the number, keep the new number, and record a 1 in the place for that power of 2.
There is Another Algorithm for Converting to Binary.
(Change from a "comment" to a "sidenote" when page is complete.) --MF, 11/20/17
3. Then, determine if the next largest power of 2 that is less than the new number, and:
   • If it does, subtract that power of 2 from the number, keep the new number, and record a 1 in the place for that power of 2.
   • If it doesn't, keep the same number, and record a 0 for that power of 2.
   Repeat this whole step with the next largest power of 2 until you have a bit (1 or 0) for all the remaining places down to and including the ones place (by which point you should have nothing left of the original number).

The string of ones and zeros you have recorded is the binary representation of your original number.