Ever wonder about binary, Boolean algebra, and bitwise operations? All of computing can be broken down to the combination of these three topics, and there are many examples of their uses in the real world. In this article I'll explain the basics of these three topics and how they're related and used in practical applications.



I'll begin by explaining binary representation for those of you who don't understand it yet. As you may already know, binary is a number system of base two. This means that it uses only the numbers zero and one. Each digit represents a power of two. This may be a difficult concept to wrap your head around if you aren't already familiar with it, so it's best to begin with an example.

Take a look at the decimal number system, i.e. a number system based on powers of ten, which you are already familiar with:
1234 is a number in base ten. The 4 on the right represents the 1's digit (i.e. 4*100) the 3 is in the 10's place (i.e. 3*101) the 2 is in the 100's place (i.e. 2*102) and the 1 is in the 1000's place (i.e. 1*103).

Binary works in the same way:
00101101 is an example of a number in binary. The 1 on the far left is 1*20 which is 1. Then we have 0*21 which is 0. Then 1*22 which is 4, and so on. When we've gone all the way through the bits (the term meaning 'binary digit') we end up with 1+4+8+32 which equals 45 in decimal.

Here are some units of measurement in binary which are useful to know:

8 bits=1byte
1024 bytes=1 kilobyte
1024 kilobytes=1 megabytes
1024 megabytes=1 gigabyte
1024 gigabytes=1 terabyte
1024 terabytes=1 petabyte

There's more after that, but you probably won't need to know about those for some time.
Also note that 1024=210

Signed Representation:

Next I'll give a brief explanation of a few different methods of signed binary integer representation that have been used in the past. The term signed refers to the inclusion of both positive and negative values whereas unsigned numbers are only positive.

This is a method of representing signed numbers in binary where the first bit means negative if it is a one and the rest of the bits indicate the magnitude, i.e. the actual numeric value. The downside with this representation is that there are two ways to make zero. For example: 1000 (which is -0) and 0000 (which is 0).

One's Compliment:
The first bit represents the negative sign if is a one. To make a number negative, you flip every bit. So, for example, 01011 (i.e. eleven) when negative is 10100. This method has the same problem as Sign-Magnitude in that there are two ways to represent zero.

Two's Compliment:
This is the method that is used in most modern computers. Here's an example:
Seven=0111 and negative seven=1001.
To convert to from positive to negative, flip the bits and add 1. In this method, the first bit (on the far left) represents -2(n-1) where n is the number of bits and all the rest of the bits represent positive numbers. So in the case of -7 (i.e. 1001), the value is (-8)+0+0+1 which equals -7. So to summarize again in layman's terms, the digit which contains the negative value is on the far left, and all other digits are positive values which are added.

Excess Bias:
In this technique, you first set a bias to make everything in the positive range.
So you would bias by (2(n-1))-1, so, for example, assuming a three bit number, you would have a bias of three.

You then offset all of the numbers by the bias, so:

111 = 4
110 = 3
101 = 2
100 = 1
011 = -0
010 = -1
001 = -2
000 = -3

This might seem like a pain in the ass to use, but we're going to come back to this later when we get into floating-point representation, so don't just brush it off.

Floating-Point Representation:

Next I'll explain floating-point numbers. A floating-point number is basically a number with a decimal point (well, I suppose it would be a 'binary point' in binary). So as you know, the digits in binary go by powers of two for integers, and they also go by powers of two for decimals too.

Say you have this number in binary: 1.111 the first digit on the left is going to be 1 (i.e. 20) and to the right of the dot you have 111. From here we just keep subtracting one from the exponent as we move to the right. So we had already 20, next is 2-1 then 2-2 and so on. So the number 1.111 will be 1+.5+.25+.125 which is 1.875. This however is referred to as Fixed-Point Representation.

We can't just represent decimals like that with a fixed-point since we have a limited number of bits that we can use. Thus the idea of the floating-point number was created. In this, you use something similar to scientific notation to move the decimal point around so that you can represent larger numbers. Hence the term floating-point.

Floating point representation uses a Sign-Magnitude representation for the fractional part (referred to as the Mantissa), and an Excess Bias representation for the exponent part. The numbers are represented in this format:

Sign * Mantissa * 2Exponent

But when written they are in the format of:


So say you have this number in binary: 0 1011 010 you are told that the exponent has four bits and the fraction has three bits.

  • The sign is zero, so the number is positive
  • The exponent is 1011 (the four bits after the sign). Since there are four bits, the bias is 2(4-1)-1 = 7, so the exponent, 1011 equals 11-7 = 4
  • The mantissa is 010 (the last three bits). In floating point representation, if the number is normalized. I'll explain that in a bit (no pun intended). Then you always assume it leads with a one, so the fraction is going to be 1.010 which equals 1.25
  • So your floating point number equals 1.25*24

If the number has an exponent of all zeroes or all ones, the number is considered denormalized otherwise it is normalized.

  • If the exponent is all zeroes, the fraction starts with a leading zero.
  • If the exponent is all ones and fraction is zero, you get infinity (positive infinity if the sign is positive, and negative infinity if the sign is negative). If fraction is not zero you get NaN (i.e. not a number).

Boolean Logic:

Your computer's integrated circuits use these things called gates that to make certain things happen. These gates are AND, OR, NOT, XOR, NAND, and NOR.

The symbols used to represent these can vary, but the ones that I will be using in this article are:
AND = &
OR = |
NOT = !
XOR = ^
NAND = !&
NOR = !|

You can write truth tables to see what the outcomes will be for each of the logic gates:

A   B   A&B
0   0    0
1   0    0
0   1    0
1   1    1
A   B   A|B
0   0    0
1   0    1
0   1    1
1   1    1 
A   B   A^B  
0   0    1
0   1    1
1   0    1
1   1    0

Note that 1 means true and 0 means false.

AND means that the outcome will be true if both inputs are true, otherwise the output is false.

OR means that the outcome will be true if either input is true. If both inputs are false then the output is false.

XOR means 'exclusive or'. It works like an OR gate except it only returns true if exactly one of the inputs is true, and it returns false if both are true or false.

NOT is a unary operator (meaning it takes just one input), and it just returns the opposite of the input.
So !1 = 0 and !0 = 1

NAND is basically just 'not and'. It gives the opposite value that an AND gate would give.

NOR is just 'not or'. It gives the opposite value that an OR gate would give.

Bitwise Operations:

Bitwise operations are very much like Boolean logic operations. Bitwise operators are used in computer programming to perform logical operations on bit strings at the level of their individual bits (rather that on the object as a whole).

The bitwise operators are:
& = AND
| = OR
! = NOT
^ = XOR
~ = One's Compliment (i.e. invert the bits)
>> = Right Shift
<< = Left Shift

You already know how the AND, OR, NOT, and XOR operators work from the Boolean Logic section, but you probably didn't know that you can apply them to full binary strings. For example, if you have 10100101 | 10110111 it would be the same as apply OR to each corresponding digit for the two strings.

10100101 OR
10110011 Equals

The same goes for all the AND and XOR operators.

The NOT operator works slightly different than you may expect. If you have any non zero number, the NOT of it will give zero, and the NOT of zero will give one.

Now, if you remember back to the beginning of this article, I briefly explained what One's Compliment was. So basically, if you use the ~ operator on a number, it will flip all of the bits, so 1011 would become 0100.

Right shift does exactly what it sounds like. It shifts the bits to the right. So, if you have the binary number 0101000, and you use the right shift operator on it like this: 0101000 >> 3, it will shift all of the bits to the right 3 places, so you will get 0000101. This is the same as dividing by 23.

So x >> n is the same as x/2n

Note that the right shift does preserve sign. So if you have a number like 1011 (this number is negative since the sign bit on the far left is one) and you shift it to the right, it will fill in the leading bits with ones. So 1011 >> 2 will be 1110. Also note that since these are integers, there will be a round-off error.

Left shift works the same way as right shift. It shifts the numbers to the left. This always fills in the new bits with zeroes though. Left shift (x << n) is the same as x*2n.

You can create all of the operators used in programming with these and the plus operator.

For example:
To make a number negative, you can do this: (~x + 1)

To see if two numbers are equal, you can do this:
Given int x and int y, the is equal operator is: !(x ^ y)
To see if a number is positive, you can do this: !(x>>31) & !!x; (note an integer is typically 32 bits)

There are many more things that you can do with these, but you are probably thinking "why the hell would I waste my time doing that crap when I can just use an if statement?" Well I was thinking the same thing until I learned that using bitwise operators is actually much faster and more efficient as far as processing time than using for loops and if statements. So if you are in dire need of efficiency, this might be the way to go.

If you're interested in seeing more about using bitwise operations, check this out: Bitwise Operations Examples.