#Bits, Bytes, Binary and Hex Math

The bitwise operators and binary math in general have always been a mystery to me. If not for studying for the Zend PHP Certified Engineer exam. I still would probably not care to learn them :) However here I am and I finally got down the mystery of bits and bytes.

What this tutorial will cover:

1. What are bits and bytes and binary math

2. PHP's Bitwise Operators

###BINARY MATH INTRODUCTION

The bitwise operators utilize something called Binary Math. If you already know binary math, skip ahead. Binary math is a BASE 2 numbering system, where as our normal everyday numbering system is a BASE 10 system. Think back to elementary school where we learned numbers this way...

Think of the number 4768. That's a normal everyday decimal number.

If you write that out full its 4 THOUSAND, 7 HUNDRED, 6 TENS, 8 ONES or (4 * 103) + (7 * 102) + (6 * 101) + (8 * 100) = 4768

The BASE 2 system has the same concept however it goes by 2's instead of 10s. So you can only have a value of 0 or 1. Lets look at a binary number 00101101 = (0 * 27) + (0 * 26) + (1 * 25) + (0 * 24) + (1 * 23) + (1 * 22) + (0 * 21) + (1 * 20) = 32 + 8 + 4 + 1 = 45

Wow that's a lot of math. Lets break it down piece by piece starting with the first parentheses.

(0 * 27) which means 0 TIMES 2 to the POWER of 7 or 0 * ( 2 * 2 * 2 * 2 * 2 * 2 * 2) which basically comes out to 0 * 128 which equals 0 (0 times anything is 0)

So going down our binary number we can see it calculates like this:

0 - (0 * 128) = 0
0 - (0 * 64) = 0
1 - (1 * 32) = 32
0 - (0 * 16) = 0
1 - (1 * 8) = 8
1 - (1 * 4) = 4
0 - (0 * 2) = 0
1 - (1 * 1) = 1

So add those all up together 32 + 8 + 4 + 1 and you get 45

What is a bit? A bit is a representation of 1 or 0. Basically OFF and ON. Why do computers use such simple math? Well computers process with electric current so if the current is over a certain level the computer considers that to mean ON or 1 and if it is under that level it is set to OFF or 0.

What is a byte? A byte is made up of 8 bits and the highest value of a byte is 255, which would mean every bit is set. We will look at why a byte's maximum value is 255 in just a second.

What does a byte look like?

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1

That is a representation of 1 Byte. The "Place Value" column represents the math we did doing 2 to the power of 7,6,5,4,3,2,1,0

So if all bits are set and the value = 255 my byte would look like this

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
	1	1	1	1	1	1	1	1	=	255

How do we get 255?

Lets take it right to left and add up all those values together 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255

What would the number 22 look like?

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
	0	0	0	1	0	1	1	0	=	22

2 + 4 + 16 = 22

Lets look at some other examples of decimal to binary, the ones on the end try for yourself.

43 = 00101011
4 = 00000100
230 = ?
65 = ?
31 = ?

###Converting from HEX to DEC

HEX numbers are composed of digits 0 through 9 like DEC but also adds A-F

HEX	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
DEC	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

###Here is a HEX number: 1E5DF

To convert this to a DEC, we need to define the base for our power function. Since HEX is based on 16 different digits [0-9A-F], our base is 16.

To convert from HEX to DEC, follow these steps:

We know that F = 15 in DEC so we use this formula (15 * 16⁰) = 15
We know that D = 13 in DEC so we use this formula (13 * 16¹) = 208
We know that 5 = 5 in DEC so we use this formula (5 * 16²) = 1280
We know that E = 14 in DEC so we use this formula (14 * 16³) = 57344
We know that 1 = 1 in DEC so we use this formula (1 * 16⁴) = 65536
Now we add all of the numbers together to get the DEC number for HEX number 1E5DF:
15 + 208 + 1280 + 57344 + 65536 = 124383

So our answer is HEX 1E5DF = DEC 124383

#BITWISE OPERATORS (PHP)

You can find complete detail on the PHP bitwise operators here: PHP'S BITWISE OPERATORS

What are going to cover? We're going to cover the bitwise operators below and see how they work

Example	Name	Result
$a & $b	And	Bits that are set in both $a and $b are set.
$a | $b	Or	Bits that are set in either $a or $b are set.
$a ^ $b	Xor	Bits that are set in $a or $b but not both are set.
~ $a	Not	Bits that are set in $a are not set, and vice versa.
$a << $b	Shift left	Shift the bits of $a $b steps to the left (each step means "multiply by two")
$a >> $b	Shift right	Shift the bits of $a $b steps to the right (each step means "divide by two")

####Lets use two variables to get started with the "And" operator &. The & or "And" operator takes two values and returns a decimal version of the binary values the left and right variable share. So using our tables above we can see that the only bit these two share is in the 8 position so $a & $b = 8.

$a =	9;
$b =	10;
echo $a & $b;

This would output the number 8. Why?? Well lets see using our table example

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a	0	0	0	0	1	0	0	1	=	9
$b	0	0	0	0	1	0	1	0	=	10

So you can see from the table the only bit they share together is the 8 bit. So 8 gets returned.. not too hard eh? Lets look at another example of the & operator.

$a =	36;
$b =	103;
echo $a & $b;

This would output the number 36. Why?? Well lets see using our table example again

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a	0	0	1	0	0	1	0	0	=	36
$b	0	1	1	0	0	1	1	1	=	103

So you can see the only bits these two share together are the bits 32 and 4 which when added together return 36. This operator is saying "I want to know what bits you both have set in the same column"

####Lets move on to the | also known as the "OR" operator.

$a =	9;
$b =	10;
echo $a | $b;

This would output the number 11. Why?? Well lets see using our table example

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a	0	0	0	0	1	0	0	1	=	9
$b	0	0	0	0	1	0	1	0	=	10

If you notice we have 3 bits set, in the 8, 2, and 1 column.. add those up 8+2+1 and you get 11. It is just saying "I want to know what bits either one of you guys have set".

####Moving on to the ^ operator also known as the "Xor" operator.

$a =	9;
$b =	10;
echo $a ^ $b;

This would output the number 3. Why?? Well lets see using our table example

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a	0	0	0	0	1	0	0	1	=	9
$b	0	0	0	0	1	0	1	0	=	10

The XOR operator wants to know "Tell me what bits you both have set but I don't want any bits you share" Notice we have 3 bits set but both $a and $b share the 8 bit, we dont want that one, we just want the 2 bit and the 1 bit that they each have set but don't share. Soooo 2+1 = 3

####OK, here is one that gets tricky the ~ operator also known as the "NOT" operator.

$a =	9;
$b =	10;
echo $a & ~$b;

This would output the number 1. Why?? Well lets see using our table example

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a	0	0	0	0	1	0	0	1	=	9
$b	0	0	0	0	1	0	1	0	=	10

The NOT operator wants to know what is set in $a but NOT set in $b because we marked $b with the ~operator in front of it. So looking at our table we can see the only bit set in $a thats not in $b is 1.

What happens if we do this...

$a =	9;
$b =	10;
echo ~$a & $b;

We get the value 2, because we want the bits set in $b but NOT set in $a this time, so since they both share the 8 bit, 2 bit is the only one $b has that $a does not.

####BIT SHIFTING TIME!!!

$a =	16;
echo $a << 2;

This would output the number 64. Why?? Well lets see using our table example

1 Byte ( 8 bits )
Place Value	128	64	32	16	8	4	2	1
$a - BEFORE!	0	0	0	1	0	0	0	0	=	16
$a - AFTER!	0	1	0	0	0	0	0	0	=	64

How do we get 64? well bit shifting tells PHP to take the variable $a which in our case is 16 and shift if 2 bits which is basically like saying take 16 and multiply it by 2 twice. So 16 X 2 = 32 x 2 = 64

Special thanks to Jim Plush