Binary, Octal & Hex Conversions

**lor** · 02-09-2010, 02:35 PM

In this tutorial I will be showing you how to translate different types of numbers into... well, other numbers and such. I will also be teaching you some of the discrete mathematics involved with computing.

Basics:

Natural Numbers:
1, 2, 3, 4, so on, these are basically referred to as "counting numbers".

Integers:
-3, -2, -1, 0, 1, 2, so on, negative and positive numbers.

Rational Numbers:
Rational numbers are numbers which can be expressed as x/y where x and y are integers and y is not equal to 0. e.g. 1/4, 1/6, 12/1, etc.

Irrational Numbers:
Numbers which can't be expressed as x/y where x and y are integers and y is not equal to 0. A common example is pi. (The latest I heard about pi is that some University students got the biggest, fastest computer in the world, calculated pi and they recorded about 1 trillion numbers or so, no one has actually worked out if pi has a repetitive system, for all we know pi could start repeating at 1 trillion and 1, we'll have to wait and see!)

Decimal Number System:
Each digit has a place value that depends on its position relative to the decimal point. Decimal number systems have 10 as its base. Example:

333.310

The first 3 (to the left) has a value of 300, the next, value of 30, the next, a value of 3, and the next, a value of 3/10.

Expanded Form:
A random number, 2461.98510 =
2x10³ + 4x10² + 6x10¹ + 1x100 + 9x10-¹ + 8x10-² + 5x10-³
20,000 + 4,000 + 600 + 10 + 9 + 0.8 + 0.05
= 24,619.85

*The reason why we multiply it by 10 is because a natural number has a base of 10.
*Any number raised to the power 0 is equal to 1, e.g. 10° = 1, 1° = 1, 3° = 1, y° = 1, so on and so forth.

Counting in Binary:
Binary has 2 different numbers therefore it has a base of 2. Be careful when writing natural numbers/binary/octal/hex as "101" could easily be mistaken for "one hundred and one" instead of the binary representation, thus why we add a small "2" down the bottom right corner.
Anyway, let's get to the counting.

Decimal: Binary:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
16 10000

If you haven’t noticed, there is an algorithm hidden within binary. If 0 is appended to its binary representation the integer is multiplied by 2. If 1 is appended to its binary representation the integer is multiplied by 2 and 1 added.
Each time a zero is added onto a decimal, the size of the binary number increases 10 times. E.g. 1, 10, 100, 1000, etc. Each time a zero is added onto a binary number, the size of the decimal number doubles.

Converting Binary to Decimal:
To convert binary to a decimal number we use the expanding technique using base 2.

1101.012
= 1x2³ + 1x2² + 0x2¹ + 1x2° + 0x2-¹ + 1x2-²
= 8 + 4 + 0 + 1 + 0 + 0.25
= 13.2510

Converting Decimal to Binary:
We basically divide by 2 and record the remainder. Once we have reached “0” we stop. We then read upwards, giving us our answer.

25
12 1
6 0
3 0
1 1
0 1

2510 = 11001²

If we have a number after the decimal point, multiply by 2 instead of dividing by 2.

0.32
0 64
1 28
0 56
1 12
0 24
0 48
0 96

0.3210 = 0.0101000²

*In order to maintain the level of accuracy, binary numbers need three places for each place in the decimal number, plus 1.

Binary, Octal And Hexadecimal:

Dec Bin Oct Hex
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10

...and so on. I’ll give you another example in Hex as it’s a bit more confusing.

0 10 20... 90 A0 B0 C0... F0 100 110
1 11 21 91 A1 F1 101 111
2 12 22 92 A2 F2 102 112
3 13 23 ... ... F3 103 ...
4 14 24 F4 104
5 15 25 F5 105
6 16 26 F6 106
7 17 27 F7 107
8 18 28 F8 108
9 19 29 F9 109
A 1A 2A FA 10A
B 1B 2B FB 10B
C 1C 2C FC 10C
D 1D 2D FD 10D
E 1E 2E FE 10E
F 1F 2F 9F AF FF 10F 11F

Alright, conversions.

Octal to Binary:
Octal: Binary:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
10 1000

When converting octal to binary we group the binary in groups of 3. For example:

6 0 1 3
110 000 001 011

Binary to Octal:
Octal Binary:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
10 1000

We divide the binary into groups of 3, basically the same as converting octal to binary but the other way round.

1 101 001 011 . 011 100
1 5 1 3 . 3 4

Hexadecimal to Binary:
Hex: Binary:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111

Keep the binary in groups of four when converting.

7 0 A 3
111 0000 1010 0011

Binary to Hex:
Hex: Binary:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111

Same rule applies when converting binary to hex as goes for hex to binary, groups of four.

1 0110 0011 1101 . 0001 1001 11
1 6 3 D . 1 9 C

Octal to Decimal:
Note that octal has 8 as its base.

374.28 = 3x8² + 7x8¹ + 4x8° + 2x8-¹
= 192 + 56 + 4 + 2/8
= 52.2510

Hexadecimal to Decimal:
Note that hex has 16 as its base.

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

E9C.816 = 14x16² + 9x16¹ + 12x16° + 8x16-¹
= 7358 + 144 + 12 + 8/16
= 3740.510

Conversion from Decimal to Octal and Hexadecimal:
Divide by 8/16 and record the remainder from the bottom upwards.

8 1459
182 3
22 6
2 6
0 2

= 26638

16 1459
91 3 3
5 11 B
0 5 5 5

= 5B316

Thanks for reading. If there are any questions be free to ask but know that I'm not a binary expert. If there are any mistakes please let me know.
Author - lor.

**whitey6993** · 02-09-2010, 03:50 PM

01010110011001010111001001111001001000000110001101 10111101101101011100000110110001100101011101000110 01010010000100100000010001110110111101101111011001 00001000000111010001110101011101000110111101110010 01101001011000010110110000101100001000000010101101 1100100110010101110000