Binary, Denary and Hexadecimal

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Binary Data storage capacities (in order)

Bit (one binary digit; 0 or 1)

Nibble (4 binary digits/half a byte)

Byte (has 8 bits)

Megabyte (has 1024 bits)

Kilobyte (has 1024 Mb)

Gigabyte (has 1024 Kb)

Terabyte (has 1024 Tb)

Binary

-only has two states: on or off

-data in a computer is stored as binary digits: 0s and 1s

Central Processing Unit

-controls and executes operations in the computer

-a chip that has millions of switches that can be either 0 (off) or 1 (on)

Denary System

0 1 2 3 4 5 6 7 8 9

-base 10 number system

In a three-digit denary number, the digit on the farthest right has that same amount of units. The next digit to its left has that number multiplied by ten as the amount of units it has. The farthest left digit has that number multiplied by one hundred as the amount of units it has.

For example: 243 = (2 * 100) + (4 * 10) + 3

Convert Binary to Denary

Position of denary numbers:

128 (÷2) → 64 (÷2) → 32 (÷2) → 16 (÷2) → 8 (÷2) → 4 (÷2) → 2 (÷2) → 1

To convert an 8-digit denary number to an 8-bit binary, check if that number can be cut down by 128. If it can, 1 will be the first bit of that binary number, if not then it will be 0. Check with 64, then 32, and so on.

For example:

For denary number 243

243 - 128 = 115 (first bit is 1)

115 - 64 = 51 (second bit is 1)

51- 32 = 19 (third bit is 1)

19 - 16 = 3 (fourth bit is 1)

3 - 8 cannot be done (fifth bit is 0)

3 - 4 cannot be done (sixth bit is 0)

3 - 2 = 1 (seventh bit is 1)

1 - 1 = 0 (eighth bit is 1)

Denary 243 = Binary 11110011

Hexadecimal -a single number or letter used to represent a nibble (four bits)

-base 16 number system

-shorter and easier to remember than binary

-can be easily converted to and from binary

-each digit represents 4 binary digits

Denary to Hex table

Denary: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hexadecimal: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Convert Denary to Hex

Position of denary numbers:

4096 (÷2) → 256 (÷2) → 16 (÷2) → 1

Similar to binary-denary conversion but starting with 1, multiply that by the value of the farthest hex digit or letter. With the value of the next hex digit or letter to the left, multiply it with 16. If necessary, multiply the next with 256, then the next with 4096. Add all the products to get the denary number.

For example:

For hex number AC

Value of C = 12

Value of A = 10

(12 * 1) + (10 * 16) = 12 + 160 = 172

Hex AC = Denary 172

Convert Binary to Hex

In any binary number, every four digits starting from the right represents how many 0s to Fs there are. For every four digits of the binary code, the value of the first number to the right is multiplied by 1. The next value to the left is multiplied by 2. The next is multiplied by 4, then 8.

For example:

For binary number 10110110

First four digits from the right = 0110

Hex value = (0 * 1) + (1 * 2) + (1 * 4) + (0 * 8) = 6

Next four digits to the left = 1011

Hex value = (1 * 1) + (1 * 2) + (0 * 4) + (1 * 8) = 11 = B

Binary 10110110 = Hex B6

Convert Denary to Hex

In a denary number, depending on how large it is, you can divide it by the powers of 16 (1, 16, 256 or 4096). If needed, convert the whole number into hex. If there is a remainder, multiply it with the number you divided the hex number with. If needed, divide it with an appropriate power of 16 then convert it to hex if needed. Depending on the power of 6 a denary number was divided by, place it in the position of that power.

For example:

For denary number 2000

2000 / 256 = 7.81

Denary 7 = Hex 7 (placed in 256ths place)

0.81 * 256 = 208

208 / 16 = 13

Denary 13 = Hex D (placed in 16ths place)

No denary remainder = Hex 0 (placed in 1s place)

Denary 2000 = Hex 7D0