Number systems

Understand binary

• Any form of data needs to be converted to binary to be processed by a computer.
• The basic building block in all computers is the binary number system.

Switch

Switch ON is 1 and OFF is 0.

Digit weight

• Every one of us is used to the decimal or denary (base 10) number system. This uses the digits 0 to 9 which are placed in ‘weighted’ columns.

10000	1000	100	10	1
104	103	102	101	100
3	1	4	2	1

• Example: 3x10000 + 1x1000 + 4x100 + 2x10 + 1x1 = 31421

Binary to denary

• Except for denary, we use binary(base 2) and hexadecimal(base 16) number system in the computer.

Denary value	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Binary value	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal value	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

• The binary system uses 1s and 0s only which gives these corresponding weightings.
• We can convert binary number to denary according to digit weightings.

128	64	32	16	8	4	2	1
27	26	25	24	23	22	21	20
1	1	1	0	1	1	1	0

• Example: 1110 1110₂ = 128 + 64 + 32 + 8 + 4 + 2 = 238₁₀

Hexadecimal to denary

• The hexadecimal system is very closely related to the binary system.
• Hexadecimal is a base 16 system.
• Because it is a system based on 16 different digits, the numbers 0 to 9 and the letters A to F are used to represent hexadecimal digits.
• A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15.
• We can convert hexadecimal number to denary according to digit weightings.

65536	4096	256	16	1
164	163	162	161	160
0	1	1	2	3

Example: 01123₁₆ = 1x4096 + 1x256 + 2x16 + 3x1 = 4387₁₀

Use of the hexadecimal system

• Areas within computer science that hexadecimal is used should be identified:

  • Error codes
  • MAC addresses
  • IPv6 addresses
  • HTML colour codes

• Hexadecimal is easier for humans to understand than binary, as it is a shorter representation of the binary.

• When the memory contents are output to a printer or monitor, this is known as a memory dump.

Denary to binary

• Converting from denary to binary is slightly more complex.
• This method involves successive division by 2 until the result is 0; the remainders are then written from bottom to top to give the binary value.

Divider	Result	Remainder	Process
2	107
2	53	1	107 ➗ 2 = 53 ……1
2	26	1	53 ➗ 2 = 26 ……1
2	13	0	26 ➗ 2 = 13 ……0
2	6	1	13 ➗ 2 = 6 ……1
2	3	0	6 ➗ 2 = 3 ……0
2	1	1	3 ➗ 2 = 1 ……1
	0	1	1 ➗ 2 = 0 ……1

• Example: 107₁₀ = 110 1011₂

Denary to hexadecimal

• Converting from denary to hexadecimal is slightly similar with denary to binary.
• This method involves successive division by 16; the remainders are then written from bottom to top to give the hexadecimal value.

Divider	Result	Remainder	Process
16	2004
16	125	4	2004 ➗ 16 = 125 ……4
16	7	13(D)	125 ➗ 16 = 7 ……13(D)
16	0	7	7 ➗ 16 = 0 ……7

• Example: 2004₁₀ = 7D4₁₆

Binary to hexadecimal

• Since 16 = 2⁴, four binary digits are equivalent to each hexadecimal digit.

Binary value	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal value	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Denary value	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

• Example: 1111100001² = 0011 1110 0001² = 3E1¹⁶

Hexadecimal to binary

• Since 16 = 2⁴, one hexadecimal digit are equivalent to four binary digits.

• Example: 45A¹⁶ = 0100 0101 1010²

Binary addition

bianry addtion

• 0 + 0 = 0
• 1 + 0 = 1 (sum 1 and carry 0)
• 1 + 1 = 10 (sum 0 and carry 1 )
• 1 + 1 + (carry 1) = 11= sum 1 and carry 1

• Add 0 0 1 0 0 1 0 1 (37 in denary) and 0 0 1 1 1 0 1 0 (58 in denary).

	-128	64	32	16	8	4	2	1
37	0	0	1	0	0	1	0	1
	+
58	0	0	1	1	1	0	1	0
	=
Carry	0	1	0	0	0	0	0	0
Sum	0	1	0	1	1	1	1	1

• The sum is 0101 1111, which is 95 in denary.

Overflow error

• Add 0 1 0 1 0 0 1 0 (82 in denary) and 0 1 0 0 0 1 0 1 (69 in denary).

	-128	64	32	16	8	4	2	1
82	0	1	0	1	0	0	1	0
	+
69	0	1	0	0	0	1	0	1
	=
Carry	1	0	0	0	0	0	0	0
Sum	1	0	0	1	0	1	1	1

• The sum is 1001 0111, which is -105 (Incorrect)

Overflow error

• The expected answer for 82 + 69 is 151, which is out of range for the 8 bits register (-128~127), this is known as an overflow error.

Logical shifts

• The positive binary integer is multiplied or divided according to the shift performed.
• Bits shifted from the end of the register are lost and zeros are shifted in at the opposite end of the register.
• The most significant bit(s) or least significant bit(s) are lost.