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Midterm schedule

EL gdwal ( second semester )

V.I

num system

Numbering systems

Decimal: {0, 1, 2, 3, 4, 5…}

Binary: {0, 1}

Octal: {0, 1, 2, 3, 4, 5, 6, 7}

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

Conversion methods

• From decimal to binary (divide the decimal number by 2 then take the remainder –up right-to be the binary representation of the original number)

Example: (13)₁₀ à(----)₂

Original #	division	Remainder
13	2	1
6	2	0
3	2	1
1	2	1
0

So the result will be (1101) ₂

• From binary to decimal: multiply each digit with 2 to the power of (0 to 7) in its order from right to left then sum the final result

Example: (1101)₂à (….)₁₀

Original #	1	1	0	1
Multiplication	1*23	1* 22	0*21	1* 20
Result for each digit	8	4	0	1
addition	13

So the result will be (13)₁₀

Special case: fraction:

Example: (1101.10)₂à (….)₁₀

Original #	1	1	0	1	.	1	0
multiplication	1*23	1* 22	0*21	1* 20	.	1*2-1	0*2-2
Result for each digit	8	4	0	1		.5		0
addition	13				.	.50

So the result will be (13.5)₁₀

• From binary to octal: each 3 binary digits represents 1 octal digit

Octal= 2³ binary

Example: (1011101) ₂à (…) ₈

Original #	1011101
Divide to 3 digits	001	011	101
Convert to decimal	1	3	5
Concatenate (result)	135

The result will be (135) ₈

• From octal to binary: convert each octal digit to its binary representation then concatenate the result

Example: (357) ₈ à (…….)₂

Original #	357
For each octal digit	3	5	7
Convert to binary	011	101	111
Concatenate (result)	011101111

The result will be (011101111)₂

• From binary to Hexadecimal: each 4 binary digits represents 1 hexadecimal digit

Hexadecimal= 2⁴ binary

Example: (1011101) ₂à (…) ₁₆

Original #	1011101
Divide to groups of 4 digits	0101	1101
Convert to decimal	5	13
Convert to hexadecimal	5	D
Concatenate (result)	5D

The result will be (5D) ₁₆

• From Hexadecimal to binary: convert each hexadecimal digit to its binary representation then concatenate the result

Example: (D7B) ₁₆ à (…….)₂

Original #	D7B
Convert each hexadecimal digit to binary	1101	0111	1011
Concatenate (result)	110101111011

The result will be (110101111011)₂

• From any numbering system to decimal: multiply each digit by the target system base to the power of (0 to 7) in its order from right to left then sum the final result

Example: (247)₈à (….)₁₀

Original #	2	4	7
The digit multiplied by the Base to the power from 0-7 from R to L	2* 82	4*81	7* 80
Result for each digit	128	32	7
addition	167

So the result will be (167)₁₀

• From decimal to any numbering system: (divide the decimal number by the base of the target system then take the remainder –up right-to be the new representation of the original number)

Example: (4037)₁₀ à(----)₁₆

Original #	division	Remainder
4037	16	5
252	16	12àC
15	16	15àF
0

So the result will be (FC5) ₁₆