Number Systems

Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal p. 33

Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal A B C D E F

Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal Etc.

Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary pp

Quick Example = = 31 8 = Base

Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide…

=>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = Base Weight

Binary to Decimal Hexadecimal DecimalOctal Binary

Binary to Decimal Technique –Multiply each bit by 2 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

Example => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = Bit 0

Octal to Decimal Hexadecimal DecimalOctal Binary

Octal to Decimal Technique –Multiply each bit by 8 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

Example => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 =

Hexadecimal to Decimal Hexadecimal DecimalOctal Binary

Hexadecimal to Decimal Technique –Multiply each bit by 16 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

Example ABC 16 =>C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 =

Decimal to Binary Hexadecimal DecimalOctal Binary

Decimal to Binary Technique –Divide by two, keep track of the remainder –First remainder is bit 0 (LSB, least-significant bit) –Second remainder is bit 1 –Etc.

Example = ? =

Octal to Binary Hexadecimal DecimalOctal Binary

Octal to Binary Technique –Convert each octal digit to a 3-bit equivalent binary representation

Example = ? =

Hexadecimal to Binary Hexadecimal DecimalOctal Binary

Hexadecimal to Binary Technique –Convert each hexadecimal digit to a 4-bit equivalent binary representation

Example 10AF 16 = ? A F AF 16 =

Decimal to Octal Hexadecimal DecimalOctal Binary

Decimal to Octal Technique –Divide by 8 –Keep track of the remainder

Example = ? =

Decimal to Hexadecimal Hexadecimal DecimalOctal Binary

Decimal to Hexadecimal Technique –Divide by 16 –Keep track of the remainder

Example = ? = 4D = D

Binary to Octal Hexadecimal DecimalOctal Binary

Binary to Octal Technique –Group bits in threes, starting on right –Convert to octal digits

Example = ? =

Binary to Hexadecimal Hexadecimal DecimalOctal Binary

Binary to Hexadecimal Technique –Group bits in fours, starting on right –Convert to hexadecimal digits

Example = ? B B = 2BB 16

Octal to Hexadecimal Hexadecimal DecimalOctal Binary

Octal to Hexadecimal Technique –Use binary as an intermediary

Example = ? E = 23E 16

Hexadecimal to Octal Hexadecimal DecimalOctal Binary

Hexadecimal to Octal Technique –Use binary as an intermediary

Example 1F0C 16 = ? 8 1 F 0 C F0C 16 =

Exercise – Convert... Dont use a calculator! DecimalBinaryOctal Hexa- decimal AF Skip answer Answer

Exercise – Convert … DecimalBinaryOctal Hexa- decimal C AF Answer

Binary Addition (1 of 2) Two 1-bit values ABA + B two

Binary Addition (2 of 2) Two n-bit values –Add individual bits –Propagate carries –E.g.,

Multiplication (1 of 2) Binary, two 1-bit values AB A B

Multiplication (2 of 2) Binary, two n-bit values –As with decimal values –E.g., 1110 x

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