1. 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

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

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... Don’t use a calculator! DecimalBinaryOctal Hexa- decimal AF Skip answer Answer

Exercise – Convert … DecimalBinaryOctal Hexa- decimal C AF Answer

Common Powers (1 of 2) Base 10 PowerPrefaceSymbol picop nanon micro  millim 10 3 kilok 10 6 megaM 10 9 gigaG teraT Value

Common Powers (2 of 2) Base 2 PowerPrefaceSymbol 2 10 kilok 2 20 megaM 2 30 GigaG Value What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

Example / 2 30 = In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties

Exercise – Free Space Determine the “free space” on all drives on a machine in the lab Drive Free space BytesGB A: C: D: E: etc.

Review – multiplying powers For common bases, add powers 2 6  2 10 = 2 16 = 65,536 or… 2 6  2 10 = 64  2 10 = 64k a b  a c = a b+c

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 3) Decimal (just for fun) 35 x

Multiplication (2 of 3) Binary, two 1-bit values AB A  B

Multiplication (3 of 3) Binary, two n-bit values –As with decimal values –E.g., 1110 x

Fractions Decimal to decimal (just for fun) 3.14 =>4 x = x = x 10 0 =

Fractions Binary to decimal => 1 x 2 -4 = x 2 -3 = x 2 -2 = x 2 -1 = x 2 0 = x 2 1 =

Fractions Decimal to binary x x x x x x etc

Homework – Convert... Please provide with your full answers steps DecimalBinaryOctal Hexa- decimal C.82

Homework Answer– Convert … DecimalBinaryOctal Hexa- decimal …35.63…1D.CC… D C C.82 Answer