Number Systems

Today

• Decimal

• Hexadecimal

• Binary

– Unsigned Binary

– 1’s Complement Binary

– 2’s Complement Binary

Decimal(base 10)

( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 )

Binary(base 2)

( 0, 1 )

Hexadecimal(base 16)

( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F )

How do we represent numbers?Decimal:

100 = 1101 = 10102 = 100103 = 1000104 = 10000105 = 100000

.

.

.

.

Example:Decimal 1024 =

(1024)10

Binary:

20 = 121 = 222 = 423 = 824 = 1625 = 3226 = 6427 = 12828 = 25629 = 512

210 = 1024 = 1 Kb220 = 1 Mb230 = 1 Gb240 = 1 Tb

Notes: Counting Chart

( 99 )10 + 1 = ??( FF )16 + 1 = ??

( 100 )10

( 100 )16

Conversion

• Decimal (Base 10) --> Binary (Base 2)

Step 1 - Divide the Number by 2

Step 2 - If the result has a remainder, --> add 1 as the current MSB

Otherwise --> add 0 as the current MSB

Step 3 - Finish when result < base --> Add Quotient as the final MSB

Example:

Convert (1000)10 to Binary (base 2)

Conversion

• Decimal (Base 10) -> Hexadecimal (Base 16)

Step 1 - Divide the Number by 16

Step 2 - Take the remainder as the current MSB

Step 3 - Finish when result < base --> Add Quotient as the final

MSB

Example:

Convert (1000)10 to Hexadecimal (base 16)

Bits & Bytes (Side Note)

• Bit

A bit is a single binary digit, a ‘1’ or a ‘0’

• Byte

A series of 8 bits ( 8 bits = 1 Byte )

Examples: ( 1010 1010 )2

( AA )16

Conversion

• Binary (Base 2) --> Hexadecimal (Base 16)

Step 1 - Make groups of 4 bits, starting from the LSB

Step 2 - Directly convert each group into Hexadecimal

Example:

Convert (1111101000)2 to Hexadecimal (base 16)

Binary Addition

Example:

Add (10011011)2 and (1110)2

Signed Binary

• MSB is the sign bit

0 <-- Positive Numbers1 <-- Negative Numbers

2’s Complement Binary

• Example: Convert (-100)10 into 2’s comp

• Example: Binary Addition

2’s Complement Binary

• Why?

– Simplifying the implementation of arithmetic on computer hardware.

– Allows the addition of negative operands without a subtraction circuit or a circuit that detects the sign of a number.

– Moreover, an addition circuit can also perform subtraction by taking the two's complement of a number