Binary system…

Binary Binary Number System Binary to Decimal Decimal to Binary Octal and Hexadecimal Binary to Hexadecimal Hexadecimal to Binary Hexadecimal to Decimal Any Number Base to Decimal Decimal to Any Number Base Binary Coded Decimal (BCD)

Topic to be discuss

A computer is a “bistable” device A bistable device:

◦Easy to design and build◦Has 2 states: 0 and 1

One Binary digit (bit) represents 2 possible states (0, 1)

Binary

With 2 bits, 4 states are possible (22 = 4)

• With n bits, 2n states are possible• With 3 bits, 8 states are possible (23 = 8)

Bit1 Bit0 State

0 0 1

0 1 2

1 0 3

1 1 4

Bit2 Bit1 Bit0 State

0 0 01

0 0 12

0 1 03

0 1 14

1 0 05

1 0 16

1 1 07

1 1 18

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From left to right, the position of the digit indicates its magnitude (in decreasing order)◦ E.g. in decimal, 123 is less than 321◦ In binary, 011 is less than 100

A subscript indicates the number’s base◦ E.g. is 100 decimal or binary? We don’t know!◦ But 1410 = 11102 is clear

Binary Number System

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Binary is base 2 Example: convert 10110 (binary) to decimal

101102 = 1x24 + 0x23 + 1x22 + 1x21 + 0x20

= 1x16 + 0x8 + 1x4 + 1x2 + 0x1

= 16 + 0 + 4 + 2 + 0

= 22 So 101102 = 2210

Binary to Decimal

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Binary is base 2 Example: convert 35 (decimal) to binary

Quotient Remainder

35 / 2 = 17 117 / 2 = 8 18 / 2 = 4 04 / 2 = 2 02 / 2 = 1 01 / 2 = 0 1

So 3510 = 1000112

Decimal to Binary

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It is difficult for a human to work with long strings of 0’s and 1’s

Octal and Hexadecimal are ways to group bits together

Octal: base 8 Hexadecimal: base 16

Octal and Hexadecimal

With 4 bits, there are 16 possibilities

Use 0, 1, 2, 3, …9 for the first 10 symbols

Use a, b, c, d, e, and f for the last 6

HexadecimalBit3 Bit2 Bit1 Bit0 Symbol0 0 0 0 00 0 0 1 10 0 1 0 20 0 1 1 30 1 0 0 40 1 0 1 50 1 1 0 60 1 1 1 71 0 0 0 81 0 0 1 91 0 1 0 a1 0 1 1 b1 1 0 0 c1 1 0 1 d1 1 1 0 e1 1 1 1 fGo back

01010110101100112 = ? in hex Group into 4 bits, from the right: 0101, 0110, 1011, 00112 Now translate each (see previous table):

01012 => 5, 01102 => 6, 10112 => b, 00112 => 3So this is 56b316

What if there are not enough bits? ◦ Pad with 0’s on the left

Binary to Hexadecimal

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Hexadecimal to Binary• f0e516 = ? in binary

• Translate each into a group of 4 bits:

f16 => 11112, 016 => 00002, e16 => 11102, 516 => 01012

So this is 11110000111001012

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Hexadecimal is base 16 Example: convert 16 (hex) to decimal

1616 = 1x161 + 6x160

= 1x16 + 6x1= 16 + 6= 22

So 1616 = 2210

Not surprising, since 1616 = 0001, 01102◦ If one of the hex digits had been > 9, say c, then we

would have used 12 in its place.

Hexadecimal to Decimal

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From right to left, multiply the digit of the number-to-convert by its baseposition

Sum all results

Any Number Base to Decimal

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Take the decimal number, and divide by the new number base

Keep track of the quotient and remainder Repeat until quotient = 0 Read number from the bottom to the top

Decimal to Any Number Base

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Why not use 4 bits to represent decimal? Let 0000 represent 0 Let 0001 represent 1 Let 0010 represent 2 Let 0011 represent 3, etc.

◦ This is called BCD◦ Only uses 10 of the 16 possibilities

Binary Coded Decimal (BCD)

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Thanks…..

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