BINARY NUMBERS

COMPUTERS

STRUCTURE(think place values)

27 26 25 24 23 22 21 20

CONVERT(from binary to decimal)

128 64 32 16 8 4 2 1

27 26 25 24 23 22 21 20

0 1 1 1 0 1 0 1

Try a couple: 10101010bin = _________dec

001100112 = _________10

128 64 32 16 8 4 2 1

27 26 25 24 23 22 21 20

CONVERT(from decimal to binary)

128 64 32 16 8 4 2 1

27 26 25 24 23 22 21 20

Try a couple: 5010 = ____________ 2147dec = ____________bin

=72

PRACTICE

10010 = _____________2

110101BIN = _________DEC

1000DEC = _________________BIN

11011010102 = ___________10

OCTAL

• OCTAL IS BASE 8

• REMEMBER, DECIMAL IS BASE 10 (DIGITS 0-9)

• BINARY IS BASE 2 (DIGITS 0,1)

• OCTAL USES 8 DIGITS: 0-7

• USED IN MEMORY REPRESENTATION BECAUSE EACH

OCTAL DIGIT CAN REPRESENT 3 BINARY DIGITS (23 = 8)

CONVERT(octal to decimal)

• 126OCT

REMEMBER, THESE ARE POWERS OF 8

(6 * 80) + (2 * 81) + (1 * 82)

(6 * 1) + (2 * 8) + (1 * 64)

86DEC

• TRY 2 MORE:

321OCT = __________DEC

25368 = _________10

CONVERT(decimal to octal)

SIMILAR PROCESS TO CONVERTING

DECIMAL TO BINARY, DIVIDE BY 8,

CHECKING THE REMAINDERS (THINK

ABOUT THE POSSIBLE DIGITS),

READING THE ANSWER DOWN:

254DEC =

• TRY A COUPLE:

732DEC = ___________OCT

95210 = ___________8

2548

31 R68

3 R78

0 R3

376oct

CONVERT(octal to binary)

• SINCE EACH DIGIT REPRESENTS 3 BINARY, YOU

CAN USE A TABLE TO CONVERT:

• 572OCT

5 → 101 7 → 111 2 → 010

101111010BIN

Oct 0 1 2 3 4 5 6 7

Bin 000 001 010 011 100 101 110 111

Try a couple:1438 = __________2

651oct = __________bin

CONVERT(binary to octal)

• GROUP BINARY NUMBERS INTO SETS OF 3

FROM THE RIGHT, USING THE CHART,

TRANSLATE EACH SET OF 3 INTO IT’S OCTAL

EQUIVALENT

10110110BIN

→ 010 110 110 → 266OCT

Oct 0 1 2 3 4 5 6 7

Bin 000 001 010 011 100 101 110 111

Try a couple:

1111100bin=_______oct

010101112=______8

HEXADECIMAL NUMBERS

• REMEMBER THAT OCTAL WAS HELPFUL SINCE EACH DIGIT CAN REPRESENT 3

BITS.

• WOULDN’T IT BE HELPFUL IF WE COULD REPRESENT MORE THAN 3? MAYBE

4?

• IF 3 BITS NEEDS 8 DIGITS (23), THEN 4 BITS NEEDS ____ DIGITS (24).

• THEN, IT ONLY TAKES 2 OF THESE NEW SETS OF DIGITS (WE WILL CALL

THEM HEXADECIMAL FOR 16 DIGITS) TO CREATE AN ENTIRE BYTE (8 BITS)

• BUT, WE ONLY HAVE 10 STANDARD DIGITS (0-9), WHAT DO WE USE FOR

THE OTHER DIGITS?

• WHY, LETTERS, OF COURSE

• A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

CONVERT(hexadecimal to decimal)

• REMEMBER, NOW WE HAVE A BASE OF 16, SO WE WILL

USE POWERS OF 16

• EXAMPLE: 8B (B = 11)

(11 * 160) + (8 * 161)

(11 * 1) + (8 * 16)

11 + 128

139DEC

160 = 1

161 = 16

162 = 256

163 = 4096

Try a couple:

A5hex =___________dec

2CE16 = __________10

CONVERT(decimal to hexadecimal)

• REMEMBER, WE ARE WORKING BACKWARDS AND OUR

BASE NOW IS 16

• EXAMPLE: 200DEC

20016

12 R816

0 R12

12, 8 = C8

Try a couple:

18410 = ___________16

1452dec = __________ hex

CONVERT(hexadecimal to binary)

• AGAIN, WE NEED OUR TABLE:

• REPLACE EACH HEXADECIMAL NUMBER WITH ITS

BINARY EQUIVALENT:

• A6HEX = 1010 0110 BIN

• TRY A COUPLE:

• D2HEX = ________________BIN

• 5A3B16 = ________________2

Hex Bin

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

CONVERT(binary to hexadecimal)

• AGAIN, WE NEED OUR TABLE:

• REPLACE EACH SET OF 4 BINARY NUMBERS

(STARTING FROM THE RIGHTMOST 4) WITH ITS

HEXADECIMAL EQUIVALENT:

• 1011 0100 BIN = B4HEX

• 0010 1110 BIN = 2EHEX

• TRY A COUPLE:

• 11100001BIN = __________________HEX

• 11101101101000012 =________________16

Hex Bin

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

CONVERT(hexadecimal to octal)

• IT IS EASIEST TO CONVERT TO BINARY FIRST,

THEN TO OCTAL FROM THERE:

• EXAMPLE: B9HEX 1011 1001BIN

010 11 1 001BIN

2 7 1 OCT

• TRY A COUPLE

• D616 = _____________8

• EEHEX = ______________OCT

Hex Bin

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

CONVERT(octal to hexadecimal)

• IT IS EASIEST TO CONVERT TO BINARY FIRST,

THEN TO HEXADECIMAL FROM THERE:

• EXAMPLE: 543OCT

101 100 011BIN

0001 0110 0011BIN

1 6 3 HEX

• TRY A COUPLE

• 243OCT = ______________HEX

• 57348 =_____________16

Hex Bin

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

PRACTICE(hw ns1)