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numbers

BY

SA

01bit

(one of two states)

number system(a scheme for symbolically and verbally

representing quantity)

unary(the tally system)

Aiwt$e#&!)9?

positional notation(represents arbitrarily large quantities

using a finite set of symbols)

decimalbase-10

(positional notation using 10 symbols)

Arabic numerals

0 1 2 3 4 5 6 7 8 9

909

…0000000009

8699987000

78seventy-eightseven eight

octalbase-8


0 1 2 3 4 5 6 7

12345671011121314

123456789101112

octalbase-8

decimalbase-10

151617202122232425262730

131415161718192021222324

octalbase-8

decimalbase-10

317777320000

61 (decimal)061 (octal)

hexadecimalhex

base-16


0 1 2 3 4 5 6 78 9 A B C D E F

123456789ABC

123456789101112

hexbase-16

decimalbase-10

DEF101112131415161718

131415161718192021222324

hexbase-16

decimalbase-10

191A1B1C1D1E1F2021222324

252627282930313233343536

hexbase-16

decimalbase-10

ADDBAD

BEADBEE

BEEF CAFE DEAD DEAF DEED FAD

FEED

0xADD0xBAD

0xBEAD0xBEE

0xBEEF 0xCAFE 0xDEAD 0xDEAF 0xDEED 0xFAD

0xFEED

0xA3BFFFFF0xA3C00000

binarybase-2


11011

100101110111

10001001101010111100

123456789101112

binarybase-2

decimalbase-10

131415161718192021222324

binarybase-2

decimalbase-10

110111101111100001000110010100111010010101101101011111000

101b (binary)101 (decimal)

10101111b10110000b

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base conversions

(30000) + (6000) + (200) + (50) + (9)

(3 * 10000) + (6 * 1000) + (2 * 100) + (5 * 10)

+ (9 * 1)

(3 * 104) + (6 * 103) + (2 * 102) + (5 * 101) + (9 * 100)

36259104 103 102 101 100

10000 1000 100 10 1

octal to decimal

(03000) + (0600) + (070) + (05)

(03 * 01000) + (06 * 0100) + (07 * 010) + (05 * 01)

(03 * 0103) + (06 * 0102) + (07 * 0101) + (05 * 0100)

(3 * 83) + (6 * 82) + (7 * 81) + (5 * 80)

(3 * 512) + (6 * 64) + (7 * 8) + (5 * 1)

(1636) + (384) + (56) + (5)

03675 =

2081

decimal to octal

(3 * 104) + (6 * 103)+ (2 * 102) + (5 * 101)+ (6 * 100)

(03 * 0124) + (06 * 0123)+ (02 * 0122) + (05 * 0121)+ (06 * 0120)

(03 * 023420) + (06 * 01750)

+ (02 * 0144) + (05 * 012)+ (06 * 01)

(072460) + (013560)+ (0310) + (062)+ (06)

36256=

0106640

hex to decimal

0x36E59

(0x3 * 164) + (0x6 * 163) + (0xE * 162) + (0x5 * 161) + (0x9 * 160)

(3 * 65536) + (6 * 4096) + (14 * 256) + (5 * 16) + (9 * 1)

(196608) + (24576) + (3584) + (80) + (9)

0x36E59 =

224857

binary to decimal

10101b

(1 * 24) + (0 * 23) + (1 * 22) + (0 * 21)+ (1 * 20)

(1 * 16) + (0 * 8) + (1 * 4) + (0 * 2)+ (1 * 1)

(16) + (0) + (4) + (0)+ (1)

10101b=21

10101101b

10101101b128 32 8 4 1

(128) + (32) + (8) + (4)+ (1)

10101101b=

173

22 423 824 1625 3226 6427 12828 25629 512210 1024211 2048212 4096213 8192214 16384215 32768216 65536217 131072218 262144219 524288220 1048576

decimal to binary

1) find biggest fitting power of two

2) subtract it out 3) repeat until left with 0

3587232768 (215)

1???????????????b

35872 - 32768=

3104

31042048 (211)

10001???????????b

3104 – 2048=

1056

10561024 (210)

100011??????????b

1056 – 1024=32

3232 (25)

10001100001?????b

32 – 32=0

1000110000100000b

why use hex and octal?

octal to binary(and vice versa)

01234567

000001010011100101110111

octalbase-8

binarybase-2

03673011110111011b

1101010110b

001101010110b01526

hex to binary(and vice versa)

0123456789ABCDEF

0000000100100011010001010110011110001001101010111100110111101111

hexbase-16

binarybase-2

0x7E9011111101001b

1100101101b0x32D

recapEach digit is a coefficient of the number base raised to an increasing power.

One technique works for all number base conversions.

Shortcut conversions:• decimal to binary (sum powers of 2)• binary to decimal (subtract out powers of 2)• octal and binary (1 octal digit = 3 binary digits)• hex and binary (1 hex digit = 4 binary digits)

Effectively, hex and octal serve as a compacter way of writing binary.

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meaning relies upon agreement

integer(whole number)

58780

-87234

signed integer

unsigned integer

0, 1, 2, 3, 4, 5…0b, 1b, 10b, 11b, 100b, 101b…

n bits = 2n values1 bit = 2 values2 bits = 4 values3 bits = 8 values4 bits = 16 values5 bits = 32 values6 bits = 64 values7 bits = 128 values8 bits = 256 values

etc…

unsigned value range1 bit = 0..12 bits = 0..33 bits = 0..74 bits = 0..155 bits = 0..316 bits = 0..637 bits = 0..1278 bits = 0..255

etc…

10000011(negative three)

00000011(positive three)

sign bit



one’s complement



two’s complement

00000101(negative three in excess-8)

00001011(positive three in excess-8)

excess-n

00100111(negative three in excess-42)

00101101(positive three in excess-42)

excess-n

sign bit: -127 to +127one’s complement: -127 to +127two’s complement: -128 to +127

8-bit range

rational numbers2/51/987/1

-61/1738

radix-point notation(ratio written as an integer component and a

fractional component, separated by a radix point)

3/41/8-7/1138/20

0.750.125-7.06.9

36.259

(30) + (6)

+ (0.2) + (0.05)

+ (0.009)

(3 * 101) + (6 * 100) + (2 * 10-1) + (5 * 10-2) + (9 * 10-3)

36.259101 100 10-1 10-2 10-3

10 1 1/10 1/100 1/1000

062.73281 80 8-1 8-2 8-3

8 1 1/8 1/64 1/512

10.111b21 20 2-1 2-2 2-3

2 1 1/2 1/4 1/8

finite rational(rational with a fractional component which can be

expressed with a finite number of digits)

3/4-7/1138/201/3

738/61

0.75-7.06.90.33

12.0983606557…

a ratio with denominator 2a5b is finite in decimal a ratio with denominator 2a is finite in octal a ratio with denominator 2a is finite in hexa ratio with denominator 2a is finite in binary

all ratios which are finite in binary are also finite in decimal some ratios which are finite in decimal are also finite in binary

rational as two integers3/4

Numerator: 00000011 Denominator: 00000100

-7/13 Numerator: 11111001 Denominator: 00001101

fixed-point

57/8111.001b

Integer: 00000111 Fraction: 00100000

(the computing equivalentof radix-point notation)

scientific notationengineering notation 362.354 = 3.62354 * 102

0.00736234 = 7.36234 * 10-3

989777.1 = 9.897771 * 105

floating-point

21/21010.1b

1.0101b * 23

Significand: 10101000 Exponent: 00000011

(the computing equivalent of scientific notation)

6.75110b + 0.?b

0.753/4

3/41/2 + 1/42-1 + 2-2

6.75=

110.11b

110.11b=

1.1011b * 22

1.1011b * 22

Significand: 11011000Exponent: 00000010

IEEE floating-point(an international standard)

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