NUMERICAL METHODS

NUMERICAL METHODS

AND

AND

STATISTICS

STATISTICS

NUMERICAL REPRESENTATIONS

NUMERICAL REPRESENTATIONS

Joanna Iwaniec

DECIMAL SYSTEM

DECIMAL SYSTEM

The decimal numeral system (also called base ten or occasionally
denary) has ten as its base.

It is the numerical base most widely used by modern civilizations.

DECIMAL NUMBER SYSTEM

DECIMAL NUMBER SYSTEM

In this system 10 symbols are used:

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

Any number can be represented as the following sum:

(a

n-1

...a

1

a

0

)

(10)

= a

n-1

*10

(n-1)

+...+ a

1

*10

1

+ a

0

*10

0

=

where:

i – position in a number,
a

i

- any digit from 0 to 9,

n - number of digits (positions) in the number

Example:

425

425

(10)

(10)

= 4*10

= 4*10

2

2

+ 2*10

+ 2*10

1

1

+ 5*10

+ 5*10

0

0

∑

−

=

∗

1

0

10

n

i

i

i

a

Position of units (0)

Position of tens (1)

Position of hundreds (2)

BINARY SYSTEM

BINARY SYSTEM

Two symbols (digits) are used:

0, 1

Any number can be represented as a following sum:

(a

n-1

...a

1

a

0

)

B

= a

n-1

*2

(n-1)

+...+ a

1

*2

1

+ a

0

*2

0

=

where: i – position in a number,

a

i

- digit 0 or 1,

n - number of digits (positions) in the number

Example:

10100

B

= 1*2

4

+ 0*2

3

+ 1*2

2

+ 0*2

1

+ 0*2

0

∑

−

=

∗

1

0

2

n

i

i

i

a

CONVERSION

CONVERSION

1.

1.

2.

2.

10100

B

= 1*2

4

+ 0*2

3

+ 1*2

2

+ 0*2

1

+ 0*2

0

=

= 1*16 + 0*8 + 1*4 + 0*2 + 0*1 = 20

D

20:2 = 10

20:2 = 10

10:2 = 5

10:2 = 5

5:2 = 2

5:2 = 2

2:2 = 1

2:2 = 1

1:2 = 0

1:2 = 0

residual=0
residual=0
residual=1
residual=0
residual=1

D

ir

e

c

tio

n

o

f r

e

a

d

in

g

D

ir

e

c

tio

n

o

f r

e

a

d

in

g

so 20

so 20

D

D

= 10100

= 10100

B

B

HEXADECIMAL NUMERAL SYSTEM

The hexadecimal numeral system, also known as just hex, is a
numeral system made up of 16 symbols (base 16):

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

Any number can be represented as:

(a

n-1

...a

1

a

0

)

H

= a

n-1

*16

(n-1)

+...+ a

1

*16

1

+ a

0

*16

0

=

where: i – position in a number,

a

i

– hexadecimal symbol,

n - number of digits (positions) in the number

Example

: 1C2

H

= 1*16

2

+ C*16

1

+ 2*16

0

∑

−

=

∗

1

0

16

n

i

i

i

a

CONVERSION

CONVERSION

1.

1.

2.

2.

1C2

H

= 1*16

2

+ C*16

1

+ 2*16

0

=

= 1*256 + 12*16 + 2*1 = 450

D

450:16 = 28

28:16 = 1

1:16 = 0

residual=2
residual=C
residual=1

D

ir

ec

ti

o

n

D

ir

ec

ti

o

n

o

f

re

a

d

in

g

o

f

re

a

d

in

g

so 450

so 450

D

D

= 1C2

= 1C2

H

H

residues are written in

residues are written in

the hexadecimal form

the hexadecimal form

CONVERSION

CONVERSION

Example values of hexadecimal numbers converted into binary, octal
and decimal:

CONVERSION

CONVERSION

1C2

1C2

H

H

=

=

= 0001 1100 0010 =

= 0001 1100 0010 =

= 000111000010 =

= 000111000010 =

= 111000010

= 111000010

B

B

111000010

111000010

B

B

=

=

=

=

000

000

1 1100 0010

1 1100 0010

B

B

=

=

= 1C2

= 1C2

H

H

ka

ka

ż

ż

d

d

ą

ą

cyfr

cyfr

ę

ę

hex

hex

. zapisujemy w

. zapisujemy w

postaci czw

postaci czw

ó

ó

rki cyfr binarnych

rki cyfr binarnych

odrzucamy nieznacz

odrzucamy nieznacz

ą

ą

ce zera na

ce zera na

pocz

pocz

ą

ą

tku liczby binarnej

tku liczby binarnej

1.

1.

2.

2.

liczb

liczb

ę

ę

binarn

binarn

ą

ą

dzielimy od

dzielimy od

ko

ko

ń

ń

ca na czw

ca na czw

ó

ó

rki ewentualnie

rki ewentualnie

dopisuj

dopisuj

ą

ą

c nieznacz

c nieznacz

ą

ą

ce zera w

ce zera w

ostatniej (pierwszej) czw

ostatniej (pierwszej) czw

ó

ó

rce

rce

ka

ka

ż

ż

d

d

ą

ą

czw

czw

ó

ó

rk

rk

ę

ę

binarn

binarn

ą

ą

zapisujemy w postaci cyfry

zapisujemy w postaci cyfry

hex

hex

.

.

Floating-point notation

In computer systems, real numbers

x

are stored as:

x = M * N

W

where:

M – mantissa of number x

,

W – exponent

(in Polish:

‘wykładnik części potęgowej’),

N – base of power

(in Polish:

‘podstawa potęgi’).

Base N

: 2 or 10 (usually 2)

In the floating point notation real number is represented by two

groups of bits:

M (0.5 ≤ M < 1): interpreted as fractional part (‘część

ułamkowa’),

W: interpreted as integer (‘liczba całkowita’)

Floating-point notation

Example 1

: the following notation:

x = (1)1101

(0)10

M

W

in binary system equals:

-0,1101 * 2

+10

= -(1/2 + 1/4 + 0/8 + 1/16) * 2

+(1*2 +0*1) =

= -3,25

Thank you for your attention!

Thank you for your attention!