number systems bangor high school ali shareef 3/10/06

Number Systems

Bangor High School

Ali Shareef3/10/06

Number Systems in History

Number system is very important The engine of mathematics Symbolic mathematics difficult to develop

without an understanding of the relationships in numerical mathematics

Number system in use today are known as the Arabic numerals Originated in India and spread west thru the

middle east and into Europe

Number Systems in History

Babylonians had a base 60 numbering system.

Other civilizations such as the Greeks assigned numerical values to their alphabets and used them as numerals.

These methods proved to be cumbersome and inefficient.

Number Systems in History Roman Numerals

Numerals I (1), V (5), X (10), L (50), C (100), D (500), M (1000) Form numbers out of combination of these

numerals No symbol for zero

IV (4), VIII (8), XXXI (?) XL (?) CCCLXIX (?) CDXLVIII + DLII = (?)

Base 10 Number Systeme.g. Decimal

A numbering system with 10 base symbols

Why is base 10 so easy to use? Base symbols (?)

Base 10 Number Systeme.g. Decimal

Why is base 10 so easy to use? Base symbols (?)

What comes next?0

1

2

3

4

5

6

7

8

9

Base 10 Number Systeme.g. Decimal

General rule: When all the base symbols have been used up, increment the digit/digits to the right and repeat the base symbols again.

00 10

01 11

02 12

03 13

04 14

05 15

06 16

07 17

08 18

09 19

Base 8 Number Systeme.g. Octal

Using the same lower 8 symbols of the decimal system.

What are the base symbols?

Base 8 Number Systeme.g. Octal

Using the same lower 8 symbols of the decimal system.

What are the base symbols? What comes next?

0

1

2

3

4

5

6

7

Base 8 Number Systeme.g. Octal

Applying the general rule.

What is the largest number?

00 10

01 11

02 12

03 13

04 14

05 15

06 16

07 17

Base 8 Number Systeme.g. Octal

Applying the general rule.

What is the largest 2 digit number? (77)

00 10

01 11

02 12

03 13

04 14

05 15

06 16

07 17

Base 16 Number Systeme.g. Hexadecimal

Using the almost the same symbols of the decimal system.

What are the base symbols?

Base 16 Number Systeme.g. Hexadecimal

Base Symbols What comes next?

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

Base 16 Number Systeme.g. Hexadecimal

Applying the general rule.

What is the largest 2 digit number?

00 10

01 11

02 12

03 13

04 14

05 15

06 16

07 17

08 18

09 19

0A 1A

0B 1B

0C 1C

0D 1D

0E 1E

0F 1F

Base 16 Number Systeme.g. Hexadecimal

Applying the general rule.

What is the largest 2 digit number?(FF)

00 10

01 11

02 12

03 13

04 14

05 15

06 16

07 17

08 18

09 19

0A 1A

0B 1B

0C 1C

0D 1D

0E 1E

0F 1F

Base 2 Number Systeme.g Binary

Number system used by computers

Using the same lower 2 symbols of the decimal system.

What are the base symbols?

Base 2 Number Systeme.g Binary

Using the same lower 2 symbols of the decimal system.

What are the base symbols? What comes next?

0

1

Base 2 Number Systeme.g Binary

Applying the general rule.

What is the largest 2 digit number?

00 10 100 110 1000

1010

01 11 101 111 1001

1011

Base 2 Number Systeme.g Binary

Applying the general rule.

What is the largest 2 digit number?(11)

Digits in binary are called bits.

00 10 100 110 1000

1010

01 11 101 111 1001

1011

Base 2 Number Systeme.g Binary

32 bit processor can process a 32 bit number at a time.

Max 32 bit number?

Base 2 Number Systeme.g Binary

32 bit processor can process a 32 bit number at a time.

Max 32 bit number? 1111 1111 1111 1111 1111 1111 1111 1111 4294967295 in decimal

Converting To Decimal

What is 10010b2 in decimal?

Converting To Decimal

What is 10010b2 in decimal? 1 x (2^4)+0 x (2^3) + 0 x (2^2) + 1 x (2^1) + 0 x (2^0)

= 18b10

Converting To Decimal

What is AAb16 in decimal?

Converting To Decimal

What is AAb16 in decimal? 10 x (16^1) + 10 x (16^0)

= 170b10

Decimal to other Bases

What is 122b10 in Octal?

Decimal to other Bases

What is 122b10 in Octal? 122 ÷ 8 = 15 Rem 2 15 ÷ 8 = 1 Rem 7 1 ÷ 8 = 0 Rem 1

= 172b8

Decimal to other Bases

What is 23b10 to binary?

Decimal to other Bases

What is 23b10 to binary? 23 ÷ 2 = 11 rem 1 11 ÷ 2 = 5 rem 1 5 ÷ 2 = 2 rem 1 2 ÷ 2 = 1 rem 0 1 ÷ 2 = 0 rem 1= 10111b2

General RulesSmaller base to Larger

base

When moving from a smaller base to a larger base: (dn+1 * bn) + (dn * bn-1) ….+ (d1 *

b0) The expansion (multiplication and

power operations) must utilize the interpretation of the base that you are moving to.

General RulesLarger base to Smaller

base When moving from a smaller base

to a larger base: N/b = Q0 + R0 Q0/b = Q1 + R1 ……. Qn/b = 0 + Rn => Where (Rn*10^n-1) + (Rn-1*10^n-2) +

…+ + (R0*10^0) Expansions and reductions must utilize the

interpretation of the base you are leaving.