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