1. Number Systems Bangor High School Ali Shareef 3/10/06

2. 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

3. 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.

4. 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 = (?)

5. Base 10 Number System e.g. Decimal A numbering system with 10 base symbols Why is base 10 so easy to use? Base symbols (?)

6. Base 10 Number System e.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

7. Base 10 Number System e.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. 0010 0111 0212 0313 0414 0515 0616 0717 0818 0919

8. Base 8 Number System e.g. Octal Using the same lower 8 symbols of the decimal system. What are the base symbols?

9. Base 8 Number System e.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

10. Base 8 Number System e.g. Octal Applying the general rule. What is the largest number? 0010 0111 0212 0313 0414 0515 0616 0717

11. Base 8 Number System e.g. Octal Applying the general rule. What is the largest 2 digit number? (77) 0010 0111 0212 0313 0414 0515 0616 0717

12. Base 16 Number System e.g. Hexadecimal Using the almost the same symbols of the decimal system. What are the base symbols?

13. Base 16 Number System e.g. Hexadecimal  Base Symbols  What comes next? 0 1 2 3 4 5 6 7 8 9 A B C D E F

14. Base 16 Number System e.g. Hexadecimal Applying the general rule. What is the largest 2 digit number? 0010 0111 0212 0313 0414 0515 0616 0717 0818 0919 0A1A 0B1B 0C1C 0D1D 0E1E 0F1F

15. Base 16 Number System e.g. Hexadecimal Applying the general rule. What is the largest 2 digit number? (FF) 0010 0111 0212 0313 0414 0515 0616 0717 0818 0919 0A1A 0B1B 0C1C 0D1D 0E1E 0F1F

16. Base 2 Number System e.g Binary Number system used by computers Using the same lower 2 symbols of the decimal system. What are the base symbols?

17. Base 2 Number System e.g Binary Using the same lower 2 symbols of the decimal system. What are the base symbols? What comes next? 0 1

18. Base 2 Number System e.g Binary Applying the general rule. What is the largest 2 digit number? 001010011010001010 011110111110011011

19. Base 2 Number System e.g Binary Applying the general rule. What is the largest 2 digit number? (11)  Digits in binary are called bits. 001010011010001010 011110111110011011

20. Base 2 Number System e.g Binary 32 bit processor can process a 32 bit number at a time. Max 32 bit number?

21. Base 2 Number System e.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

22. Converting To Decimal What is 10010b2 in decimal?

23. 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

24. Converting To Decimal What is AAb16 in decimal?

25. Converting To Decimal What is AAb16 in decimal? 10 x (16^1) + 10 x (16^0) = 170b10

26. Decimal to other Bases What is 122b10 in Octal?

27. 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

28. Decimal to other Bases What is 23b10 to binary?

29. 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

30. General Rules Smaller 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.

31. General Rules Larger 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.