1. Number SystemsNumber Systems Modified By: AM. Sihan (Hardware Engineering)

2. TYPE OF NUMBER SYSTEM -Binary Number System -Octal Number System -Decimal Number System -Hexadecimal Number System -Binary to Decimal -Octal to Decimal -Hexadecimal to Decimal -Decimal to Binary -Hexadecimal to Binary -Octal to Binary -Decimal to Octal -Binary to Octal -Decimal to Hexadecimal -Binary to Hexadecim

3. Binary :0,1 –Base 2 Decimal :0,1,2,3,4,5,6,7,8,9 – Base 10 Octal :0,1,2,3,4,5,6,7 – Base 8 Hexadecimal :0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F – Base 16 2 10 8 16 For Ex: Binary : 0,1,1,10,11,100,101,110,111,1000,1001,1010,1011, … Decimal: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20… Octal : 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,22,23… Hexadecimal : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,20,21,22,23,24,25,26,27,28,29,A,B,C,D,E,F30…

4. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

5. Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177

6. Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal 81000108 91001119 10101012A 11101113B 12110014C 13110115D 14111016E 15111117F

7. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base DecimalBinary Octal Hexadecimal

8. Binary to Decimal Technique ◦ Multiply each bit by 2 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

9. Example 101011 2 => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = 32 43 10 Bit “0”

10. Octal to Decimal Technique ◦ Multiply each bit by 8 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

11. Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10

12. Hexadecimal to Decimal Technique ◦ Multiply each bit by 16 n, where n is the “weight” of the bit ◦ The weight is the position of the bit, starting from 0 on the right ◦ Add the results

13. Example ABC 16 =>C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 = 2560 2748 10

14. Decimal to Binary Technique ◦ Divide by two, keep track of the remainder ◦ First remainder is bit 0 (LSB, least-significant bit) ◦ Second remainder is bit 1 ◦ Etc.

15. Example 125 10 = ? 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 125 10 = 1111101 2

16. Octal to Binary Technique ◦ Convert each octal digit to a 3-bit equivalent binary representation

17. Example 705 8 = ? 2 7 0 5 111 000 101 705 8 = 111000101 2

18. Hexadecimal to Binary Technique ◦ Convert each hexadecimal digit to a 4-bit equivalent binary representation

19. Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2

20. Decimal to Octal Technique ◦ Divide by 8 ◦ Keep track of the remainder

21. Example 1234 10 = ? 8 8 1234 154 2 8 19 2 8 2 3 8 0 2 1234 10 = 2322 8

22. Decimal to Hexadecimal Technique ◦ Divide by 16 ◦ Keep track of the remainder

23. Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4

24. Binary to Octal Technique ◦ Group bits in threes, starting on right ◦ Convert to octal digits

25. Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8

26. Binary to Hexadecimal Technique ◦ Group bits in fours, starting on right ◦ Convert to hexadecimal digits

27. Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16

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