1. Number Systems Binary and Hexadecimal

3. Base 2 a.k.a. Binary  Binary works off of base of 2 instead of a base 10 like what we are taught in school  The only numbers that are able to be represented are 1 and 0  Binary numbers are read right to left (inverse way of reading, normal way of reading numbers)

4. How to read binary numbers  Since binary is base 2, every bit that follows the first number in the sequence represents the previous number raised to the power of 2  So 100011101 = 256+0+0+0+16+8+4+0+1 = 285

5. Another way Repeat Division  Basically divide by 2 a lot  If the quotient has a remainder of 1, write down 1, if not write down 0  Keep dividing until you reach zero  Keep in mind, do not automatically put the remainders in fraction form

6. Example of Repeat Division  117  ÷2 58 remainder 1  ÷2 29 remainder 0  ÷2 14 remainder 1  ÷2 7 remainder 0  ÷2 3 remainder 1  ÷2 1 remainder 1  ÷2 0 remainder 1

7. Two’s complement  On the IB exam they will probably ask you to write a number using two’s complement  Two’s complement is a way to write negative numbers in binary  Basically you take the last number in the sequence (the largest number), and make it negative  You can then create any negative number less than the absolute value of the largest number

8. Example of two’s complement

9. How to write decimal points in binary (Floating Point)  Simply add the decimal, and after the decimal follow the same pattern as you would with numbers greater than 1  Instead of each bit after the decimal being 2 n, it is 2 -n.

10. Example MSB LSB ==

12. Adding in Binary  Remember these: 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 10 1 + 1 + 1 = 10 + 1 = 11  In the case of a 10 or 11, “carry the one” one digit to the left, just like in normal (base 10) addition.

13. Examples 1 11 1 11 101 + 1 +11 + 10 =10 =110 =111 1 1 1001010 +1101101 =10110111

14. Subtracting in Binary  Remember these: 0 – 0 = 0 1 – 0 = 1 1 – 1 = 0  0 -1 is a special case. Essentially, it requires you to carry a 1 from the left, just like in normal subtraction.

15. Examples 2 02 002 02 100 1100101 - 10 - 110010 = 10 =0110011

17. Base 16 A.K.A. Hexadecimal  Hexadecimal works off a base of 16.  It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen.  In base 10 (normal) numbers, for example, 14 means (1*10) + 4. In Hexadecimal, 1D means (1*16) + D (which is 13), or 29. 0 = 0 1 = 1 2 = 2 3 = 3 4 = 4 5 = 5 6 = 6 7 = 7 8 = 8 9 = 9 A = 10 B = 11 C = 12 D = 13 E = 14 F = 15

18. Some Hexadecimal Examples  F is 15  10 is (1*16) + 0, or 16.  1F is (1*16) + 15, or 31.  FF is (15*16) + 15, or 255.  1FF is (1*16 2 ) + (15*16) + 15, or 511.  etc.

19. Binary/Hexadecimal Conversion Examples Binary Hex 0000 0 1000 8 0001 1 1001 9 0010 2 1010 A 0011 3 1011 B 0100 4 1100 C 0101 5 1101 D 0110 6 1110 E 111 7 1111 F

21. How to add hexadecimal  Remember to think in base 16 when doing Hexadecimal Math.  If the value is greater than or equal to 16 you carry a 1 over to the next column, and write down the value you received from the addition minus 16  If the number that you receive from addition is greater than 32, then you subtract 32, write down the value, and carry a two over to the next column  Etc.

22. Examples 11 1 1 12 91A 2F A AF 1F2 +3A +B F +E37 69 16 +FA 1943 1B8

23. Subtraction in Hexadecimal  Subtraction works very similar to subtraction with decimal values  Just remember that if you borrow a 1 from a column to the left, the borrowed 1 is equal to 16 (not 10).

24. Example subtraction F F 18 5 10 10 12 7 12 E 8 11 6 0 0 2 B 8 2 F 9 1 - 3 4 7 A 8 -1 5 9 E B 2 B 8 8 3 6 D 5 A 6