Number SystemsBinary and Hexadecimal

Explaining Binary

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)

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

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

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

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

Example of two’s complement

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 2n, it is 2-n.

Example

MSB LSB

==

Binary Math

Adding in Binary

Remember these:0 + 0 = 01 + 0 = 10 + 1 = 11 + 1 = 101 + 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.

Examples

1 11 1 11 101+ 1 +11 + 10=10 =110 =111 1 1 1001010 +1101101=10110111

Subtracting in Binary

Remember these:0 – 0 = 01 – 0 = 11 – 1 = 0

0 -1 is a special case. Essentially, it requires you to carry a 1 from the left, just like in normal subtraction.

Examples

2 02 002 02 100 1100101- 10 - 110010= 10 =0110011

Explaining Hexadecimal

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

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*162) + (15*16) + 15, or 511. etc.

Binary/Hexadecimal Conversion Examples

Binary Hex 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

Hexadecimal Math

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.

Examples

11 1 1 12 91A 2F A AF 1F2 +3A +B F+E37 69 16 +FA1943 1B8

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).

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