1. Discrete Mathematics
Numbering System

2. Number systems Binary digits: 0 and 1, called bits.
Binary (2 digits), octal (8 digits) and hexadecimal (16 digits) numbering systems.
We already know about decimal (10 digits) system:
Example: 45,238 is equal to
8 ones 8 x 1 = 8
3 tens 3 x 10 =
2 hundreds 2 x 100 =
5 thousands 5 x 1000 =
4 ten thousands 4 x =

3. Binary number system From binary to decimal:
The number is equivalent to
1 one 1 x20 = 1
1 two 1x21 =
0 four 0x22 = 0
1 eight 1x23 = 8
0 sixteen 0x24 = 0
1 thirty-two 1x25 =
1 sixty-four 1x26 =
107 in decimal base

4. From decimal to binary The number 7310 is equivalent to
remainder 1
remainder 0
remainder 0
remainder 1
remainder 0
remainder 0
remainder 1
 7310 =
(write the remainders in reverse order preceded by the quotient)

5. Binary addition table  1 10

6. Adding binary numbers Example: add 1001012 + 1100112
 carry ones

7. Hexadecimal number system
Decimal system 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A B C D E F
Hexadecimal system

8. Hexadecimal to decimal
The hexadecimal number 3A0B16 is
11 x 160 = 11
0 x 161 = 0
10 x 162 =
3 x 163 =

9. Decimal to hexadecimal
Given the number
remainder 9
remainder 2
remainder 9
is equivalent to the hexadecimal number 92916

10. Hexadecimal addition Add 23A16 + 8F16 1 23A16 + 8F16 2C916
What actually happen?
(A16 = 1010) + (F16 = 1510) = 2510
Then 2510/16 = 1 remainder 9, meaning 2510 = 1916.
= 12
12 in hexadecimal = C16.

11. Conversion from Binary to Hexadecimal
One of the easiest to convert binary to hexadecimal is by grouping the binary digits from right to left in a group of four.
Convert each group to a hexadecimal digit. Eg = ? 16
E = 16E 16

12. Conversion from Hexadecimal to Binary
One of the easiest to convert hexadecimal to binary is by converting each hexadecimal digit to a group of four binary digits from right to left. Convert each group to a hexadecimal digit.
Eg A C 16 = ?2
A C16
= = = 1210 =