1. Ours is not to reason why Ours is to Invert and Multiply…
Numbering Systems
Ours is not to reason why
Ours is to
Invert and Multiply…

2. Numbering Systems
Since early times mankind has used many different types of symbols to represent numbers
In North America we use the Base 10 system.
There are several different types of number systems. Each is used for different purposes and each is different, yet similar.
Binary (used in computers) Base 2
Hexadecimal(used in computers) Base 16
Decimal (used in North America) Base 10

3. Decimal (Base 10)
It consists of 10 digits (hence the name decimal).
The digits, from smallest to largest are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Now, any number can be broken down into columns.
From right to left, the first column is the 1's column, then the 10's column, then the 100's, then the 1000's, etc..
0 X 1000 = 0 (0 thousands) 1 X 100 = 100 (1 hundreds) 3 X 10 = 30 (3 tens) 7 X 1 = 7 (7 ones)
= 137

4. An Example lets try the number 77: 0 X 1000 = 0 (0 thousands)
0 X 100 = 0 (0 hundreds)
7 X 10 = 70 (7 tens)
7 X 1 = 7 (7 ones)
= 77

5. Binary (Base 2) Your computer works using the binary numbering system.
The binary numbering system is ideal for representing these two states because it consists of only two digits.
Once again, any number can be broken down into columns.
Using the binary numbering system, from right to left, the first column is the 1's column,the 2's column, the 4's, the 8's, the 16's column, the 32's column, etc..

6. Let's look at the number 137 again
1 X 128 = 128 (1 one hundred twenty-eights)
0 X 64 = 0 (0 sixty-fours)
0 X 32 = 0 (0 thirty-twos)
0 X 16 =0 (0 sixteen's)
1 X 8 = 8 (1 eights)
0 X 4 = 0 (0 fours)
0 X 2 = 0 (0 twos)
1 X 1 = 1 (1 ones)
128 64 32 16 8 4 2 1

7. Converting Decimal to Binary
There are a number of ways to convert between decimal and binary.
Lets start with converting the decimal value 254 to binary.
Method 1: Use the binary calculator. What we have been doing before
Method 2: Divide the number by 2. Then divide what's left by 2, and so on until there is nothing left (0).
Write down the remainder (which is either 0 or 1) at each division stage.
Once there are no more divisions, list the remainder values in reverse order. This is the binary equivalent.

8. 254 / 2 giving 127 with a remainder of 0
Reading in reverse order(Bottom to top)
Tada.. Not too shabby!

9. Another example, 132 decimal
132 / 2 giving 66 with a remainder of 0
66 / 2 giving 33 with a remainder of 0
33 / 2 giving 16 with a remainder of 1
16 / 2 giving 8 with a remainder of 0
8 / 2 giving 4 with a remainder of 0
4 / 2 giving 2 with a remainder of 0
2 / 2 giving 1 with a remainder of 0
1 / 2 giving 0 with a remainder of 1
Thus the binary equivalent is

10. = 137
Thus, the binary number is equal to 137 decimal.
A single digit (0 or 1) is called a 'bit' (binary digit).
The table above contains 8 bits. Each column can contain either a 1 or a 0 ( 'cause there is only 2 digits in the binary numbering system).
So, as you can see, it takes 8 bits to represent the decimal number 137.
lets try the number 77:
128 64 32 16 8 4 2 1

11. 0 X 128 = 0 (0 one hundred twenty-eights)
1 X 64 = 64 (1 sixty-fours)
0 X 32 = 0 (0 thirty-twos)
0 X 16 =0 (0 sixteen's)
1 X 8 = 8 (1 eights)
1 X 4 = 4 (1 fours)
0 X 2 = 0 (0 twos)
1 X 1 = 1 (1 ones)
= 77
Thus, the binary number is equal to 77 decimal

12. Solve the first two rows using a binary calculator Then solve the last 2 rows using division.
254 8
127 13
255
1020
397
9999 20 5 16 99
178 33
207
3578

13. Hexadecimal Number System [Base-16]
The hexadecimal number system uses SIXTEEN values to represent numbers.
The values are A B C D E F
With 0 having the least value and F having the greatest value.
Hexadecimal is often used to represent values [numbers and memory addresses] in computer systems.

14. Converting hexadecimal to decimal
Convert 176 in hexadecimal to decimal
Each column represents a power of 16,  
176 = * = 6
* = 112
* = 256
= 374
Convert 11 in hexadecimal to base 10
11 = 1 * 160
= * = = 17

15. Practice Hex to Decimal
3DA D9E F3A
A9 645
3D 14DE F309 B8A4

16. Converting binary to hexadecimal
Convert to hexadecimal.
Each hexadecimal digit represents 4 binary bits.
Split the binary number into groups of 4 bits, starting from the right.
= 1 = 6
= 16 in hexadecimal.
Try base 2 to base 16
What did you get ??

17. Practice Bin to Hex

18. Converting decimal to hexadecimal
Convert 232 decimal to hexadecimal.
Use the same method used earlier to divide decimal to binary, but divide by 16 this time.
232 / 16 = 14 with a remainder of 8
8 / 16 = you can’t have since 8 is smaller than 16. So the 8 becomes the last digit.
So you get 14 and a 8. Remember…
(14 decimal = E)  = E816
A=10, B=11, C=12, D=13, E=14, F=15

19. Practice Decimal to Hex

20. Notation
To avoid confusion, we often add a suffix to indicate the number of the base.
162h h means hexadecimal
means base 16 
162d d means decimal
means base 10
162o o means octal
means base 8 
101b b means binary
means base 2

21. More Converting For Review, Work on the handout called “Digital 1”
Specific problems or questions come and ask me.
Copy and complete the following charts in your books.

22. Binary
Hex
Decimal

23. Hex
Binary
Decimal
345
AF56
78C1
B0D5
7156 9
185
FFFF
FFF FF

24. Decimal
Hex
Binary
345
32768
255
1289 15
1000
1024
999
1678
2000