1. Binary, Hexadecimal,
and Base 10

2.
656
DECIMAL
SYSTEM OR
Base 10
12875

3. The place values are powers of 10: 1’s  100 10’s  101 100’s  102
The decimal system is also referred to as base 10
A 10 in subscript is put beside any decimal number to show it is in base 10 (ex )
Base 10 is the number system we normally use
In base 10 the largest digit is 9
The place values are powers of 10:
1’s  100
10’s  101
100’s  102
1000’s  103
10,000’s  104
Etc.

4. 1 1 1 1 1 1
BINARY 1 1 1 1 1 1 1

5. Binary is also referred to as base 2
A 2 in subscript is put beside any decimal number to show it is base 2 (ex )
In base 2, the only digits used are 1 and 0
The place values are powers of 2
first place value is 20  (1)
second place value is 21  (2)
third place value is 22  (4)
fourth place value is 23  (8)
fifth place value is 24  (16)
Etc.

6. Binary Sample Place Value Chart
128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

7. Converting Base 2 To Base 10
To convert base 2 (binary) to base 10 (decimal),
Place the binary number from right to left in the place value chart.
If the digit is a 1, then add the value of that place from the place value chart.
If the digit is 0, add 0

8. 16 8 4 2 1 1 1 2 = 2 = = = + 16 = 18 10

9. 1 1 1 1 2 = = 4 = 8 = 16 = = + 64 = 92 10

10. Converting Base 10 To Base 2
To convert from decimal to binary we break up the decimal number into binary place values
Find the largest place value you can divide the decimal number into
Put a 1 in that place value
Find out how much is left over, then see the next largest place value you can divide the leftover into (any number that can not divide into the number, put a zero)
For example:

11. 1024
512
256
128 64 32 16 8 4 2 1 1 1 1 13 =
Remainders 5 1 1 1 1 1 1
114 = 50 18 2 2 2 1 1 1 1
420 =
164 36 36 4 4 4 1 1 1 1 1 1 1
1694 =
670
158
158 30 30 30 14 6 2

12. 27A29B
83B38A298E
2784B47A
HEXADECIMAL
F83B9464
34B123E3
635F048B

13. Hexadecimal is also referred to as base 16
A 16 in subscript is put beside any hexadecimal number to show it is base 16 (ex. 836F89E16)
In base 16, the digits 0-9 are used, as well as the letters A-F to represent the numbers 10-15
A=10 B=11 C=12 D= E=14 F=15
The place values are powers of 16
The first place value is  (1)
The second place value is 161  (16)
The third place value is 162  (256)
The fourth place value is 163  (4096)
Etc…

14. Base 10 vs. Base 16 1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F

15. Hexadecimal Place Value Chart
65536
4096
256 16 1
165
164
163
162
161
160

16. Converting Base 16 to Base 10
To convert a number from hexadecimal to decimal, find the value of each digit in the number, then multiply it by the value in the place value chart
Then add all these values to get a total
For example:

17. 4096
256 16 1 8 3 2 D 16 2 13 = 32 1 =
768 = 1 +
32768 = 3 3 5 8 1 10

18. 4096
256 16 1 2 A F 16 15 = =
2560 = +
8192 = 1 7 6 7 10

19. Converting Base 10 to Base 16
To convert a number from decimal to hexadecimal, find the largest place value that can fit into the number
Find out how many times it can go into the number and write that digit
If the digit is larger than 9, convert it to its corresponding letter
Find the remainder (by subtracting the previous amount by the new amount) then repeat the above steps until there is no remainder
For example:

20. 65536
4096
256 16 1 3 3 5 8 1 8 3 2 D = 10
32768
768 32 13
Remainder
813 45 13

21. 65536
4096
256 16 1 2 4 8 3 1 6 F F = 10
24576
240 15
Remainder
255
255 15

22. Converting Base 16 To Base 2
To convert from hexadecimal to binary, take each digit in the base 16 number and write out the binary equivalent
Each digit must have four binary digits after converting it
Combine all the binary numbers you just wrote out

23. Base 2 vs. Base 16
0000
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

24. 318 = 16 3
0011 1
0001 8
1000
0011 11
0001
1000 2

25. Converting Base 2 To Base 16
To convert binary to hexadecimal, first break the binary number into groups of four digits
Convert these groups to decimal numbers
Use these numbers as the digits for the hexadecimal number
If the number is greater than 9, change it its corresponding letter (A - F)

26. 0011
1101
1000 2 3 13 8 D 3 D 8 16