1. NUMBER SYSTEMS

2. DEC BIN OCT HEX DEC BIN OCT HEX
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
10000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 1 2 3 4 5 6 7 8 9 A B C D E F 10 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
10001
10010
10011
10100
10101
10110
10111
11000
11001
11010
11011
11100
11101
11110
11111
100000 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 40 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20

3. CONVERSIONS
FROM ONE BASE
TO ANOTHER BASE

4. Decimal to Binary 9410 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | 94/64 = 1R30 1
BASE TWO PLACE VALUE CHART
| 64 | 32 | 16 | 8 | 4 | 2 | 1 |
9410
94/64 = 1R30 1
30/32 = 0R
30/16 = 1R
14/8 = 1R
6/4 = 1R
2/2 = 1R
0/1 = 0R =

5. Decimal to Binary 7210 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | 72/64 = 1R8 1
BASE TWO PLACE VALUE CHART
| 64 | 32 | 16 | 8 | 4 | 2 | 1 |
7210
72/64 = 1R8 1
8/32 = 0R
8/16 = 0R
8/8 = 1R
0/4 = 0R
0/2 = 0R
0/1 = 0R =

6. Decimal to Binary 5110 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | 51/32 = 1R19 1
BASE TWO PLACE VALUE CHART
| 64 | 32 | 16 | 8 | 4 | 2 | 1 |
5110
51/32 = 1R
19/16 = 1R
3/8 = 0R
3/4 = 0R
3/2 = 1R
1/1 = 1R =

7. BINARY TO DECIMAL
Simply add up the binary place values where there is a one. Try it out on the last three slides and see how it works.
Try converting the following
Base two place value chart
| 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
11610
4310
21010
15810
Good Job

8. EASIEST CONVERSIONS
Binary to Octal and Octal to Binary
Each Base 8 digit has no more than 3 bits in the binary version.
Binary to Octal
Starting from the right, take groups of three bits, convert to base 8. =
Octal to Binary
Write three bits for each Base 8 digit
173608 =

9. EASIEST CONVERSIONS 11010000101111002 1101 0000 1011 1100 D 0 B C
Binary to Hexadecimal and to Hexadecimal to Binary
Each Base 16 digit has no more than 4 bits in the binary version.
Binary to to Hexadecimal
Starting from the right, take groups of four bits, convert to base 16.
D B C
= D0BC16
Hexadecimal to Binary
Write three bits for each Base 16 digit
3A4F16
A F =

10. ONCE BASE 2 CAN BE REACHED FROM THE OTHER BASES IT IS EASY TO GET FROM ANY BASE TO ANY OTHER BASE.
9810 = = =
= =
= =
You try these.
DEC BIN OCT HEX
_________ _____ _____ EC
____ __________ _____ A9
____ __________ _____ F

11. ADDING AND SUBTRACTING IN OTHER BASES
ADD SUBTRACT BASE 16
2A D96
+97F AE
93E E
+6DA DA
REMEMBER TO
THINK AND SPEAK
WITH BASE 10 NUMBERS
(WE HAVE THE IDEA 10 )
BUT
SEE AND WRITE
BASE 16 NUMERALS
(WE SEE AND WRITE A )
C23
3E8
1018
264

12. ADDING AND SUBTRACTING IN OTHER BASES
ADD SUBTRACT BASE 8
REMEMBER TO
THINK AND SPEAK
WITH BASE 10 NUMBERS
(WE HAVE THE IDEA 8 )
BUT
SEE AND WRITE
BASE 8 NUMERALS
(WE SEE AND WRITE 10 )
620 57
752
233

13. ADDING AND SUBTRACTING IN OTHER BASES
ADD SUBTRACT BASE 2
REMEMBER TO
THINK AND SPEAK
WITH BASE 10 NUMBERS
(WE HAVE THE IDEA 8 )
BUT
SEE AND WRITE
BASE 2 NUMERALS
(WE SEE AND WRITE )
11000
1101
To simplify the ALU (arithmetic logic unit) in the CPU,
the only operation the computer performs is addition.
How?
By using one’s and two’s complement.---

14. ONE'S COMPLIMENT AND TWO'S COMPLEMENT
First, let’s look at base 10 ten’s and nine’s complement
Instead use the ten’s compliment
which is the next higher power of
Ten’s complement - > 717
Add the 10’s compliment to the
top number and throw away the
carry
1279
Subtract: 562
-283
279
Easier way: Add 9’s complement So 562
This can be
done easily Nine’s complement->
in your head Same as above
717

15. Now for the one’s and two’s complement
Subtract:
- 101
100
Instead use the two’s compliment
which is the next higher power of
-101
Two’s complement - > 1011
Add the 2’s compliment to the
top number and throw away the
carry
10100
Easier way: Add 1’s complement So 1001
This can be
done easily One’s complement->
in your head Same as above
1011
In the CPU the inverter or NOT Circuit
performs the 1’s complement.
Easiest method of all coming up next >

16. EASIEST WAY TO GET THE TWO'S COMPLEMENT
Subtract the long way.
Rules for the easiest method
1. Copy the subtrahend exactly as it
until and including the first one (1) 10
The easy method +
2. After the first one, invert (or NOT)
all bits to the one’s complement. 1 1 1 10
3. Now we have the ten’s complement.
Add this to the top number.
4. Discard the carry.

17. Note: Negative numbers in the computer are stored in
For Practice
Write the two’s complement of these numbers.
Good Job!
Note: Negative numbers in the
computer are stored in
two’s complement form.
Also: In positive binary numbers the last bit to the left is 0.
In negative binary numbers the last bit to the left is 1.

18. THE END