SLIDE 1
ICS3U
- Mr. Emmell
Try these, in decimal
5 + 3
- 8
5 + 3 ----- Try these, in decimal 5 + 3 ----- 8 Try these, - - PDF document
5 + 3 ----- Try these, in decimal 5 + 3 ----- 8 Try these, - - PDF document
Addition / Subtraction ICS3U Mr. Emmell Try these, in decimal 5 + 3 ----- Try these, in decimal 5 + 3 ----- 8 Try these, in decimal 8 + 4 ----- Try these, in decimal 8 + 4 ----- 1 2 How did you do that? When one column
SLIDE 2
SLIDE 3
Try these, in decimal
8 + 4
- 1 2
When one column overflowed…. It incremented the next column
How did you do that?
SLIDE 4
When one column… tried to have a higher value than our number system allowed!
How did you know it
- verflowed?
Then you
- Subtracted ten
- Wrote the remainder
- Added the carry
Try these, in octal
28 + 48
SLIDE 5
Try these, in octal
28 + 48
- 68
Try these, in octal
58 + 58
SLIDE 6
Try these, in octal
58 + 58
- 5 + 5 = 10!!!!
That’s too much This is base 8 so… 10 – 8 = 2
Try these, in octal
58 + 58
- 1 28
SLIDE 7
Try these, in octal
1 58 + 58
- Try these, in octal
1 58 + 58
- 2 28
SLIDE 8
Try these, in octal
3 58 + 4 68
- Try these, in octal
3 58 + 4 68
- 1 0 38
SLIDE 9
Hexadecimal works the same way!
316 + 616
- Hexadecimal works the same way!
316 + 616
- 916
SLIDE 10
Hexadecimal works the same way!
716 + 616
- Hexadecimal works the same way!
716 + 616
- D16
SLIDE 11
Hexadecimal works the same way!
B 916 + A16
- Hexadecimal works the same way!
B 916 + A16
- Remember!
In decimal, this is: 9 + 10 = 19
SLIDE 12
Hexadecimal works the same way!
B 916 + A16
- C 316
Hexadecimal works the same way!
2 616 + 7 E16
SLIDE 13
Hexadecimal works the same way!
2 616 + 7 E16
- A 416
Don’t forget binary too
1 0 0 12 + 1 0 1 12
SLIDE 14
Don’t forget binary too
1 0 0 12 + 1 0 1 12
- 1 0 1 0 02
Good news! We can’t… Not really. We need to learn about complements first
Now… how about subtraction?
SLIDE 15
If it is 11:00 now, then three hours later it will be 2:00
11 + 3 = 14 ≡ 2 (mod 12) 2 o’clock
So instead, we add the complement.
Consider a 12 hour clock.
1 − 4 = −3 ≡ 9 (mod 12) 9 o’clock
So instead, we add the complement.
Similarly, if it is 1:00, then 4 hours ago it was 9 since
SLIDE 16
Notice that subtracting 4 hours on the clock is the same as adding 8 hours (12 – 4). In particular, we could have computed it as follows:
1 − 4 ≡ 1 + 8 = 9 (mod 12)
So instead, we add the complement.
Use an 8-bit number as an example Those numbers range from 0-255 If we have 255 and add one, then we get zero because it ‘cycles back’.
Binary numbers work the same way
SLIDE 17
How do we find the complement for a number in binary? We know that 200 – 50 is the same as 200 + (255-50) (Because we roll over after 255!)
So….?
⚫ Define how many bits you are using! 8-bit?
⚫ Always need to have leading zeroes.
⚫ Invert all the bits ⚫ Add one ⚫ BOOM – Two’s complement. Add normally
using binary addition.
HOW TO FIND “TWO’s COMPLEMENT”
SLIDE 18
100 – 58 = 42 Now for this operation, we are really saying: 100 + (-58) = 42 Let’s convert to binary
Example
100 = 011001002
Remember! 8-bits What about -58?
Example
2 100 2 50 0 2 25 0 2 12 1 2 6 0 2 3 0 2 1 1 0 1
SLIDE 19
58 = 001110102
Remember! 8-bits What about -58?
Example – Start with 58
2 58 2 29 0 2 14 1 2 7 0 2 3 1 2 1 1 0 1
001110102 //58 110001012
//FLIP
+1 //Add one 11000110
- 58 = 110001102
Example
SLIDE 20
0 1 1 0 0 1 0 02 1 1 0 0 0 1 1 02
Now we do the ‘addition’
0 1 1 0 0 1 0 02 1 1 0 0 0 1 1 02 1 0 0 1 0 1 0 1 02
If there is a leading one, we drop it (remember! 8-bits) Answer = 001010102
(42 !)
Now we do the ‘addition’
SLIDE 21