1. Topic 3: Data
Binary Arithmetic

2. Data: Binary Arithmetic
Like denary numbers, we can add, subtract, multiply, and divide binary numbers
We’ll look at addition and subtraction
We’ll start simple with addition
It works exactly like adding denary numbers
We work from the least to the most significant digits, and add the digits on the same column
Data: Binary Arithmetic

3. Binary Arithmetic (Addition)
Here’s an example
Here we’re adding to 00110
We work our way, from the right to the left, adding the digits
Any time there’s too much we carry it over to the next digit
Essentially, long addition! 1 + =
Carries
Data: Binary Arithmetic

4. Binary Arithmetic (Addition)
We can always check our work afterwards
Convert binary numbers to denary
Including our answer
Add up denary numbers
See if the two answers match 1 + =
Carries
→ 1110
→ 610
= 17
→ 1710
Same answer, therefore we are correct!
Data: Binary Arithmetic

5. Binary Arithmetic (Addition)
There is a problem with adding binary numbers
In a computer, numbers usually have a set size
A set number of digits it can use
Sometimes adding two numbers could make a number larger than the set size 1 + = 1?
Carries
Data: Binary Arithmetic

6. Binary Arithmetic (Addition)
This is called an overflow
If we don’t have a set size, we can add the 1 as the most significant digit
If we do have a set size, we ignore it, ending up with an answer that isn’t accurate 1 + = 1?
Carries
Set Size No Set Size
Answer: Answer:
In Denary: 1 In Denary: 33
Data: Binary Arithmetic

7. Data: Binary Arithmetic
Add the following binary numbers
Show the answer in binary as well
If No Set Size, include the overflow
If Set Size, ignore the overflow
ANSWERS
From top-left to bottom-right:
No Set Size
Set Size (4 bits)
Data: Binary Arithmetic

8. Data: Binary Arithmetic
Binary Logic
If you’ve programmed before, you may recognise these operators
They handle logic
AND – both values need to be true
OR – either value needs to be true
We can do similar things, bit to bits
Rather than booleans
These operations are called bitwise AND, and bitwise OR
bitwise AND: &
bitwise OR: |
&& AND
| | OR
Data: Binary Arithmetic

9. Data: Binary Arithmetic
Binary Logic (AND)
Let’s say we have two binary numbers
Performing an AND on them gives us the following:
Can you tell what’s happened? 1
AND =
Can you tell what’s happened?
We’ve created a new binary number where we’ve only kept the 1’s that appear in the same digit in both operands (numbers in the operation).
Data: Binary Arithmetic

10. Data: Binary Arithmetic
Binary Logic (AND)
We go through both numbers
Digit by digit
For each resulting digit, we write:
0 if either of the digits is 0
1 if both of the digits is 1
Uses
In networking, we have a subnet mask.
It defines what part of an Internet Protocol (IP) address we are interested in.
If we AND the subnet mask with the IP address (essentially a long binary number), the only 1’s that will be left are the ones we are looking for.
For example:
Address:
Subnet Mask:
AND Result:
See how the only 1’s left are the ones also in the subnet mask?
Num1 1
Num2
Result
Data: Binary Arithmetic

11. Data: Binary Arithmetic
Perform a bitwise AND on the following numbers
If the numbers are in binary, output the answer as binary
If the numbers are in denary, output the answer as denary
You will need to convert the numbers to binary to perform the bitwise AND
ANSWERS
From top-left to bottom-right:
0 1 0
1101 & 0101
0001 & 1010
0000 & 1111
12 & 2
1 & 3
5 & 10
Data: Binary Arithmetic

12. Data: Binary Arithmetic
Binary Logic (OR)
Let’s take the same numbers from before
Performing an OR on them gives us the following:
See the difference? 1 OR =
See the difference?
This time, we’ve created a new binary number that has 1’s whenever a 1 appears in a digit in either operand.
Data: Binary Arithmetic

13. Data: Binary Arithmetic
Binary Logic (OR)
We follow the same steps as before
For each resulting digit, we write:
0 if neither of the digits is 1
1 if either/both of the digits is 1
Uses
We can use bitwise OR to handle inputs on a controller.
Imagine we have a box with two buttons – left and right.
Pushing the left button returns a value of 1, and pushing the right button returns a value of 2. Both buttons return 0 of they aren’t pressed.
Imagine these as two binary numbers: 01 and 10.
See how they inhabit different bits?
If we wanted to tell which buttons are being pressed in the game, we can bitwise OR them together to get a single number.
We can then look through that number, bit-by-bit, to see what buttons we’re pressing.
Num1 1
Num2
Result
Data: Binary Arithmetic

14. Data: Binary Arithmetic
Perform a bitwise OR on the following numbers
If the numbers are in binary, output the answer as binary
If the numbers are in denary, output the answer as denary
You will need to convert the numbers to binary to perform the bitwise AND
ANSWERS
From top-left to bottom-right:
1101 | 0101
0001 | 1010
0000 | 1111
12 | 2
1 | 3
5 | 10
Data: Binary Arithmetic

15. Data: Binary Arithmetic
Binary Logic (Shifts)
The final form of ‘logic’ is shifts
We can take any binary number and shift it to the left or right
It makes the number look like it has moved
Shifting left ‘adds’ a 0 least significant bit
Shifting right ‘removes’ a 0 least significant bit, and adds a 0 most significant bit
Original Number: 0110
Shifted Left (1): 1100
Shifted Right (1): 0011
Data: Binary Arithmetic

16. Data: Binary Arithmetic
Binary Logic (Shifts)
There is one important thing to remember about shifts
We can shift too much
Any time we shift numbers ‘off’ the space we have, we’re replacing it with a 0
Original Number: 1111
Right Shift (1): 0111
Right Shift (3): 0001
Original Number: 1111
Left Shift (1): 1110
Left Shift (3): 1000
Original Number: 1111
Left Shift (2): 1100
Then Right Shift (1): 0110
By ‘shifting too much’, I mean completely throwing away any significant bits. In the bottom example, we shift left and then shift right. Because we shifted left too much, the most significant 1’s were thrown away (as they didn’t fit in the set-size binary numbers). When we shift right, we’ll notice that those 1’s have turned to 0’s.
Data: Binary Arithmetic

17. Data: Binary Arithmetic
Binary Logic (Shifts)
The best use of shifts is when we multiply a binary number by a power of 2
We shift left the same number of times as zeroes in the power
21 (shift once) 22 (shift twice) 23 (shift three times)
Examples
x 102 (1310 x 22) x 1002 (310 x 410)
= (2610) = (1210)
All we did here is shift the number left once Here we shifted left twice
Data: Binary Arithmetic

18. Data: Binary Arithmetic
Perform the following shifts and convert the result to denary
For each one, use a binary number size of 4 digits
Right Shift (2): 0011
Right Shift (1): 1001
Right Shift (3): 1110
Left Shift (1): 0110
Left Shift (3): 0111
Left Shift (2): 1101
ANSWERS
From top-left to bottom-right:
Data: Binary Arithmetic

19. Data: Binary Arithmetic
Complete the following exercises
Each row in the table asks for something different
Convert each answer to denary
Add (Set Size of 6 Bits)
Bitwise AND (&)
24 & 12
10 & 8
32 & 16
Bitwise OR (|)
24 | 12
10 | 8
32 | 16
Convert Units
2.1KiB to KB
0.5MB to MiB
500GB to TiB
ANSWERS
From top-left to bottom-right:
8 8 0
2.1504KB MiB TiB
Data: Binary Arithmetic