1. Unit 18: Computational Thinking
Number systems

2. Addition The rules of binary addition are: 0 + 0 = 0 0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 and carry 1
When there is a carry bit, then
= 1
= 0 and carry 1
= 1 and carry 1

3. Example of Addition 0 1 1 0 1 + 1 0 1 1 1 ----------- = 1 0 0 1 0 0=
1 + 1 = 0 and carry 1
= 0 and carry 1
= 1 and carry 1

4. Addition examples
= = =

5. Negative numbers
We can use the “sign and magnitude” method to indicate negative numbers
We can split 8 bits into
One bit (the leftmost, and sign bit) is 0 for a positive number and 1 for a negative number
The remaining 7 bits give the value (or magnitude)
So = + 127d, and
= - 127d

6. Sign/magnitude examples
For example
is -43d
What is -83 in binary?
What is in decimal?

7. Subtraction
Implementing subtraction in a computer is easier if we can just use addition!
The “two’s complement” method allows this.
In this case a negative number is the “two’s complement” of its positive value

8. Two’s complement To write -28, we write out 28 in binary form 00011100
Then we invert the bits. 0 becomes 1, 1 becomes 0.
Then we add 1.
This represents -28

9. Two’s complement If the leftmost bit is 1, the number is negative
To reverse the process, start with
Reverse the bits
Add 1
Which is decimal 28

10. Two’s complement example
What is the binary for -30?
What is in decimal?

11. Addition Adding -28 to 30 (=2) 11100100 (-28) 00011110 (30) 100000010
But we only have 8 bits, so we ignore the left most (carry) bit, and the answer is
(2)

12. Subtraction 28 – 30 = -2 Take the two’s complement of 30
(flip the bits)
(add 1)
Add this to the binary for 28
(28)
(-30)
a negative number, which we can work out to be:
(flip the bits)
(add 1, to get decimal 2)

13. Subtraction example
Subtract 69 decimal from 12 decimal using the “twos complement” method.
Show the result in binary and decimal

14. Subtraction example result
= -57 d