1. Topic 3: Data
Signed Binary

2. Positive and Negative Numbers
There are all kinds of numbers out there
Positive ones Negative ones
Whole ones Fractional ones
Rational ones Irrational ones
Real ones Imaginary Ones
A computer needs to know how to store each of these in binary form
We’re going to focus on positive and negative numbers

3. Positive and Negative Numbers
We know, almost instantly, when a number is positive or negative
10: Positive -5: Negative
More technically, we can tell if a number is negative if there is a minus sign in front of it
If there is no sign, we assume the number is positive
Computers can’t assume
It needs to know, definitively, if a number is positive or negative
To do this, scientists have come up with different methods

4. Signed/Unsigned Integers
Let’s start with the idea behind signed and unsigned integers
Integer: whole number
If an integer is unsigned, it does not care about positivity or negativity
Unsigned integers can only ever be positive
No bits are wasted for a sign
If an integer is signed, it can be positive or negative
A bit needs to be used for the sign though

5. Signed/Unsigned Integers
Take this example
We have a number: 610 (is 1102 in binary)
If this is an unsigned integer, we can leave it as 1102
If it is signed, we need to use 01102
The most-significant bit acts as the sign
0: Positive
1: Negative
Ultimately, state that the unsigned version only takes 3 bits to store, but the signed version needs 4 bits.
If space on a computer is especially tight (not so much of a problem these days), then using unsigned integers can save a bit of space now-and-again.

6. Sign and Magnitude
The signing system earlier is called sign and magnitude
When we use 1 bit for the sign, and the other bits for the number
The sign bit is always the most-significant bit
Here are some examples 1
Sign Bit
Usual Binary Value
Note that, for each of the examples shown, an extra bit is added on the far left for the sign.
1010 = = 11111
-4 = = 11000

7. Write the following denary values in sign and magnitude binary15 25
-50
100
-255 32
-16 -1
ANSWERS
From top-left to bottom-right:

8. Two’s Complement
The range of values allowed in sign and magnitude can be found using the following
So a 4-bit sign and magnitude binary number has the range
-7 to 7
There is another system we can use
Two’s Complement
𝑟𝑎𝑛𝑔𝑒=± ((2 𝑛 −1 ) −1)
𝒏 is the number of bits

9. Two’s Complement
The Two’s Complement and sign and magnitude are nearly identical
Both use most-significant bit for sign
However, in Two’s Complement, the most- significant bit is not just the sign
It represents the negative of that bit’s value
Note that we calculate the result as usual – by adding up the denary value of each bits. However, in this case, the most-significant bit is the negative of its usual value.
We can calculate the denary value of the table on the slide by adding the denary value of each bit (as usual): (-8 x 1) + (4 x 0) + (2 x 0) + (1 x 1) = = -7
Denary -8 4 2 1
Binary

10. – + Two’s Complement Here’s an example of some signed binary numbers
We can still have positive numbers in this system
0110
0010
0000
1000
0100
1100
1010
1110
1011
1001
1101
1111
0101
0111
0011
0001 – + +6 +2 +4 +5 +7 +3 +1 -8 -4 -6 -2 -5 -7 -3 -1 -8 4 2 1
Denary 3 -4 -1

11. Two’s Complement
Converting any denary number into Two’s Complement is easy
First convert to unsigned binary
If the number is negative, then
Flip all the bits (0’s to 1’s, and 1’s to 0’s)
Add 1 to least significant bit
Negative Example
Denary Number: -4
1) Unsigned Binary: 100
2) Two’s Compliment: 0100
Still add 0 as we haven’t added the minus sign yet
3) Flipped Bits:
4) Add 1 to LSB: 1100
Positive Example
Denary Number: 4
1) Unsigned Binary: 100
2) Two’s Compliment: 0100
Added 0 to show that number is positive

12. Convert the following denary numbers into Two’s Complement signed binary numbers8 7 -3
-14 1 -1
ANSWERS
From top-left to bottom-right:

13. Two’s Complement
We have two methods for converting Two’s Complement binary numbers to denary
Method 1: Multiply digits by their worth, and add together
Method 2 (only needed if negative): Flip bits, add 1, and then use Method 1
Method 1
Two’s Compliment:
1) Find values: (32 x 0) + (16 x 1) + (8 x 1) + (4 x 0) + (2 x 1) + (1 x 0)
2) Sum values: = 26
Method 2
Two’s Compliment: negative number, so we use Method 2
1) Flip bits:
2) Add 1 to LSB:
3) Find values: (32 x 0) + (16 x 0) + (8 x 0) + (4 x 1) + (2 x 1) + (1 x 0)
4) Sum values: = 6
5) Add sign: -6 needed as binary number was negative
We don’t have to use Method 2 if we don’t want to. We can just as easily sum the denary value of the binary digits manually (as shown on Slide 9).

14. Convert the following Two’s Complement signed binary numbers to denary
1100
0101
1001
010011
110011
111111
ANSWERS
From top-left to bottom-right: