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2.3 : Addition and Subtration
2.4 : Addition and Subtraction
It's finally here! We finally made it to the part where we add and subtract! For our purposes and for ease, we'll be using two's complement when dealing with negative values. Also, for ease, we will only be adding integers. Integers are numbers without decimals and fractions.
Addition
Addition is handled exactly the same way as in base-10. We go bit by bit, starting with the smallest and working up. So 0 + 0 = 0. 0 + 1 = 1. 1 + 0 = 1. 1 + 1 = 0 and then we carry the one to the next bit. Let's add two 8 bit values together. Let's add 00011101 and 10011010.
| Carry | 1 | 1 | ||||||
|---|---|---|---|---|---|---|---|---|
| Value 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
| + Value 2 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
| = Result | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
So this means 000111012 + 100110102 = 101101112. Let's check our work though. As far as I know, our binary skills are not perfect. We don't know off the top of our heads if this is correct. Let's convert these numbers to base 10 then check. 000111012 is 2910. 100110102 is 15410. 101101112 is 18310. 2910 + 15410 = 18310. Looks like we did a good job!
Subtraction
Subtraction, believe it or not, works just like base-10. You negate the value, then add. Of course, since we're working with base-2, we negate using two's complement. So this time, let's subtract 00011101 from 10011010 First let's negate 00011101.
Now lets add with our table again. 00011101 + 11100011
| Carry | 1 | |||||||
|---|---|---|---|---|---|---|---|---|
| Value 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| + Value 2 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
| = Result | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
Our result is 01111101. This, in base 10 is 125. If we check our work again 154 - 29 = 125. One thing to notice is the leftmost bit. As you can see 1 + 1 is equal to 0 with a carry of 1. However, we don't have room for a carry! In the next section on overflow, I'll cover this. Right now, let's ignore it.
Overflow
So overflow is one of the many reasons we use twos complement when doing math. I'm going to explain using 4 bit numbers for ease of demonstration. Let's look at our twos complement number.
| MSB | Value | Value | Value |
|---|---|---|---|
| 0 | 1 | 1 | 0 |
So the MSB, as we mentioned earlier in the [Negative Binary Section](binary/negativebin), is the most significant bit that tells us if the number is positive or negative. The other values are values that actually matter when we add and subtract. So let's do a problem in twos complement.
0110 + 0100 = 1010
6 + 4 = -6
So how did two positive numbers add together to create a negative number? So while a 4 bit number may have a range of 0-15 unsigned, when we sign the number, the range is now -8-7. This is much easier to think of as a wheel.
In this wheel I just counted up one by one in binary. If we consider these to be twos complement numbers, we get the following base ten values.
So when we add or subtract we work our way around the wheel. Once again, lets consider 0110 + 0100. If we start at the 6 position on th circle, we step around four places. We then get to -6. If we keep adding, we'd eventually get a 5 bit number, however, since we're only working in 4 bits, we'd lose those bits.
Practice Problems
What is the 4 bit value you get when you add 0011 and 1010?
What is the 4 bit value you get when you subtract 0011 from 1010? Make sure you check your work!