Chapter two binary arithmetic. In any number system when carrying out arithmetic operations, you have to follow a set of rules. And when adding to single bit binary numbers, this is the binary addition rules that you must follow. The rules for binary numbers are quite straightforward and similar to those used in decimal arithmetic. The rules for addition of binary numbers are when you have two binary numbers, zero plus zero adds to zero. When you have two binary numbers.

When you add one to zero, the result is one. If you add zero to one result is one. And if you add one to one, the result is zero, but you must carry a one. So in binary addition, when adding to N bit binary numbers, the binary addition rules are add the individual bits propagate the carries to the next highest significant digit. Pretty simple. So let's look at an example.

Here we have the binary number 10101, adding it to 110011, which happens to be 21 and 25. In the decimal system, if you add starting from the least significant digit, which is the immediate right, one plus one is equal to zero and you carry the one. zero plus zero is zero, but you're carrying one so the answer is one Learn from the right is one plus zero is one. The next digit higher is zero plus one is one. The next digit is one plus one is zero, but you carry one, and one is now in the next significant column. So you end up with a number 101110, which happens to be 46.

And it is the decimal equivalent of adding 21 plus 25. So it all works. So looking at another example, we can say that binary addition is carried out much like we carry out addition with a decimal number by adding up the columns starting at the right and working column by column towards the left. Just as in decimal addition, it is sometimes necessary to carry and the carry is added To the next column. So if you are going to add a binary number two, which is one zero to a binary number one, which is 01, you'd end up with 310 plus one is one, one plus zero is 01. The answer is one one, which is equal to three.

So this example works. Okay, so let's do a little bit more complex of an example. By adding two four bit binary numbers, you're going to add the numbers 01112, the number 1010. Starting from the least significant digit, the right hand side, zero plus one is zero. Moving to the next significant digit, we have one one, so one plus one is zero. But we must remember to carry the one and we just put it up on top there to keep track of it.

So the next significant digit is one plus zero plus one is the same as one plus one, which is zero. Carry one, and we put the carry up on top. So now we're left with one plus one plus zero. So it's one plus one is another zero, but you carry one. So the answer is 10001. So, if we are going to check this out, we can check it out with a decimal equivalent.

So the number 1010 is the number 10 in decimal. The number 0111 is the number Seven in in a decimal equivalent. You add those together and you get 17. And if you look at our chart 17 was 10001. So again, the rules work. Let's move on to binary subtraction.

There are four subtraction combinations possible when dealing with two single binary digits. Rules number one, two and three are straightforward are and are the same as if you were dealing with two decimal numbers. Rule number one, zero minus zero is zero. Rule number two, one minus one is zero. And rule number three, one minus zero is one. Rule number four, zero minus one is one.

Now that needs it. Little bit of explanation because normally, we cannot subtract something from nothing. It is similar to subtracting one from one zero. In order to subtract one from zero, you must first borrow a one from the next most significant digits. What we see here is only the end result of the formal process. Another way of looking at it is subtracting a binary number one from the binary number one zero, but because we are only dealing with a single digit, we are only able to see the zero however, we must remember that we borrow the one from the next most significant digits column.

So we can check this out with decimal numbers. If we take one zero and replace it with a decimal number, that's the decimal number two One course equates to the decimal number one. So you take one from two and you're left with one, of course in the decimal system. In the case of the binary system, one zero minus one is one. So again, the rules work. When subtracting multi digit numbers, it helps to consider them in terms of their base and weight values.

If a one appears in the least significant position, the decimal equivalent is one times two to the zero power, which is equal to one. If a one appears in the next significant position, the decimal equivalent is one times two to the first power, which is two. If a one appears in the next significant position, the decimal equivalent is one times two to the second power two squared, which is four. And if a one appears in the next significant position, that decimal equivalent is one times two to the third power, or two cubed, which is eight. Now, if we borrow an eight from the most significant bit register, it is equal to two fours in the next significant bit register, or two binary ones in that register. If we borrow four from that register, it is equal to two twos in the next significant bit register, or two ones in that register.

Last Lastly, if we borrow it too from that register is equal to two ones in the next significant bit register, which is equivalent to two binary ones in that register. So keeping that in mind when subtracting we, we subtract from right to left, so we begin with the least significant column. This means when we subtract one from zero, we cannot subtract one from nothing, so we must borrow a one from the next significant column on the left. However, the next column occupies the two place and it is actually consists of two ones. So we now have a column if we borrow one from the next significant number Have two ones that we've borrowed in the least significant digits, so we actually have three ones there. So now if we subtract one from the three, we're left with one.

As we move to the left, we see that after borrowing one, we are left with subtracting zero from zero, which is zero, and the process is finished. No matter the place value, whenever we borrow from the next significant column containing a one in binary, we always borrow a two, a one zero and splitting the two into two ones. From the bureau operation, we always record two additional ones in the needed column. Let's check this out with a decimal equivalent. converting the top number one zero to a decimal number gives us the number two and converting bottom number 01, which we're subtracting. That gives us the number one which checks out because two minus one is one and that's what we are left with in the binary operation.

Let's walk through another example. Calculating the least significant column yields us a one. There is nothing left in the next column. So that yields us zero. Calculating the next significant And takes us through a borrowed concept and yields one. And there is nothing left in the next column.

So that also leaves leaves us with zero. Going through the checking process using decimal equivalence, converting 1010 it comes to 10. And converting 0101 that gives us the decimal equivalent five, and five from 10 gives us five and the 101 in the binary arithmetic yields us a five so it checks out. So I'm not going to take a long time to go through this. So just let's quickly go through the process very fast. We can see here That, going through the process of borrowing and subtracting we're left with 0011.

And when we check that against our decimal equivalent, 1000 is 80101 is five. So five from eight leaves three, and one one checks out to be three. So again, our process checks out. The multiplication rules for two single bit binary values are much the same as what they are with decimal numbers. If you have two numbers, the a column and the B column, a being zero being zero, you take zero times zero, you're left with zero. In the a column you has zero B column you have one, so zero times one is zero.

In the a column, if it's one in the B column is zero, you have one times zero, it's still zero. If you have a one and a one, one in the a column one in the B column, one times one is equal to one. No big surprise there. Let's go through the calculation of multiplying two, four bit numbers together and see what the answer is. The two four bit numbers are 1110 times 1011. And we approach this much in the same manner as we do with a a digital multiplication.

Start with the least significant bit, and you follow the rules. One times zero is 01 times one is one, one times one is one, one times one is one, so the number is 1110. Now when we multiply with the next significant number, we have to shift everything over because we're in the next significant bit, but the rule still hold. One times zero is 01 times one is one, one times one is one, one times one is one. So now we have 1110. The next significant bit is a zero.

So when we multiply any number by zero, we come up with a zero, we have to shift one register to the left, because it's now the third most significant digit, but they're all zeros. Go away. For the last significant bit, we still multiply using the same rules. And no big surprise we end up with a number 1110 but everything is shifted one value to the left. Now, we add up all of those columns together and we get 01. And we get a one and a one which gives us a zero to carry one.

So you get a one and a one plus a carry one is one. And a one and a one with a carry one is one to carry one and one and zero is plus a carry is zero, carry one. One and one is with a carry is zero, and you have a carry of one good follow up. We were just adding up the columns vertically. Whenever we get a zero in the column, of course it's zero. Whenever we get three ones, we have to carry one forward to the next column.

So the answer is 1001010. So what does that convert to in decimal numbers? Well, we can do it by using the power of the weight value. So we're going to have nothing for the first number because it's zero. The second number is two to the first power. The second number two to the second power is zero.

So that doesn't show up as significant. The next significant one is two to the third power. The next significant one is to to the fourth power that we have The fifth and the sixth power are zero. So we don't have to calculate those, but the last number is two to the seventh power. Now, if you multiply those powers out, add them together, you get 128 plus 16 plus eight plus two is equal to 154. The numbers that we started with 1110 is 14 1011 is 11.

And you multiply What? 14 times 11. And you come up with 154. So no big surprise that our theory of our rules for multiplication, check out. This ends chapter two