Learn how computers add numbers and build a 4 bit adder circuit

so if we want to add 28 plus 22 it's pretty simple 8 plus 2 is 10 so we put the zero down here carry the 1 1 plus 2 plus 2 is 5 it's 15 okay simple enough now if we want to do that with binary it actually works pretty much the same way so we we line up all of our columns and so 28 in binary is 1 1 1 0 0 22 is 1 0 1 1 0 so we've got that converted we line up the columns we just go column by column so 0 plus 0 is 0 0 plus 1 is 1 1 plus 1 in binary is is 2 but we don't have a 2 in binary so it's we write it as 1 0 so the same thing we did here with the 10 here we do it the same thing so we put a 0 down here and carry a 1 and that's how we write a 2 here again we've got 1 plus 1 plus 0 which is 2 so we write that in binary again by carrying the 1 and putting the 0 down here and then finally 1 plus 1 plus 1 is 3 and in binary a 3 is 1 1 and here we we end up with the binary representation of 50 so this the same algorithm works in any base to kind of cool and so now if we want to build a you know a circuit that does this for us really there's only it's actually a lot easier I think then then regular arithmetic because you only need to know a couple facts and this is everything you need to know and so if you can build a circuit that does all of this then you can add a single column because you've got the a and B which is the just the first number in the second number and then you might also have this this one that sort of carries in from the previous so if you've got all that then then we should be able to to add some numbers so let's think about how we would build a circuit that it kind of does this and one way to do it is just kind of break down you know input and output so we have the inputs a and B and then we have these two outputs which are the sum that you put down here at the bottom and then the carry I'll call this the carry bit because you would carry that bit over to the next column so if we look just at the case here where we've got two numbers that we're adding so we haven't carried something in from a previous column and we look just at the number that we're putting down at the bottom here this actually looks a lot like in exclusive-or right because if a or B is a 1 then this last bit here is a 1 if a or b is a 1 but not both so we can actually start to build this right so we can have an exclusive or gate here this is how you draw an exclusive or and we have our a and our B coming in and then coming out is going to be this right here now what about the the second bit the carry bit here well this is also pretty simple because it's always going to be 0 unless a and B are 1 and so that's just a NAND gate so I can have a NAND gate here and gate and just take this same a and B input that we had over here bring it into the and gate and this will be this this bit here that we would potentially carry over into the next column and so this actually is enough to add two numbers this can this covers all of these cases here because we have our a and our B input our two inputs and we have our two outputs here this is the this first output and then this here is the the second output it gets a little bit more complicated now if we have a carry coming in from a previous column because it's not enough to just be able to add to two digits because as soon as we could to this one plus one then we have to carry a 1 to the next column and in the next column we have to deal with three numbers so let's just going to add this case where we have this carry that might be a 1 well in that case again if we look at the last column here it's it's actually the inverse of the case here so really what we want to do is we want to invert this output here whenever the carry bit is set and a clever way of doing that is with another XOR gate because when you XOR something with a 1 it has it will invert it so if we have our carry coming in here we send that into the XOR gate and this will be the sum that we put at the bottom here because what happened if this is a one then you know a0x or one is going to be 1 a 1 XOR 1 is a 0 a 1 XOR a 1 again is a 0 and then here we have a 0 X or a 1 is a 1 so this this will will give us this this last column now for this this first column which is the carry bit that we are the carry you know that we might put into the next column we've got these cases covered right because if a and B is a 1 then we we get our 1 and we even have this case covered here because if a NB is a 1 then we have a 1 here as well but we don't have covered is these 2 cases here where a or B is a 1 and we have this carry and so what we can do is we can actually say well if we have the carry exclusive or A or B we have we already have that exclusive or a and B so if a or B is a 1 and we have the carry then we want this one so we can actually just and the a or b so this is this is this this you know a or B we already have and the carry bit is set then we get this one here so now we have this case now we have this these two cases covered by this output of this and gate oops we want to bring the carry bit in there as well and we have these two cases where it's 1 covered by this and gate and so really what we want to do is we want to say that this bit is going to be a 1 either time this condition or this condition is met so we can use an or gate so this is an or gate this is an and gate this is an XOR so you guys don't get confused here and then this is the carry and so this actually is what is called a full adder because we have a and B so we have two numbers that we're trying to add we potentially have a carry from a previous edition and then we get a sum out which is the the number we put here at the bottom plus we get a carry that goes into next column and so with this sort of simple circuit if we build one of these circuits for each column we can add you know as large a number as we like as long as we keep building you know as many of these circuits as we like and so for example I've drawn up this diagram here which shows that same circuit sort of replicated four times and in this case you have an A and a B coming in here and the carry coming out goes into the carry of the next circuit so this is sort of like you know when you're actually doing the arithmetic here you have a 1 plus a 1 you have a carry it goes in as an input to the next addition and so that carry comes into here along with a new a and a new B which is going to be the a and B for this column for the second column and then you have a something coming out which would be the number down here and then you have that carry going over which goes into the next column plus the new a and the B for that column carry comes out and so forth and so here I've drawn four of them so we can add two 4 bit numbers but we we could keep building more and more of these if we wanted to so to actually build this we can use some some pretty simple components because we have XOR gates and these these chips here actually implement XOR gates and here is the datasheet for them and so you can see from this datasheet basically each of these chips has four XOR gates built onto it and so you know to build a 4-bit adder like I've drawn out here you need one two three four five six seven eight XOR gates so we have two of these chips each one has four we also needed some and gates and so there's another chip that has and gates we need eight of those and so each of these chips has four and gates as you can see on the datasheet so we got those we also need some or gates here we just need four of those so there's another chip that has four or gates so we got that and then a couple other things that we that we need we need a breadboard to build this on so we're going to build this on this breadboard here and for the inputs I have these little little dip switches and so each of these has four little switches on it and that will let us set the for a 0 a 1 a 2 and a 3 for the one number that we want to add and then another one for the B zero one two and three for the other number that won't add together and then finally a couple other things we need are some kind of outputs here so we have our our sums coming out here and so I just have some little LEDs that will light up for each of these digits and I have five of them because there's four outputs plus the final carry which is you know because if you're adding two 4 bit numbers you might end up with a 5 bit output so that last carry will will have another LED for that and then I'm just going to power it with a little bit of a little USB this is just a standard USB charger that I've kind of hacked the end off of and just taking the red and black wires out of there and that will give us 5 volts so I'm going to go ahead and start getting this thing built on this on this little breadboard here and I'll just start by sticking this stuff on to the onto the breadboard okay at this point I've got everything on the board and all the chips are powered so each of these chips has a VCC which is the the positive voltage and then the ground so that each of those is connected for each of these chips everything's on the board so I'll go ahead and just actually just wire up the first of the adders and we play with that one first and then we'll do the rest you okay so I've got the first stage of this wired up I think we'll see we'll power it up and see what happens but this so I'm using the one of the using to the XOR gates on this chip two of the and gates on this chip and then the or gate is over here so we have some kind of wires running back and forth that I guess as I as I do the additional bits hopefully the wires I get a little shorter so it all it all fit but any event this these this is the the sum of the first two bits over here and this will be the carry so let's get the power of n already plugged in so I'm going to just hook this up here and see what happens and that's not good oh I'm not done yet okay now I think I'm done I just have to hook up a couple extra connections here because the switches weren't fully wired in okay so now we add power and everything is off because these first two bits are off and if I toggle either of these on hopefully yes we see 0 plus 1 equals 1 so this is a binary 0 1 so this is the answer there if we turn this off and turn the other one on we see again 1 plus 0 also equals 1 that's good and if we turn both of them on we get 1 plus 1 equals 2 so it looks like this is working so now I'll go ahead and just wire up I'll actually wire up the all of the remaining 3 adders or the remaining 3 adders for the remaining 3 bits here you okay so I just finished this so let's see if it works I've been testing it as I as I go but I haven't tested the whole thing all together but let's let's see how it does if we plug this in we've got zeroes and 0 so we've got all zeroes on the output there so let's see if we want to do some some arithmetic here with this let's imagine we want to add 10 plus 7 and get 17 so in binary this is 10 this is 7 so we would add 0 & 1 then we get 1 and we add 1 & 1 and that's 2 so it's a zero carry a 1 1 and 1 again is 2 so 0 carry a 1 101 again is is 2 so that's a 1 0 in binary so this is what we would expect so let's go ahead and give it a try so 10 is 1 0 1 0 so we can do 1 0 1 0 and already you see the output is 1 0 1 0 because we're adding that to 0 so 10 plus 0 is 0 so that's so far so good and then 7 we can do 0 1 1 1 so if we do 0 1 1 1 there we go we get 1 0 0 0 1 1 0 0 0 1 so it looks like it's working

Channel: Ben Eater

Views: 4,475,055

Keywords: Bit (Unit Of Data Size), Adder, 4-bit, Computer (Industry), Central Processing Unit (Computer Peripheral), Arithmetic Logic Unit (Computer Processor), Computer Hardware (Industry), Electronic Circuit (Literature Subject)

Id: wvJc9CZcvBc

Length: 13min 38sec (818 seconds)

Published: Mon Mar 02 2015