- 5/18/2025
Want to conquer negative numbers in binary? This fun, beginner-friendly tutorial breaks down two's complement with step-by-step examples (-109, -29) and shows how to convert, pad, and subtract in binary. Learn sign bits, carry bits, and avoid common mistakes. Subscribe for more coding tips and tech tutorials that make learning a blast! Scan the QR code for more resources and join our community!
Introduction to Two's Complement 00:00:00
Signed vs. Unsigned Integers 00:00:28
Sign Bit Explanation 00:01:55
Positive and Negative Representation 00:02:06
Range of Signed Integers 00:02:48
Padding Signed Integers 00:05:36
Converting to Negative (Example: -109) 00:07:00
Binary Addition and Carry Bits 00:10:16
Correcting Conversion Mistakes 00:16:38
Converting Negative 29 00:13:32
Subtraction Using Two's Complement 00:18:21
Adding Binary Numbers (109 - 29) 00:20:41
Verifying Results 00:23:56
Conclusion and Call to Action 00:25:40
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Find us on other social media here:
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Please help support us!
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Introduction to Two's Complement 00:00:00
Signed vs. Unsigned Integers 00:00:28
Sign Bit Explanation 00:01:55
Positive and Negative Representation 00:02:06
Range of Signed Integers 00:02:48
Padding Signed Integers 00:05:36
Converting to Negative (Example: -109) 00:07:00
Binary Addition and Carry Bits 00:10:16
Correcting Conversion Mistakes 00:16:38
Converting Negative 29 00:13:32
Subtraction Using Two's Complement 00:18:21
Adding Binary Numbers (109 - 29) 00:20:41
Verifying Results 00:23:56
Conclusion and Call to Action 00:25:40
Thanks for watching!
Find us on other social media here:
- https://www.NeuralLantern.com/social
Please help support us!
- Subscribing + Sharing on Social Media
- Leaving a comment or suggestion
- Subscribing to our Blog
- Watching the main "pinned" video of this channel for offers and extras
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TechTranscript
00:00Hey there, let's talk about representing negative numbers in binary.
00:06We're going to use a system called two's complement, which is going to allow us to represent signed
00:11integers, which means there will be a positive sign or a negative sign on the integer, and
00:16we'll still be able to do it in pure binary.
00:24So let me talk a little bit about what I mean first here.
00:27Okay, so what do I mean by signed and unsigned?
00:29Okay, so if I just type the number 14 by itself, that's an unsigned integer.
00:36We don't really know if it's positive or negative, like do we know that it's negative 14 or do
00:41we know that it's positive 14, right?
00:42So the sign is what lets us know if something is positive or negative.
00:47Okay, so if I type a number in binary, let's just do a pure binary number, 1, 2, 3, 4,
00:535, 6, 7, 8.
00:54So this is 8 bits or just one byte.
00:57If you know how to convert binary to decimal, you probably recognize right away that this
01:01is just the number zero.
01:02Even if we add a couple of bits here, that's just like the number three.
01:07And so this is the number three, but so far, if you haven't learned signed integers in
01:15two's complement or in binary, then you don't really know that there's a sign.
01:19You just kind of assume that the number is positive.
01:22By default, if you're not using two's complement and you're just saying like, well, let's just
01:26do a bunch of binary digits, then yeah, it's a safe assumption that the sign is positive.
01:33But we'll use two's complement, which starts with the idea that the highest bit, the leftmost
01:39bit, the bit with the most power, that's going to turn into the sign.
01:44That's going to turn into the plus or the minus.
01:47So like plus or minus is going to be the sign.
01:55So this means that we have to decide, you know, does a zero mean positive or does it
01:59mean negative or what means what?
02:01So in two's complement, we'll say that zero is positive and one is negative.
02:06So you can see right here that we're still actually representing the number three, but
02:11it's positive three for sure, because this sign bit right here is zero.
02:15That's positive.
02:16Once we decide to represent two's complement, then we can say that for sure.
02:20On the other hand, if we put a one here, then we definitely know that this number is now
02:24negative.
02:25However, we don't actually know that it's a three because the numbers don't work out
02:28the same anymore in two's complement.
02:30So the positive numbers, they will mostly look the same.
02:33The negative numbers, they'll look a lot different, but they'll still be valid in order, you know,
02:37in terms of us being able to add them together, subtract them from each other and things like
02:41that.
02:42So two's complement is pretty cool.
02:44Let's do, let's see, what else can I tell you?
02:47Oh, one thing to understand is that in an eight bit unsigned integer in binary, let's
02:52say 1, 2, 3, 4, 5, 6, 7, 8, let's say unsigned bits.
03:01Maybe I'll just put like, maybe just some like V for value.
03:07All the bits are values.
03:08That means the range is zero to 255 with 256 total combinations.
03:16But if we, if we want to use a signed number, then we'll actually have the sign bit be the
03:22first bit and all the rest will be value bits, 1, 2, 3, 4, 5, 6, 7.
03:27In this case, the range kind of goes down because if you think about it, every single
03:31bit, you know, represents like, you know, it helps towards the highest possible value
03:38that you can represent.
03:40If we're using the leftmost bit, then for an eight bit integer, that's 128.
03:45That's a value of 128.
03:47So we lose a lot off of the maximum integer that we can represent.
03:51So the range here is, I think it's negative 128 to positive 127.
03:58Just keep in mind, you'll have to trust me in terms of why is the negative 128 bigger
04:02and the positive 127, you know, smaller, but that's just the way it is.
04:07So that means we only get these value bits here.
04:09And if you, if you look at it real fast, you can see that, let's see if I can do this quickly
04:15enough without screwing it up.
04:16We'll say 127 minus, minus negative 128, I guess that's 255 and then also a zero.
04:28But in terms of positive and negative, the way this is going to work out as a zero will
04:33probably show up twice because, well, that's just the way it is.
04:39Notice how I can represent 255 possible combinations with seven bits.
04:47Normally you would have a range from negative, or sorry, from zero to positive 255.
04:52And in this case, we're just, you know, losing our range, but we can represent negative numbers.
04:58Anyway, okay.
05:00So let's talk about doing an example real fast.
05:03Let's see.
05:04I've got like a little notes to tell me what I should do.
05:08Okay.
05:09So it's important to understand that normally when you have, let's say, let's say you have
05:14like a, an eight bit number and you go 1, 2, 3, 4, 5, 6, 7, 8, and then you have two
05:20bits right there.
05:21And so this is like, you know, positive three.
05:24If you wanted to copy that number into more bytes, like for example, if you wanted to
05:28take a two byte integer and have it copy the value of a one byte integer, then it's pretty
05:34easy.
05:35Let's just pad zeros to the left, just say 1, 2, 3, 4, 5, 6, 7, 8.
05:40That would work perfectly.
05:42However, if you did this with a negative number, let's say that we have a negative number.
05:46I'm going to put some of the random patterns that you don't think it's the three.
05:50Let's see.
05:511, 2, 3, 4, 5, 6, 7, 8.
05:53That's eight total.
05:54So I'm going to make that a one and then get rid of that.
05:55So we have like eight negative number.
06:00If we were going to copy that negative number from one byte into a two byte integer, then
06:05we would copy paste it to start, but then you'd have to pad with ones to the left, 1,
06:112, 3, 4, 5, 6, 7, 8.
06:15Same negative number, more bits.
06:19So be very careful about how you pad.
06:21If you're padding an unsigned integer, then yeah, you'll always pad zeros to the left,
06:26no matter what is happening.
06:27But if you're padding a signed integer, then you have to pad differently.
06:31When you're padding a signed integer, you have to pad with whatever the highest bit
06:34is.
06:35So in this case, the highest bit was a zero.
06:37So we do pad with zeros, but then in this case, the highest bit was a one.
06:40So we have to pad with ones.
06:41If you don't do that, you're going to end up with a number that doesn't actually make
06:44sense.
06:45Okay.
06:46So now let's work on actually converting a number to a negative number or representing
06:53a negative number in two's complement.
06:54Okay.
06:55So I'm going to write two's complement here, and then let's start off with the number negative
07:01109.
07:02Okay.
07:03How do we do this?
07:05The first thing is convert it to its positive form.
07:08Take the absolute value.
07:09Okay.
07:10So we're really just, you know, take positive form here, and that's just going to be positive
07:15109.
07:16Okay.
07:17No problem.
07:18I'll say start with negative 109, and then we'll take the positive form 109.
07:24Now we'll convert it to binary.
07:26Okay.
07:28This is not a video that teaches you how to convert to binary.
07:31So I'm just going to try to do this in my head real fast here.
07:34It's going to be, let's see, we got eight bits.
07:40We'll use just eight bits to store the number.
07:44And because, you know, it's low enough and I don't want to use that many bits.
07:48So it's odd.
07:49So I can add a one there.
07:50Maybe for now I should find the highest bit that is less than the actual number.
07:56So this is the 128 bit right here.
07:58So that's going to be a zero.
07:59This is the 64 bit.
08:00So I'm going to put a one there and I'm just going to say, maybe like, this is like, maybe
08:08not the smartest way to do it.
08:09I'll say 109 minus 64 because I put a one there and then 45.
08:14So now this is 32.
08:16That's less than 45.
08:17So I'll put a one there and then it's going to be minus 40, sorry, 32.
08:24So it goes from 45 to 13.
08:27So then that was the 32.
08:30Okay.
08:31So 128, 64, 32, and then 16.
08:33Is 16 less than 13?
08:35No, it's not.
08:36So we'll put a zero here and then the next one is going to be, it's going to be just
08:42four.
08:43Four is definitely less.
08:44So I'll put a one bit there and then I'll subtract four and then one, two, four, oh,
08:51wait a minute, one, two, four, eight, sorry.
08:54That was supposed to be subtracted, subtracting eight because that was the eight bit.
09:00Then I want to get five.
09:01So here is the four bits.
09:04So I put a one there and then I'll just put a one here because four plus five is equal
09:08to, sorry, four plus one is equal to five.
09:11So let's see, it is, let me just double check here.
09:21One, let me just double check my conversion real fast.
09:25It's going to be the one bit plus two, four plus eight, 32 plus 32 plus 64.
09:33Did I get 109?
09:34Yeah.
09:35Okay.
09:36So I guess I did it right.
09:37So convert to binary and that's going to be this.
09:41The next thing we'll do is we will invert the bits.
09:43I made a little edit jump here because I inverted the bits incorrectly on the first try, which
09:47is sad, but hey, it happens.
09:49The next thing we'll do is we'll invert the bits.
09:51So basically, you know, we take this original sequence here and I'll just turn every single
09:55bit.
09:56I'll flip it.
09:57I'll flip ones to zeros and zeros to ones.
10:00So I'm going to go one, zero, zero, one, zero, zero, one, zero.
10:03Okay.
10:04So now we have this number.
10:06Invert the bits.
10:07Then we just have to add one.
10:10So we'll add positive one to that number in binary.
10:14And you can tell right away that it's just going to be a one there at the end.
10:18But sometimes that might not be the case because what if we already had a one there and we
10:22had to add one to that?
10:24Well, we'd add one to the right side and it would turn into a zero and then it would carry
10:28a bit to the left.
10:30That would become a zero and then the carry bit would show up all the way over there.
10:34So I just want you to be aware of the fact that when you add two binary numbers together,
10:39you have to be careful.
10:40You have to add them the same way you would add decimal numbers.
10:42For each digit's position, you have to add the two numbers together.
10:47And then if they overflow, then you just kind of wrap around to the lowest number again,
10:52subtracting the highest value, or sorry, subtracting the base.
10:58Like in decimal, if you add nine and nine, the answer is 18.
11:02But you're not going to write 18 in that one position.
11:04You're going to subtract the base, which is 10.
11:07So it'll actually be eight.
11:08And then you'll carry the one over the next position.
11:11So it's like nine plus nine is equal to eight.
11:14Carry the one, right?
11:15So we'll do the same thing in binary.
11:16We'll say, if we ever get a one plus one when we're adding, the answer will be two.
11:21But then we subtract the base, which is two.
11:23So the answer is actually zero.
11:24Carry a one bit.
11:26Keep that in mind.
11:27I'm just going to write it all out for you so you can kind of get a little bit of practice.
11:32It's important to practice this because it's easy to get wrong.
11:36I'm going to put a bunch of dashes up at the top to represent carry bits.
11:41And I'm going to say we're going to add one number plus another number, put a little plus
11:45symbol over there, just to try and make sure we do it the right way.
11:48Okay, so how do we add these?
11:50Well, we'll just go to the right side.
11:52Zero plus one is one.
11:54No carry bit.
11:55One plus zero is one.
11:56No carry bit.
11:57Then we got a couple zeros here.
11:59No carry bit, of course.
12:01One plus zero is one.
12:02No carry bit.
12:03And then a couple zeros.
12:04And then another one with no carry bit.
12:06Okay, so, you know, we could have done that pretty easily, but well, now we're getting
12:11a taste for binary addition that might be harder later.
12:14So we'll just do that for now.
12:17And now this is the two's complement representation of negative 109.
12:26I'll say now we have negative 109 is equal to that.
12:35Notice how the leftmost number, the most powerful, sorry, the leftmost bit, the most powerful
12:39bit is a one.
12:40Remember, one always indicates negative.
12:42So when you're looking at it, if your leftmost bit turns out to be a zero, then you probably
12:47did something wrong or you had an overflow maybe.
12:51And again, if we were going to try to, you know, send this number into a two byte number
12:55or an eight byte number or whatever, then we would just have to pad with the sign bit.
13:00So one, two, three, four, five, six, seven, eight.
13:03I'm going to put a space to help my brain.
13:05So this was, this is the way it would look in a two byte number.
13:07This is the way it would look in a three byte, a four byte and five, six, seven, how many?
13:14One, two, three, four, five, six, seven.
13:16Okay, one more.
13:17This is what it would look like as an eight byte number or a quad word, 64 bit number.
13:22Yeah.
13:23Okay.
13:24So now we know how to do negative 109.
13:27Okay.
13:28So now let's do a number that's a little bit smaller.
13:30Let's do a negative 29.
13:33So let's say convert, or how about represent negative 29 in two's complement.
13:41Okay.
13:42So first we, you know, first get the absolute value.
13:49So just 29.
13:52And then we have to invert the, sorry, we have to get that into binary.
13:55So next convert to binary.
13:59And I'm going to start with zeros.
14:00One, two, three, four, five, six, seven, eight.
14:02For this video.
14:03Remember we're choosing to use one byte integers, but if you wanted to do a bigger one or you
14:07had to do a bigger one, then just, you know, keep that in mind.
14:11Okay.
14:12So 128 is not a smaller than 29, 64 is not, 32 is not, 16 is though.
14:18So I'll put a 16 bit there and I'll just subtract a 16 from 29, 29 minus 16.
14:24Now we've got 13 left.
14:27So six, four, 32, 16, eight.
14:31Okay.
14:32So now I'm going to put a one bit there and I'm going to subtract eight from the remainder.
14:37And then we got a five, which is pretty easy to do.
14:39Eight, four, and then a one.
14:42So now we have a, well, a zero, zero, zero, one, one, one, zero, one.
14:48That's the binary number, the positive or unsigned representation.
14:54AKA just positive or representation.
14:58So now we'll add one.
15:02Positive one.
15:04And let's try to do this the right way so that we can practice carry bits with addition
15:07a little bit.
15:08Notice how this one is already there on the right side.
15:10So it's going to, we're going to have at least one carry bit for sure.
15:13So then I'm going to go doop like that.
15:16And then I'm going to say that we have like, you know, what is the result, put a positive
15:20sign there.
15:21I'm going to put a bunch of dashes just to remind me to do my carry bits because I can,
15:25I can forget that pretty easily.
15:27So the first thing is we add one and one.
15:30The answer is two, but we can't put the number two here because it's binary.
15:34Instead we need to subtract the base, which is two.
15:37So two minus two is equal to zero, but then we have a carry bit of one.
15:41So I'm going to put a one there.
15:43The first carry bit will stay as a dash for this whole exercise because you're not going
15:48to carry on to the first digit.
15:52Okay.
15:53So now we have what would have been just zero plus zero.
15:54And now we have one plus zero plus zero.
15:56So that means this is going to be one and then the carry bit is just going to be zero
16:01because we don't actually carry anything.
16:03So then we have zero plus one plus zero.
16:05So that's going to be a one.
16:06And then a zero carry it.
16:10And then a zero one, zero is just going to be one.
16:13And then there's going to be no carry bit.
16:14And in zero one, zero again, no one, no carry bit, because we didn't actually overflow.
16:19then just zeros. So 0, 0, 0 and then I'm just going to put zeros here. Okay, so now we have
16:26successfully added... I blew it. Totally blew it. I always forget steps. Don't forget the steps.
16:36This is a good lesson. I'm going to leave this in the video because I want you to see that
16:40everybody makes mistakes and you got to practice, practice, practice,
16:43especially before you have to actually do this in real life or for a quiz or an exam or something
16:48like that. Next, convert to binary. Before you add one, I'm going to just remove this. Oh my gosh.
16:58Next, flip the bits, which is going to be 1, 1, 1, 0, 0, 0, 1, 0. Okay,
17:06so now we take this bit flipped number and we will add one. So that's going to be
17:130, 0, 0, 0, 0, 0, 0, 1. Oh, it's too easy. Maybe I got excited and I thought, oh, it's carry bit time.
17:19But even though the last addition that I did was wrong because I forgot to carry the or flip the
17:23bits, you still at least saw a little bit about how to carry the bits, right? Okay, so it's just
17:29going to be 1. Let me start from the right side. 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1. Let me just double check here.
17:391, 1, 1, 0, 0, 0. Okay, so I got that. Now we have
17:46negative 29 in 2's complement. Again, notice if we actually tried to add those numbers up
17:53to be like an unsigned binary number, they're not really going to make sense because this is
17:57like 64 plus 32 plus 3. So what would that end up being? Let's just double check here.
18:0264 plus 32 plus 3, 99. That's not actually the number, but the number is 29. So
18:09keep in mind, you can't just look at this and know what number it is unless you're like really,
18:13really practiced. Okay, and again, notice that the number is 1 at the very left, indicating that
18:19it's a negative number. Now let's look at how to subtract one number from another using 2's
18:27complement. Okay, so what I want to do is I want to subtract, let's say, 29 from 109. Okay,
18:36so let's subtract and I'll just say 109 minus 29. And how I would do that is basically I'll start
18:46by just taking 109. Let's see, convert to binary. So I'm just going to copy paste that number.
18:52So 109 is this, and then 29, take the positive version, it is just, let's see, before we flip
18:59the bits or anything, let me make sure that I grab the right one, invert the bits. Okay,
19:04so positive 29 is this number. Whoops, let me put parentheses around that so it's easy to tell,
19:13and then I'll put a positive sign there. Like that. Maybe like that. Nope, nope, nope, nope,
19:22nope, nope. How about this? Okay, so now we have both of these numbers in positive form. So now
19:28if we added 109 plus 29, that wouldn't be the problem that we're trying to solve.
19:32If we added 109 plus negative positive 29, that would be what we wanted, right? Because really,
19:40if you're subtracting, I'll say aka positive 29 plus negative 29, all we really need to do
19:51is invert the 29 and then add the result to 109. So that means we'll turn positive 29 into negative
20:0129 using two's complement. I'm not going to go through the steps again, but basically,
20:06I'll put it there, but basically, you know, that's negative 29. So say positive 129 is equal
20:15to what I just put up here. And then negative 29 is equal to, maybe I should do the parentheses
20:22again for clarity, is equal to this. Okay, so you can tell that positive 29 is pretty,
20:31pretty different from negative 29, but now we have both of those numbers. So let's see,
20:35zero, one, one, zero, and then we're ready to add. Okay. All we got to do is add them together.
20:42Next, add them together. Maybe I should write the steps up here on top.
20:49Positive versions of both, whoops, converts 29,
20:57positive 29 to negative 29 using two's complement and then next add them together.
21:07So then I'm going to copy paste the bits here and it's going to be this plus this
21:15and do a positive plus sign just to remind ourselves that we are actually adding.
21:19And then I'm going to put a bunch of placeholders for sign bits up at the top.
21:23And now we'll have a little bit more fun adding numbers together. Maybe I'll drag this down.
21:27Oh my God. Oh, there we go. Okay. So I'll start with the one on the right,
21:37the position on the right. That's going to be one plus one equals two, but then that's an
21:42overflow. So I'm going to subtract the base. So it's going to be zero. And then don't forget to
21:47carry the one. Oh, cool. More interesting. So we have one plus zero plus one. That's going to be
21:52another two carry the one. I'm sorry. That's going to be another two minus two carry the one.
21:56So it's going to be zero and then carry the one again. So I'm gonna put the carry bit up there.
22:01And then again, we have one plus one is equal to two. So it's going to be zero carry the one
22:07again, zero carry the one. And then finally, we don't really have a carry bit.
22:13So we'll just have like a one and there's, there's no carry. So it's going to be, you know,
22:18carry a zero and then we add these two together. So it's going to be a zero carry the one. And
22:23then now we have a three. Oh, that's kind of nice. So this is an interesting, uh, edge case,
22:29kind of one plus one plus one is three. But if we, uh, subtract two, the base from it,
22:34you know, three minus two, it's going to be one, not zero. So it actually is going to be
22:39a one and then carry the one on top of that. Uh, then for here, uh, let me space this over
22:46a little bit so that I can illustrate what's going on a little bit better. We're going to
22:50have one plus zero plus one. That is definitely going to be zero and then carry the one,
22:56but there's no bit where that carried one can, uh, can show up on, right? So that one overflows,
23:02it falls off the edge. If this was a bigger number, uh, then, okay. Uh, we, you know,
23:09if we had more bits to this number, then sure, we would just keep carrying over, over to the left.
23:14But remember we said before that when we have a very big number, let's see
23:18if we have like a negative number with more bits, it's just ones all the way to the side
23:23that will actually help us make sure that, uh, if our final number is actually going to end up
23:27being positive, that everything kind of like dominoes, like carry the one, carry the one,
23:31carry the one, carry the one, carry the one all the way until one of the ones falls off.
23:37Think about it. So anyway, this one just is gone. We don't really care about it anymore.
23:42The result is going to be just only eight bits because that's the number that we started with
23:46zero one, zero one, and then a bunch of zeros. Uh, what is, you know, the final answer?
23:54Let's just compute this real fast to decimal. So this is one 28 and then 64. So it's going to be
24:0064 plus not 32, but 16. So 64 plus 16, that's going to be oops, 16. That's going to be 80.
24:11And now we just have to ask ourselves again, as like a final step to double
24:14check yourself, to make sure that you understand what you're doing
24:17and that you got it right is, uh, just punch up one Oh nine minus 29, just to make sure
24:24one Oh nine minus 29. Whoops. What happened here? One Oh nine minus.
24:32Oh, I think I stole my subtraction key for the annotator. One Oh nine minus 29 is 80.
24:39So again, you know, if you're, if you're trying to like, you know,
24:41write something down to do some work at work or, you know, taking an exam or something,
24:46you definitely want to double check yourself in several ways.
24:49Uh, as you can tell from this video alone, I got, uh, one of these things wrong. Cause
24:53I forgot to input the bits as a step before adding one. Um, so, you know, your final step
24:59should be actually trying to add two numbers together or subtract numbers or whatever you're
25:03doing to make sure that you got the binary correct. So let's see.
25:14I guess maybe your first indication that the result was going to be positive would be that
25:18there's a zero there. And just, you know, as a sanity check, you look at the top and you're
25:22like, well, I was going to subtract a small number from a larger number. So the result
25:27should probably be positive, right? Like 29 is like way lower than one Oh nine. So it should
25:33be positive, which means the final result should have a zero at that left-most position.
25:38Okay. So that's a two's compliment, how to convert numbers from a positive to negative
25:44in two's compliment. Uh, you know what the sign bit means and all that stuff
25:48and how to perform subtraction via two's compliment. I hope you enjoyed this video.
25:53Thank you for watching. I hope you learned a little bit of stuff and had a little bit of fun.
25:57See you in the next video.
26:00Hey everybody. Thanks for watching this video. Again, from the bottom of my heart,
26:04I really appreciate it. I do hope you did learn something and have some fun.
26:08If you could do me a please, a small little favor, could you please subscribe and follow
26:13this channel or these videos or whatever it is you do on the current social media website that
26:18you're looking at right now? It would really mean the world to me and it'll help make more videos
26:23and grow this community. So we'll be able to do more videos, longer videos, better videos,
26:27or just I'll be able to keep making videos in general. So please do, do me a kindness and, uh,
26:33and subscribe. You know, sometimes I'm sleeping in the middle of the night and I just wake up
26:37because I know somebody subscribed or followed. It just wakes me up and I get filled with joy.
26:41That's exactly what happens every single time. So you could do it as a nice favor to me, or you
26:46could, you could troll me if you want to just wake me up in the middle of the night, just subscribe.
26:49And then I'll, I'll just wake up. Uh, I promise that's what will happen.
26:53Also, uh, if you look at the middle of the screen right now, you should see a QR code,
26:57which you can scan in order to go to the website, which I think is also named somewhere at the
27:01bottom of this video. And it'll take you to my main website where you can just kind of like,
27:06see all the videos I published and the services and tutorials and things that I offer and all
27:11that good stuff. And, uh, if you have a suggestion for, uh, uh, clarifications or errata or just
27:20future videos that you want to see, please leave a comment, or if you just want to say, Hey, what's
27:23up, what's going on? You know, just send me a comment, whatever. I also wake up for those in
27:28the middle of the night. I get, I wake up in a cold sweat and I'm like, it would really,
27:33it really mean the world to me. I would really appreciate it. So again, thank you so much for
27:37watching this video and, um, enjoy the cool music as, as I fade into the darkness, which is coming
27:45for us all.
28:15So
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