[Intro music playing] So, welcome to this TeachElectronics.com presentation

of Binary to Decimal Conversion. What we’re gonna do is we’re gonna teach you

how to convert, and there’s only one learning objective: at the end of this lesson, you’re

gonna be able to convert binary numbers to their decimal equivalents. So take a look

at the examples at the bottom. Notice that there are five decimal numbers:1, 7, 10, 64,

and 100. And over to the right are their binary equivalent. So the number on the right, seven

zeroes and a one, that’s the same equivalent, that’s the same number as 1 in decimal. If

I were to write the second number down in binary as 111, those three ones represent

the number in binary that is decimal 7. So what we’re gonna do is we’ll pick a couple

of random binary numbers, and we’ll walk through how it works. Now you probably noticed in

the binary column that there’s a lot of leading zeroes. And you also know that in the decimal

system, we don’t really use leading zeroes. Writing 00046 is not the way we write the

number 46. Over on the right are the same numbers as the binary numbers in the center

column, but they have no leading zeroes. And we’re going to see a lot of leading zeroes,

so let’s talk about how that works. In computers, most binary numbers, and we don’t store decimal

numbers — there’s no place to store a 7 or a 6 or a 4 in a computer, because the only

thing you can store in a computer are the numbers 1 and 0. So you have to figure out

kind of a secret code that changes the numbers we know of as 6 and 4 and 3,000 into only

ones and zeroes. And, of course, that’s the binary code that you see here. In computers,

binary numbers are always stored in multiples of 8 bits, or 8 digits. I’m gonna interchange

the word bit and digit; they both mean the same thing. It’s one of the digits of a binary

number. Any number in a computer is stored as a sequence of ones and zeroes, but they’re

always stored in multiples of 8. We either store 8 bits or 8 ones and zeroes, or 16 bits

or 32 bits. We usually store a number of bits divisible by 8. When we store the number 1

or 7 or 10, in binary, which is the third column over on the right, when we think about

it, we don’t think about those leading zeroes, but when we store it, we’re gonna store them

in a storage place that requires 8 ones or zeroes, because each of those eight memory

storage places has to be either a one or a zero. So we use those leading zeroes to say

there’s no 1 here; there’s no 1 here, there’s no 1 here, but they still have to be there

because you can’t have nothing in a computer. I can’t store seven nothings and a one. I

can only store eight ones or zeroes or any combination of eight ones or zeroes. What

we do is we put the leading zeroes in whenever we store any number that has less than eight

bits. And the same thing with 16 bits. If we have 16-bit numbers and our number is less

than that, we zero-fill, or left-fill as zeroes, any binary numbers to get to that even number

of eight or sixteen. Although the numbers on the right are the real binary numbers,

you’ll often see them with the leading zeroes, meaning that this is how we’re gonna store

it in a computer. So get used to seeing both, or either. They’re both correct. I’m gonna

use the leading zeroes in this presentation, because we’re going to act like our binary

numbers are gonna be stored in a computer, so I want you to get used to seeing ’em. Well, we’re gonna talk about column weights

in this presentation, so we need to understand what that means. What I’ve done, I’ve put

the decimal number 101 and the binary number 101 up on the screen, so I wanna talk about

that a little bit. You all know if I tell you what does the number 101 — how much is

that?– that’s really easy. You’re gonna say one hundred and one. You’re gonna say that

because you understand the decimal system. You’ve been trained since you were a tiny

child how the decimal system works, and you’ve been trained with column weights. You’ve been

trained that the column weight of the 1 on the right is one — it represents one thing,

or one unit, or one dollar, or one of whatever we’re counting. The zero in the center, you’ve

been trained to realize that that zero is really the tens column, and this tells you

how many tens are in the number. In this particular number, there are no tens. And of course,

in the third column, you’re taught that this column, a number in this column, represents

the number of one hundreds in your number, so in this case, you can look at that number

and say there’s one one-hundred, there’s no tens, there’s one one. So we have the number

one hundred and one. Binary counting over on the right isn’t anything like that. We

don’t use column weights of ten; we use column weights of two. So our column weight columns

go one, two, four, eight, sixteen, thirty-two, instead of one, ten, one hundred, one thousand,

ten thousand, and so on. So you need to get used to that column weight system and realize

that number 101 is a different quantity than the decimal 101. And another thing we do quite often is we

provide subscripts. I’m gonna put the subscripts up here. You can see the subscript in decimal,

“10,” and the subscript in the binary number of “2.” Here’s what that means: The subscript

“10”says, “I am a base-ten number. I am a decimal number, and the decimal numbering

system has a base of 10. Because everything, each of the columns is a multiplier or 10.

So it’s a base-10, and we have ten numbers that we use to count. On the right-hand side,

you’ll see the subscript 2 on the 101, and what that tells you is this is a base-2 number.

Everything in this system is based on powers of two instead of powers of ten. The first

column on the right represents a one; the next column to the left represents a two,

then a four, then an eight, then a sixteen, and so on. We’ll talk more about that in a

minute. For right now, just so that we’re not confused, when I show you the number 101,

I’m gonna put the 10 up there, or the 2 up there, to remind you that the 101 base-10,

or the 101 subscript 10, is a decimal number, and on the right, the 101 base 2, or the 101

subscript 2, represents a binary number. How do we determine what a number is? The decimal

is easy. Again, you’ve been trained in that and taught on that every since you were born.

Now, you probably weren’t taught exactly this way, but I’m gonna lay this out for you in

a powers-of-ten format. What’s in the green box on the left represents the third digit.

What’s the third digit represent?And we can see here it says one times ten squared, or

one times ten to the second power. Well, ten squared is one hundred, so one times one hundred

is one hundred. So in decimal, a 1 in the third position from the right represents the

quantity 100. If it were a 2 inside the green box here, we would write “two times ten squared”

which would make it worth 200. That’s the column weight. The column weight of this column

is in the hundreds. Well, looking at the center column, in this case, there’s no tens, and

that’s ten to the one. This says zero times ten to the one, meaning it’s a zero because

that’s what’s in that digit, the zero. And we multiply it times ten because it’s the

ten column, and ten to the one, or ten to the first power, is ten. So this says zero

times ten is zero, so we don’t really add anything, so so far we’ve come up with one

hundred from the first digit, nothing from the second digit, and the third digit, the

one, or the first digit on the right, represents one times ten to the zero, and ten to the

zero is one, so one times one is one. So the third digit is a one. This tells us that we

have one hundred plus zero plus one, or one hundred one. That’s how we figure out what

the number is, or how we figure out how large the number is, or what its quantity or size

is. Well, in the binary numbering system, it’s exactly the same. The only difference

is notice that the exponents are still the same: two and one and zero. Notice that the

one times, or zero times, are still the same. But notice the number now becomes TWO. TWO

to the second power. TWO to the first power. TWO to the zero power. Why? Because it’s a

base-2 system. Later, when we learn octal, which is a base-eight system, those twos are

all gonna get replaced with eight. And later, we’ll get into the hexadecimal system, which

is a base-16 system, and we’ll replace all of those twos with the number 16. So they’re

all the same; it’s just that each column has a different weight based on the power of the

base. This one in the third column says “We’re base 2 (that’s the two)” and two squared is

two times two is four, meaning that the third column has a weight of four. So one times

four is four. So the one in this column represents the quantity four. Well, we move over here,

same idea with zero — there’s nothing — and this happens to be the two-to-the-ones column,

or the twos column. This says there’s no twos, ’cause zero times anything is zero. So there

are no twos in this number, and then lastly, the far-right-hand digit is a one, but it’s

two to the zero, and anything to the zero is a one. So one times one is one. So what

this number says is “add one times two squared, which is four, plus nothing, which is nothing,

we still got four, and this last equation over here, one times two to the zero represents

the number one.” So this means four plus one. So when we do that, we lay it out. I’ll lay

’em both out here for you. We’ll say that the decimal number 101 represents 100 + 0

+ 1, or what we know of as one hundred one. The 101 in binary means something totally

different, because we used different column weights. And this number says 4 + 0 + 1, or

5. In the decimal system, the number 101 represents one hundred one. In the binary system, and

I made a mistake there — it’s a 10. That should be a two. Should be 5 with a 2 subscript,

made a mistake. So the 5 subscript 2 represents that this number is 5 in the base-2 system. So that’s how the base systems work. Let’s

figure out how we change the numbers. We start out with a blank piece of paper. Start out

by writing the number 1 at the top right, as I’ve done here. On here, I’ve made it brighter.

I’ve dimmed the numbers that are coming up. After you write 1, go to the left of it, write

a 2, and then you’re gonna keep doubling the number all the way as far to the left as you

wanna go. Huh. So we double the 2, we get a 4. Double 4 we get 8, double 8 we get 16,

double 16 we get 32, and we continue doing that. Now, how long do we do it? Well, since

we’re working with 8-bit numbers, we’ve gotta have 8 numbers up top. We have to have one

to represent every bit in the 8-bit number. So as you can see, the 1, 2, 4, 8, 16, 32

represents six numbers, but we’re gonna use 8 bits, not 6 bits, so we’re gonna write 2

more numbers, so we’ll double the 32 to 64, double it to 128. Now, these numbers will

stay at the top of the page and we’ll write our binary number under it and then figure

out what its value is. Pick a number, I just picked a couple of random numbers. Down at

the bottom of the page, you’ll see 10101010. I just picked that ’cause it’s a nice pattern.

Okay, and I picked the one to the right ’cause it’s just a number. It’s ones and zeroes.

So what we’re gonna do is we’re gonna take each of these two numbers. We’re gonna walk

through the conversion process so that you can easily figure out what the number 10101010

base 2 represents in the decimal numbering system. So here we go. Well, let’s work on the first number, the

10101010. That’s the number. We’re trying to solve the problem, what is that number

worth in the decimal numbering system? Remembering that each of the digits in a binary number

has a specific column weight. What that 1, 2, 4, 8, 16, 32 up top says is the bit, the

farthest bit on the right in an 8-bit binary number, has a column weight of 1. The next

number, the 2, means the next number over, the second from right in a binary number,

has a column weight of 2. The next bit a 4 and the next bit 8. And so one. So we’re gonna

walk through one bit at a time and we’re gonna add up those numbers to see what our total

is. We’re gonna add up the column weights and you’re gonna notice these zeroes. We’ve

got some zeroes in our binary number. We’re not gonna add anything if we hit a zero. So let’s start with the right-hand bit. The

right-hand bit is a 1, and down at the bottom, in our binary number, it says, well, the binary

number that we’re solving for has a zero in that bit, so there are no ones in that number.

So we don’t add anything to our total. We start our total at zero, of course. And we

say, well, how much do we have to add up to get to the number we’re looking for? This

zero in bit position one tells us, well, there’s nothing here, there’s no ones. So let’s move

to the next bit. Well, the next bit is a one, and if we look up top, we see that that’s

column weight of 2, meaning there is a 2 in this number, so we need the quantity 2 to

be in our number. So we’re gonna start adding numbers up, and the first number we’re gonna

add is 2. We’ll put a box over on the right, and we’ll say, okay, because there’s a 1 in

the second bit, and the weight of the second bit is 2, we know our number, our total, has

a 2 in it. So we’ll place the 2 in the box. And we’ll keep looking for numbers in this

box until we’ve added up whatever we need to get the correct total. Now, as we move

left from the second bit, you’ll see we have three zeroes in different positions in the

8-bit number. We can ignore those, because if the zero’s under a 16, it means there’s

no sixteens in the number, you don’t have to worry about it. If the zero’s under a 32,

then you say, well, we don’t have to worry about that. So we can ignore the zeroes and

just look for the value of the ones. So here’s one. Now, this one is located in position

4, if I’m counting from the right, one two three four. It’s the fourth number over from

the right. It’s a one. Well, what’s the fourth number at the top of the page? It’s an 8.

This column, or this one, represents the column weight of 8. So that means there’s an 8 in

our number that we have to add. So we’re gonna add that 8 to our total. So far we’ve got

a total of 10. We’re gonna keep going as long as we have ones, and we’ll eventually get

the correct total that’ll gie us the sum of all these numbers, therefore, what its decimal

value is. Well, the next one over is in the sixth position

from the right, and if there’s a one in the sixth position from the right, it means that

that’s the 32 column-weight column, so we’re gonna need to add 32 to our number to find

out what the total is. Let’s move the 32 into our sum. We’re now — we’ve now summed three

numbers — the 2, the 8, and the 32. We only have one more one, so we can move to that

one, and we can say the last one in the number, the far-left bit, the eighth bit of a binary

number has a column weight of 128. Therefore, we need to add 128 to our number. So when

we add 128, we wind up with a grand total of 170, and 170 is the correct answer for

what is the decimal value of this binary number. So down the bottom, we can replace our answer

of three question marks with 170 subscript 10, meaning in the decimal system, that’s

the subscript 10, this number has a value of 170. So we’re done filling out this number.

Now, I did it kind of the long way. When we do the next number, I’ll show you a shorter

way. And it’s quicker once you get used to it. But I wanted to get you to understand

that each column weight goes up:1, 2, 4, 8, 16, 32. If I was teaching the decimal system

right now, guess what I’d do? I’d write 1, 10, 100, 1000, 10000, 1000000, okay? I’d keep

adding a zero to each of the number, or multiplying by 10 each time. That’s what the decimal number

means, base 10. Each column goes up times ten. This is a base-2 system. Each column

goes up times two. So as long as we write these numbers down, 1, 2, 4, 8, 16, 32, 64,

and we put a 1 or a 0 under each of those numbers, we can figure out what the total

of the number is. Let’s take a look at the right-hand number

on the bottom, 00110110, subscript 2, meaning it’s the binary number, so we have this 8-bit

binary number. It’s got 4 ones in it. What columns do those 4 ones represent? Well, the

two ones on the right represent the 2s column and the 4s column. Remember now, if this was

decimal, that would be the 10s column and the hundreds column. You need to be able to

shift back and forth between the powers that we’re talking about. Right now, we’re talking

powers of 2. So we know we have a 1 in the 2s column, 1 in the 4s column, 1 in the 16s

column, and one in the 32s column. So of course, what that means is to find out what this number

is, add the numbers 32, 16, 4, and 2. And we’ll have our total. And so, sure enough,

if we add up 32, 16, 4, and 2, we’re gonna get the number 54. Two plus four is six, plus

six is 12, plus 2 is 14, carry the 1, 3 + 1 + 1 is 5. So the number is 54. We can now

at the bottom right of the screen replace those three question marks with the number

54, meaning 00110110 is the equivalent of decimal 54. That’s it! That’s all you have

to do to convert binary to decimal. Write 1, 2, 4, 8, 16, 32, and so on up top for however

many bits are in the binary number. Write the zeroes and ones underneath it, and every

number that has a 1 underneath it, add that to your total, and you’ll come up with a winner

every time. So here, I wrote the actual ones and zeroes

right underneath the number, and that’s probably how you’d do it, if you were doing it on a

piece of paper. I did it ’cause it was easier to keep the slide straight if I did it this

way. But if I was doing it on a whiteboard right now, I’d write those numbers on the

board, 1, 2, 4, 8, 16, 32, 64, 128, and then underneath it, I’d write 10101010 and I’d

say “What does that represent in decimal?” And you could just look at that and say, well,

oh, I got it! I’d take a 128, ’cause there’s a 1 under it, I add a 32, ’cause there’s a

1 under it, I add an 8 and a 2 because there’s ones under it, and I don’t add any of the

other numbers, and if I add all those numbers up, I’m gonna get 170. I’m done. It’s that easy! Just run out the ones and

zeroes, add up all of the column weights in the top that have a 1 under ’em, and you got

a winner. So let’s try that second number. Here’s the second number. The 00110110. I’ve

written that one out the exact same way. But on this one, you can say, “Well, this is easy,

too. I got one 32, plus one 16, that’s, it looks like 48, plus one 4, that’s 52, plus

one 2, that’s 54. If I add those four numbers up, 32, 16, 4, 2, I get 54, and I’ve got the

correct answer. That’s it. That’s all there is to it. It’s that easy. And you’ll find,

when we get to hexadecimal and octal, it’s the same idea. We just put different column

weights at the top of the screen, and we’re still gonna get the right answer each time. So here’s the deal. Time for you to practice.

I’m gonna have a slide that pops up next and it’s going to have five 8-bit binary numbers

on it. Now what I want you to do is pause the slide, figure out what the answer to each

of those 8-bit binary numbers is, write the answer down, and while the video’s paused.

When you’ve done and got all your answers, just forward the video and in about 5 seconds,

I’ll walk through the answer to each one of the problems. So what’s gonna happen when

the next slide pops up, you’ll see on the screen a countdown, that does 5, 4, 3, 2,

1, 0. If you pause it at any time while it’s counting down, you’ll pause the video. You

can try and work out the numbers for yourself. Okay? If you don’t choose to do that, if you

just wanna go right to the answers, just let it scroll for five seconds, and then the answers

will pop up and I’ll walk you through each answer. So, you got the idea? Next slide,

pause the video before the countdown gets to zero, and you can work ’em out for yourself,

or just let the video run, your choice. Here we go. There’s your 5 numbers. I’m gonna count down.

There’s 5, 4, 3, 2, 1, and 0. So now I’ll give you the answers. Okay, so here’s the

first number. 00000101. Pretty easy. There’s only two ones. The far-right-hand 1, if I

were to write all the ones and zeroes up top, write underneath the column weights that I

showed here, I’d have a 1 under the number 1 and a 1 under the number 4. So what’s the

answer? Easy. 4+1. Next number. Well, if I wrote that 8-bit number one bit at a time

underneath the numbers at the top of the screen, I’d have a 1 under the 64, a 1 under the 16,

a 1 under the 4, a 1 under the 1. If I added all of those guys up, I’d get 85. See the

number 85 over on the right. That’s how I got the decimal equivalent, or that’s how

I got the number that 01010101 represents. Pretty easy, I think. I hope you see that

now. Well, the next number. Again, let’s see, we got 4 ones, and if we lay out the ones

underneath the column weights up top, we’re gonna have a 1 underneath the 128, the 64,

the 8, and the 4, and way over on the right in green, I add those 4 numbers up, and this

represents the number 204. Next number. Only 3 ones here, and they would be under the number

128, 16, and 2. What’s the answer? 128 + 16 + 2 or 146, so this 8-bit number represents

the decimal number 146. And the last one, of course, worst case, what happens if everything’s

a 1? Well, if everything’s a 1, I’ve got a 1 under every single number, gotta add ’em

all up, and it turns out when I add up 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1, I get 255.

So that’s what 8 ones represents in decimal, when I write 8 ones in binary. That’s is.

That’s how it works. So, write your numbers up top, put the ones and zeroes underneath,

starting at the right, line them up, one number per column, one bit per column, keep ’em lined

up correctly, and then if you just add up those top numbers that are on top of ones,

you’ll always get the correct answer. Here’s the procedure. you could take a picture

of this slide if you wanted to. You could write this down, pause the video and write

it down, whichever is the most convenient for you. You may already remember and know

exactly what this procedure is and you’re good. So whatever’s best for you, but I’ll

walk through it one time quickly. 1. At the top of a piece of paper, write the

number 1. That’s what we’ve done. Keep doubling it to the left until you have

as many numbers as the number of ones and zeroes in your binary number. So if you only

have a four-bit binary number, like 1010, you only have to write 1, 2, 4, 8, because

you don’t need to go any farther than the number of bits you have. Step 3. Place the

right-hand bit — and that’s important. Gotta be the right-hand bit, not the left-hand.

Place the right-hand bit of the binary number under the one on the right, up at the top

of the page. Continue placing, step 4, each one or zero of the binary number under the

decimal numbers going off to the left. Keep the bits in their original order. When you

read the numbers back underneath these red numbers at the top, the ones and zeroes should

be the same sequence that you were reading when you put ’em up there. And of course,

when you’re done, go to every number up here at the top in the top row that has a one underneath

it. Add those numbers up. That’s gonna give you the sum. That sum is the number. So I hope that this has helped out. I hope

that you understand this. If not, you can certainly email me. I’m [email protected]

I’m also [email protected] Either one, you can email or you can go to the TeachElectronics.com

site and find a place to ask a question. So, glad to have you. I hope you now understand

better, or really well, how to convert binary numbers into decimal. Thanks for watching. [Outro music playing]