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## Computers and the Internet

# The binary number system

Binary numbers form the basis of computing systems. Binary numbers contain only the digits 0 or 1, or bits, where each bit represents a power of two. To convert binary to decimal, multiply each bit by its corresponding power of two and add the results.

. Created by Pamela Fox.## Want to join the conversation?

- 5:21"It took us 8 bits to represent a number that only took 2 digits to represent in the decimal system."

01010101 Uses 8 bits, but I can make it with 7: 1010101.

So if we can add 0's in the front, then can't I just say,

"It took us 8 bits to represent a number [01010101] that takes a whole 10 digits to represent [0000000085] in the decimal system."

It would be nice if the video at least pointed out that you could do it in 7... anyone with me?(15 votes)- From the author:That's a really good point. YouTube doesn't make it particularly easy to edit videos, but I'll try to edit it if I get the chance. I very much agree.(18 votes)

- How many bytes does it take to equal one KB(1 vote)
- Why do they leave the 0 there, typically its not supposed to do that, like I wouldn't say 010, I would just say 10.(0 votes)
- Binary numbers are generally only used in the context of computers which use a fixed number of bytes/bits to represent numbers. For example, a computer may use 32 bits to represent the number 13 even though 13 would only need 4 bits. This leaves 28 preceding 0s in the binary representation of 13. So, in general, we are less concerned about getting rid of preceding 0s when we write out binary numbers.(10 votes)

- *Humming intesifies(2 votes)
- how can represent the number 10389810 in binary language(2 votes)
- 100111101000100100110010(0 votes)

- how do you figure out how many bytes does it take to equal one KB?(1 vote)
- How we can represent 53426 into binary numbers??(1 vote)
`53426`

= 2^15 + 2^14 + 2^12 + 2^7 + 2^5 + 2^4 + 2^1

= 1*2^15 + 1*2^14 + 0*2^13 + 1*2^12 + 0*2^11 + 0*2^10 + 0*2^9 +

0*2^8 + 1*2^7 + 0*2^6 + 1*2^5 + 1*2^4 + 0*2^3 + 0*2^2 +

1*2^1 + 0*2^0

= 1101000010110010(1 vote)

- how do you figure out how many bytes does it take to equal one KB?(1 vote)
- 1 kilobyte = 1000 bytes. Kind of like how one kilometer = 1000 meters.(1 vote)

- If you're making a binary number can you just make them all ones? Like if I wrote "1 1 1 1" would that be equal to 20, or would that not work, the numbers need to be ones and zeros?(1 vote)
- A binary number can consist of entirely 1s, so 1111 is a valid binary number. (However, 1111 is not equal to 20; it equals 15.) In fact, any number equal to 2^x - 1 (for any positive integer x) will be entirely 1s when written in binary.(1 vote)

- How are negatives represented in the binary system? Does the negative symbol (and other symbols and special characters for that matter) have its own binary code?(0 votes)
- To write a negative number represented in binary, we simply write a negative sign in front of it, like normal. Of course, computers can only store 1s and 0s so they cannot store a negative sign. Instead, computers can either use a single bit to represent a positive/negative sign, or use 2's complement representations.(5 votes)

## Video transcript

- [Instructor] The binary number system works the same way as the
decimal number system. The only difference is what
each of these places represents. This is a four digit binary number 1010. Each of these digits can
also be called a bit, since a bit represents zero or one. Now this four digit number
equals the decimal number 10. And we're going to see why. This first place, this is the ones place, just like in decimal. Now there's zero here, so that means we're gonna multiply zero times one, get the very exciting value of zero. The second place, the second bit here, that's where things get more interesting. This is actually the twos
place, not the tens place, and there's a one here, so we're gonna multiply one times two. So so far, this number equals two. The third place, the third bit, this is zero and this is the fours place. So we're going to
multiply zero times four, which is once again just zero. And we're still looking at the number two. Finally, this fourth bit here,
this is the eights place, and there's a one here. So we're gonna multiply one times eight. Add that to everything else, we end up having eight plus two, which equals decimal 10. In the decimal system, each of these places
represented a power of 10. But in the binary system, each place represents a power of two. The first bit is two to
the zero power, it's one. The second bit is two to the
power of one, the first power, which is two. The third bit is two squared,
two to the power of two, which is four. And this fourth bit is two cubed, two to the power of three, that's eight. That's the only difference
between decimal and binary, what each place represents. Once we know that, we see
what digit is in each place, we multiply them together,
and we get our final value. Now let's try to convert another binary number to decimal. We can erase this work here. And then for the number,
let's just flip the bits. Zero becomes one, one becomes
zero, zero becomes one, and one becomes zero. What do you think this equals and decimal? I'll give you a second to think about it. (hums) All right. Now let's figure it out together. There's a one in the ones place. So this so far equals one times one, that's one. There's a zero in the twos place. So we add zero times two, that's zero. There's a one in the fours place. So we add one times four, that's four. So so far, we've got four plus one, and then there's a zero
in the final place, the eights place, that's
zero times eight or zero. So we've got zero plus
four plus zero plus one, which equals five in decimal. Let's try converting a bigger
binary number to decimal. We know that this equals five. What happens if we add four more digits to the left of these four digits? Let's start by figuring out
what each place represents. One way of doing that is just
to double the previous places. So this place here is 16, this place is 32, this place is 64, and this place is 128. Another way of doing this is
to calculate the powers of two. So this place is two to
the fourth, which is 16. This is two to the fifth, which is 32. This is two to the sixth, 64. And this is two to the seventh, 128. Okay, so now we know what
each place represents. Now we can add these numbers up. We know already that
these one's equal five. We already did that. Now, let's add in what we've added here. So we have a one in the 16. So that means we're going
to add 16 to the number. Then there's a zero in the 32. So that's just gonna be zero. There's a one in the 64,
so that's going to be 64. And there's a zero in the
128, so that'll be zero. So now we've got 64 plus 16 plus five. So that's 80 plus five, which is 85 in decimal. So it took us eight bits
to represent a number that only takes two digits to represent in the decimal system. That's because each bit
represents less information, just zero or one instead of zero to nine. But it's easier for computer hardware to store binary values. So binary numbers and
the binary number system are the foundation of all computing.