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NATE HARDISON: Back when you learned how to read and write

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numbers, you learned about the digits 0 to 9.

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To write whole numbers larger than 9, you learned that all

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you had to do was use some combination of these digits,

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as in 52 and 437.

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So this way of writing numbers has a name, decimal notation.

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>> Why decimal?

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Well, the Latin root of decimal, "decem," means 10.

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And when you have 10 digits in your notation system, 10

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becomes a rather special number.

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Let's look at the number 437 written in decimal notation to

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understand why.

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We can first break up 437 into 400 plus 30 plus 7.

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>> We can take it apart even more so that we've got 4 times 100

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plus 3 times 10 plus 7 times 1.

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Remember learning about the ones place, the tens place,

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the hundreds place, and so on?

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This is exactly where that comes from.

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And finally, we can see how we've got a bunch of powers of

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10 embedded in here.

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We've got 4 times 10 to the 2 plus 3 times 10 to the 1 plus

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7 times 10 to the 0.

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So now you see why 10 is a special

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number in decimal notation.

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In fact, we've got a name for it, it's called the base since

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it's the base of the exponent in our arithmetic here.

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>> Decimal notation is not the only way to represent numbers.

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In fact, even if we get rid of the digits 2 through 9, we can

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still represent all of the numbers that

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we could with decimal.

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So now that we have two digits, 0 and 1, 2 is our

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special number, the base of our notation system.

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The name of this notation system is called binary, since

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the prefix "bi" means two.

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So instead now of having a ones place and tens place and

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so on, we now have a ones place, a twos place, a fours

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place, and so on, going up by powers of two.

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So let's see this by doing some counting.

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So 0 still 0, and 1 is still 1.

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>> However, now that we've got a twos place instead of a tens

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place, 10 represents the number 2.

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To get 3, we add 1 to that and get 11.

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4, since there's now a fours place, is represented by 100.

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Five is 101.

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6 is 110.

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7 is 111.

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8, again, has its own place, so it's 1000.

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And I think you get the point.

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Let's take a stab at reading a large binary number and

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turning it back into decimal notation, since that's what

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we're used to.

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This number, in binary, reads 101110011.

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>> To figure out its decimal representation, let's start by

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writing the places under each of the digits.

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To start, we have the 2 to the zeros place on the far right,

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followed by the 2 the ones place, 2 to the twos place, 2

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to the three, 2 to the four, 2 to the five, 2 to the six, 2

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to the seven, and finally, all the way up to 2 to the eight.

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Now if we do the math, that's the ones place, the twos

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place, the fours place, the eights place, the 16ths place,

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the 32nds place, 64ths place, 128ths place, and finally the

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256ths place.

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Whew.

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So now, if we start multiplying everything

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together, we see we have 1 times 256 plus 1 times 64 plus

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1 times 32 plus 1 times 16 plus 1 times 2 and 1 times 1.

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>> So if we sum all of that together, we get to 256 plus

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64 plus 32 plus 16 plus 2 plus 1, all for a

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grand total of 371.

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Translating from decimal notation to binary notation is

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somewhat tricky, since we need to go from a number that's

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based on powers of 10 to one that's based on powers of 2.

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Let's give it go.

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Here we have the number 237 in decimal notation.

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To translate into binary notation, I start by finding

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the largest power of 2 that's less than it, which is 128.

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I put a 1 in the one hundred twenty-eighths place down here

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in my binary number.

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And then I subtract 128 from 237, and I get 109.

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Then I just repeat the process.

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The largest power of 2 that's smaller than 109 is 64, so I

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put a 1 in the 64ths place and subtract 64

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from 109 to get 45.

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Again, the largest power of 2 that's less than 45 is 32, so

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put a 1 in the appropriate slot and subtract 32--

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I'm going to move up here--

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to get 13.

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>> Moving on, I get 8 as the largest power

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of 2 now, not 16.

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So I put a 0 in the 16s place, a 1 in the 8s place,

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subtract, and get 5.

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Then 4 is the largest power of 2.

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I subtract and get 1.

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Now I can finish off the translation easily.

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I put a 0 in the twos place, and put a 1 in the ones place.

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The result, 11101101.

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>> One thing you might not have expected is that all of the

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algorithms you learned to add, subtract, multiply, and divide

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in decimal notation work in binary notation as well.

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We'll do an example of addition.

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Here we've got 1101101 plus 1010110.

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Just as in decimal addition, we'll start from the right and

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work our way to the left.

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The only difference is that we carry a 1 if the two digits

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we're adding have a sum greater than 1, instead of a

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sum greater than 9, as in decimal.

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So on the right, we have 1 plus 0, 1.

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>> Moving to the left, we have 0 plus 1, again 1.

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Moving left again, we have 1 plus 1, we write a 0,

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and we carry a 1.

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Then we have 1, 1, 0, so we have a 0, carry a 1.

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Then 1, 0, 1, again 0, carry a 1.

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1, 1, 0, 0 again, carry a final 1.

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>> And finally, 1, 1, 1, so we have a 1 and a

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final 1 on the left.

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The result, 11000011.

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And that concludes our quick

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introduction to binary notation.

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>> My name is Nate Hardison, and this is CS 50.