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![]() What does each one of those digits mean? Well the rightmost 2 means… 2. If we want to write the number one thousand, three hundred, and twelve, we need four digits: 1,312. The way those work isn’t as complicated as it might seem.Think about how you write a number in decimal digits using place value. So instead of that, we use powers of 2. The only difference is, since binary only has two different digits, 0 and 1, it doesn’t make any sense to use powers of 10 there’d be loads of numbers we couldn’t write, anything greater than 1 but less than 10 couldn’t be represented. And the leftmost 1 gets multiplied by 1000.Guess what? Binary numbers work the same way. The 3 doesn’t mean either 3 or 30, it means 300, because it’s being multiplied by 100. And then as you go through the rest of the digits, you go up by another power of 10 for each one. You take the digit itself and multiply that by 10 to get the value that’s being represented there. For each digit, we multiply its value by the power of 2 for that place value location (and the multiplication is pretty darn easy, because the only digits are 0 and 1), and then add up all those results. I’ll write out the whole thing this time: 1100There we go! We got 12. Why? Well, let’s do the same thing we did with our decimal example, multiply each digit. Here’s the number twelve in binary: 1100. X 1 2 X 4 0 How To Do ThatWe’ll add them up the same way we add decimal numbers by hand: 1100The first two digits are easy, 0 + 1 = 1, and 0 + 0 = 0.But now we have 1 + 1. First we have to write both of those numbers in binary we already know 12 is 1100, so 13 should just be one more than that, 1101. Except again it’s even easier because there’s only two different digits.Let’s have another simple example, 13 + 12. If you read the last section, you can probably guess what I’m about to say: that’s exactly what you do in binary too. Let’s say we want to add something to a binary number.Well, we know how to do that in decimal: you add up each digit starting from the lowest one and carry over into the next digit if necessary. That’s how you add numbers in binary. And to prove we got the right answer, let’s convert 11001 back to decimal, the same way we did before: 11001So now we know we were right 12 + 13 = 25, and 1100 + 1101 = 11001. This is the last column now, so we don’t have to worry about carries anymore, we can just write that down: 1And we’re done! 1100 + 1101 = 11001. So really we have to do 1 + 1 + 1, which is 3 but in binary that’s 11. Again, it’s 1 + 1, but now we have a 1 carried over from the previous digit. Well, just like in decimal, we have to carry out of that digit the sum of 1 and 1 in binary is 10 (because that’s just binary for 2), so that means we need to write a 0 in that column and carry the 1. Invert all the other bits in the number, meaning make every 1 a 0 and vice versa. Discard the sign bit before doing anything else. But, when the sign bit is 1, we need a couple extra steps to convert between binary and decimal. If the sign bit is 0, the number is nonnegative (either zero or positive), and if the sign bit is 1, the number is negative. “Signed” means that the number can be either negative or positive (or zero), and “unsigned” means it can only be positive (or zero) 3.What we’ve been talking about up to now are unsigned integers, so how do signed integers work? To start with, the first bit of the number isn’t part of the number itself anymore, it’s now the “sign bit”. But you don’t want every number to potentially be negative a lot of the kinds of things that you need to keep track of in a program just cannot possibly be negative, and sometimes (as we’ll see) allowing certain things to be negative can be actively harmful.So, computers (and many languages, including C) provide two different kinds of integers that the programmer can select between whenever they need an integer: “signed” or “unsigned”. 0101001Now just remember to add back the negative sign, and we get -42. Now we convert that to decimal and add 1. First, we need to flip all the bits except the sign bit, so that gets us 0101001. What is that in decimal? Well, we see right away that the sign bit is set, so we need to take the two’s complement. Let’s go through another simple example.Let’s say we have a signed 8-bit integer and the value is 11010110. Mini toy pianoThat means the hardware is simpler, which means it’s smaller, cheaper, and faster. The exact same hardware and logic can do arithmetic on both unsigned numbers and signed two’s complement numbers 5. Why?Why do we bother with any of this? Why not do something simple like have the sign bit and then just the regular number 4? Well, the two’s complement representation has one huge advantage: you can completely disregard it while doing basic arithmetic. It turns out all we have to do to make a two’s complement integer wider is copy over the same value into the low bits and then fill in the new high bits with copies of the sign bit. If it’s unsigned, that’s easy copy over the same value into the lower (right-hand) bits and then fill in the new high bits with 0’s, and you’ll have the same value, just now with more bits.But what if we need to widen a signed integer? Two’s complement’s here to save the day with a solution called “sign extension”. So to balance those concerns, languages like C give the programmer access to a few different bit widths to choose from for their integers.So, what happens if we need to do some arithmetic between integers that are different widths, or just pass an integer into a function that’s narrower than the function expects? We need a way to make an integer wider. Wider integers can represent larger (or more negative) numbers, but take up more space in the computer’s memory. Integers in computers have a fixed “width”, or number of bits that are used to represent them. Sign ExtensionThere’s one other neat trick two’s complement let’s us do that we need to talk about.
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