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ILikeTofuuJoe

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From: Obvervable Universe

Joined: 2018-06-04

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Binary Calculation Tutorial v1.1

Binary is the most compact way of representing numbers. But its calculation is a bit complex.
Binary Addition

In this section we will use 5+7 as an example. 5 = 101 and 7 = 111. Add them up together in binary and you'll get 1100(12). Let's take a look at how the binary calculator is going to accomplish that. So first of all, you need to let your calculator know that 1+1=10,1+0=1,0+1=1,and0+0=0. You need to do that for every bit. 1+1 is fairly tricky compared to the others. Because it advances a bit. You'll need to check that digit to see if it equals 1. If yes, change that to 0 an

▼View

In this section we should calculate 00000111 + 00000101. 0 + 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 10. If you calculator finds a bit calculation that results in 10, tell it to advance a digit and calculate 1 + whatever the next bit is. Also tell your calculator to leave a 0 in its 10's original bit. So it might look something like this:

if(result > 1) {
nextbit++;
nextbit.check10;
} else if(result == 1) {
bit = 1;
} else {
bit = 0;
}

Not precise but you'll get it. So let's finally calculate the above expression. Our calculator already finds an overflow of bit on the one's bit. So now it checks the next bit's sum while changing the one's bit to 0. Here a temporary bit will be used to store the overflow. The two's bits' sum is 1, so our calculator finds that another overflow is created. It now checks the four's bit. Another overflow is found! Our calculator stores that in the second overflow bit. Since the calculator discovered that the first overflow bit can be used, it resets it to 0 and changes the fours bit to 1. The same goes for the second overflow bit. But before that, the second overflow bit is moved to the first overflow bit. Now it changes the eights bit to 1. And we got out result: 00001100.

Binary Subtraction
this section we will use 9-3 as an example. 9 = 1001, 6 = 110, 9-6 = 11This is a bit trickier than the addition, but is simpler than multiplication and division. First, know that 1-1 = 0, 1-0 = 0, 0-1 = -1, 0-0 = 0. Subtract each bit and you'll get 10(-1)0. Since -1 is not a part of binary bits, We'll have to get rid of it. From the -1, check the left number. If it is 1, change it to 0 and the bit itself to 1. If it is -1, then continue looking left, when 1 is found, drop down the chain of -1s until your original bit. If it is 0, or if your chain of -1s has 0s in it, continue searching. When a 1 is found, change it to 1 if there are -1s after it(It gives 1 to the -1), If there aren't -1s after it, don't do anything. Alright let's test! Step 1: Subtract bits 1001 - 110 = 1(-1)(-1)1. Step 2: Clearing -1s: rightmost -1: the left is a -1, So it has to be 1. Leftmost -1: The right is a -1, the left is a 1. Take the 1 and give it to the right -1. Step 3: Answer! 1001 - 110 = 11!
P.S.: I forgot to take screenshots...

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#Joe Griffin

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JadElClemens

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From: Colorado, USA

Joined: 2015-02-15

Posts: 4,559

Re: Binary Calculation Tutorial v1.1

Good post! I'll add that computers largely do not actually subtract binary values. Negative values in binary are, in most cases, represented as a two's complement value, where the most significant bit (the leftmost bit in most representations) is the sign bit - 0 for positive, 1 for negative. When using a signed integer of this type, subtraction is performed as the addition of two numbers where at least one is negative, because the rules are actually the same.

▼Quick rundown on calculating two's complement values

Normally with binary, the value of a given position x in the number is 2^{x} - the LSB is at index 0 so it has value 2^{0} = 1. The next bit is index 1 and has value 2^{1} = 2, and so on. So the binary value 1010 is, reading from right to left, 0*2^{0} + 1*2^{1} + 0*2^{2} + 1*2^{3} = 2 + 8  = 10 in base 10.

With two's complement values, this convention actually still holds, but the MSB (the sign bit) is a negative value. So while decimal 10 is 01010 in two's complement binary, 11010 is not -10.

Take the signed two's complement number 110 (shorter this time because I don't want to type too much more) - This is 0*2^{0} + 1*2^{1} - 1*2^{2} = -4+2 = -2. Note the negative sign for the MSB.

My teachers taught me some convenient methods for finding the binary representations of decimal values by hand but I've forgotten those. What I do for signed two's complement values is

• If the decimal value is positive, the MSB should be 0. Find the rest of the binary representation as normal.

• If negative, the MSB should be 1. Take your decimal value and add the value of the MSB to it - for example, say we're trying to represent -5 in 4 bits. The MSB has value 2^{3} = 8, and -5+8 = 3. This is the value that you need to represent using the rest of your bits to get to the desired value. Thus, -5 becomes 1011. 1+2-8 = -5, as expected.

For example, take the calculation 10 + (-3). Normally you could represent 10 in 4 bits, but since we need the MSB to be a sign bit we have to use at least 5.

  01010 (10)
+ 11101 (-3)
-----------------
  00111 ( 7) 

Technically this does overflow (the output value is 100111 = -25, but since we're using 5-bit hardware in this example we have to throw out the MSB). At this moment I don't remember the rules for when overflow should be ignored.

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ILikeTofuuJoe

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From: Obvervable Universe

Joined: 2018-06-04

Posts: 1,770

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Re: Binary Calculation Tutorial v1.1

^ Alright! I'll add some text for calculation negative numbers (as soon as i figure out how to do that).
So for binary numbers, value = (non sign bits' sum) - sign bit?

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LukeM

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From: England

Joined: 2016-06-03

Posts: 3,009

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Re: Binary Calculation Tutorial v1.1

^ Alright! I'll add some text for calculation negative numbers (as soon as i figure out how to do that).
So for binary numbers, value = (non sign bits' sum) - sign bit?

The most significant bit counts for negative what it used to, yes. So if it was an 8 bit number then the most significant bit would be -128 instead of 128, and the rest of the bits would stay the same.

The easiest way to negate a number that I know of is to flip all the bits then add one (effectively flipping all the bits after the first one from right to left):
34 = 00100010
   -> 11011101
     + 00000001
     = 11011110
= -128 + 64 + 16 + 8 + 4 + 2 = -34