Binary SubtractionDate: 7/25/96 at 15:27:39 From: Norman or Gwen Houston Subject: Binary Subtraction Dr. Math, Please explain and show examples of how to subtract in binary. I understand the rules of adding, but how in the world do you subtract? Thank you, Gwen Houston Date: 7/25/96 at 19:9:53 From: Doctor Robert Subject: Re: Binary Subtraction Actually, you subtract in binary pretty much the same way that you do in base 10. You subtract digit by digit starting on the right side. If the subtraction cannot be made (for example, you cannot subtract 1 from 0), you must then "borrow", just as you do in base 10 subtraction. But when you borrow a "one" from the 4's digit, it turns into two 2's. This borrowing by two's (rather than 10's) is what makes it quite different from base 10 subtraction. Perhaps an example is in order: 10101 - 1011 Starting on the right, 1-1 = 0, so the rightmost digit in the answer is 0. Moving left, we cannot subtract the 1 from the 0 (in the 2's position), so we borrow on from the 4's position giving us TWO 2's. Then 2-1 = 1 so that there is a 1 in the 2's position of the answer. Moving left 0-0=0 (remember we already borrowed the 1 in the 4's position of the top number) So 0 is the answer digit in the 4's position. Moving left, we cannot subtract 1 from 0 so we borrow a 1 from the 16's position giving us 2 in the 8's position. 2 - 1 = 1 so the answer is 1010. It would look like this when you are done 10101 - 1011 ----- 1010 Just to check, in base 10 this would be 21 - 11 = 10. It checks! Hope that this helps. -Doctor Robert, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 7/25/96 at 20:13:50 From: Doctor Anthony Subject: Re: Binary Subtraction You can use the same method as subtracting in base 10, except that the carry takes place when you subtract 1 from 0, (which is made up to 2 by borrowing from the column next on the left). However a method of subtraction, by adding the 'complement' is particularly easy in binary arithmetic. To complement a number in base 10, you subtract from a row of 9's. Likewise in base 2, the complement is obtained by subtraction from a row of 1's; e.g. the complement of 10011101011 is 01100010100 All you have to do, in fact, is interchange 0's and 1's. It is important, however, that before you complement a number which has to be subtracted, you add leading zeros to make it up to the same number of digits as the number from which you will be subtracting. There is a further problem concerning a 'carry' digit at the extreme left after adding the complement, but I will show you the method in an example. Example: Find 1101011001 - 0010111010 (note the leading zeros) Complement the second row and add: 1101011001 + 1101000101 ------------ (1)1010011110 Now you notice that there is an embarrassing (1) at the left of the row. The rule now is to take this 1 across to the first column on the right and add it there, as shown below: 1010011110 1 ----------- 1010011111 That completes the subtraction. With practice, the method of subtraction by adding the complement is quicker and less error-prone, but it is of course a matter of choice. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 7/26/96 at 4:31:58 From: Norman or Gwen Houston Subject: Re: Binary Subtraction Dr. Math, Thank you so much for your informative answer and examples. This information has helped me better understand the subtraction process of binary numbers. Regards, Gwen Houston |
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