Operations in Nondecimal BasesDate: 10/16/1999 at 20:02:32 From: Tommy Paley Subject: Operations in non-decimal bases Hi doctor, I understand how to add numbers in other bases, but I was wondering if it was possible to subtract, multiply, and divide numbers in other bases? I tried to multiply the base 9 numbers: 35 x 28 ----- When I get 5 x 8 = 40, where do I go from there? I was thinking that 40 = 4(9^1) + 4(9^0). So I thought I'd place a four in my answer, and carry the other 4: 4 35 x 28 ----- 4 and then I went 3 x 8 + 4 = 28 = 3(9^1) + 1(9^0). Is this right so far? If it is, I think the answer should be 1124. Can you show me how to divide and even square numbers in other bases? Thanks in advance. I am looking forward to teaching my math 9 classes about different bases. Tommy Date: 10/16/1999 at 22:58:12 From: Doctor Peterson Subject: Re: Operations in non-decimal bases Hi, Tommy. You'll want to look through our FAQ on bases, http://mathforum.org/dr.math/faq/faq.bases.html It includes some help on operations in various bases, especially binary, as well as a long discussion of "adding in hexadecimal." Some of the other information on the meaning of bases and how to convert may be of use to you as well. The operations all work the same in any base; the only things that are different are the tables. Your multiplication is exactly right; it might go faster if you actually wrote up a multiplication table in base 9, so you could see instantly that 5 * 8 = 44 (base 9). That will especially help if you try dividing in base 9 - a good table is a must, as you may recall from your early experiences with base 10 division. Actually, the operations are almost trivial in binary, because there's practically no table to learn; that allows you to focus on the algorithm alone, so it can be a great teaching tool for kids who follow binary well in the first place, but need to see the multiplication and division algorithms more clearly. I sort of taught my son to multiply in binary before I taught him to do it in decimal. Since we already cover operations in binary pretty well in our archives, let's try a different small base. I'll use 3. First we make the tables: + | 0 | 1 | 2 | * | 0 | 1 | 2 | --+---+---+---+ --+---+---+---+ 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | --+---+---+---+ --+---+---+---+ 1 | 1 | 2 |10 | 1 | 0 | 1 | 2 | --+---+---+---+ --+---+---+---+ 2 | 2 |10 |11 | 2 | 0 | 2 |11 | --+---+---+---+ --+---+---+---+ Now for subtraction, the only trick is borrowing. In decimal, you borrow by subtracting one from the column to the left, and adding 10 to the column you are working in. Here, you do the same thing, remembering that 10 now means 3. Or if you want, you can add 3 in decimal to the digit, since within one digit the base doesn't matter. Let's subtract 10 from 15: 13 // 15 --> 120 - 10 --> - 101 ---- ----- 012 --> 1*3 + 2 = 5 You've shown me how to multiply, but I'll write one out in base 3 so I can use it for division: 20 --> 202 * 15 --> * 120 ---- ----- 000 1111 202 ------ 102010 --> ((((1*3 +0)*3 +2)*3 +0)*3 +1)*3 + 0 = 300 (Here I've used one of the easy conversion methods in the FAQ.) Now let's divide: _____120_ 202 ) 102010 202 --- 1111 1111 ---- 00 Since there's such a small table, this was pretty easy - all I can multiply a number by is 0, 1, or 2, so it's easy to guess a digit for the quotient. To square a number, of course, you just multiply it by itself. Hope this helps. Let me know if you need any more ideas for teaching this subject - it's one of my favorites. (I work with binary and hexadecimal all the time in my programming job.) - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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