Divisibility and Subtraction in Other BasesDate: 07/28/99 at 00:37:11 From: Agnes Morrison Subject: Number bases other than 10 In base ten you can tell when a number is even by looking at its last digit. Can you recognize an even number in other bases, for example base 2 or 3 or 4? Thank you for the help. Date: 07/28/99 at 11:34:10 From: Doctor Rick Subject: Re: Number bases other than 10 Hi, Agnes. This question is a nice topic for exploration! Look for yourself. I'll write out some numbers in base 10, 2, 3, and 4: base 10 base 2 base 3 base 4 ------- ------ ------ ------ 1 1 1 1 2 10 2 2 3 11 10 3 4 100 11 10 5 101 12 11 6 110 20 12 7 111 21 13 8 1000 22 20 9 1001 100 21 In base 2, all even numbers end in 0 (the only even digit) and odd numbers end in 1 (the only odd digit), so the rule works in base 2. What about bases 3 and 4 - which bases work this way? Can you come up with a hypothesis (an idea that you can test) about which bases the rule works for? Let's not stop here. Evenness is the same as divisibility by 2. What about divisibility by other numbers? In base 10, you can test divisibility by 5 in the same way as divisibility by 2: if the last digit is divisible by 5 then the whole number is divisible by 5. Do you notice something here? The base, 10, is divisible by 2 and by 5. Those are the two numbers whose divisibility you can test by looking at the last digit of a number in base 10. Can you guess what numbers you can do this with in base 3? Base 8? Base 12? Test your hypotheses by writing numbers in these bases as I did. Then try to understand why it works. I'd be glad to correspond with you further if you have ideas. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 07/28/99 at 16:31:55 From: Marc Morrison Subject: Re: Number bases other than 10 Dear Dr. Rick, Thank you for the neat answer. Can the idea be that if the base can be divided evenly by two, then you can tell by looking to see if the last digits are even numbers that the number is an even number? And too, if the base is evenly divisible by 2, and the last digit is an odd number, then the number is odd? If the base number is not evenly divisible by 2, then you can't tell by looking at the last digit whether the number is even or odd. Does this work? Thank you again for your help. Agnes Date: 07/28/99 at 18:04:48 From: Doctor Rick Subject: Re: Number bases other than 10 Hi, Agnes - Yes, you've got the idea. What do you think about divisibility by 3? And why do you think this works? (Hint: 5274 = 527 * 10 + 4; have you learned about the distributive property yet?) - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 07/28/99 at 18:11:00 From: Marc Morrison Subject: Re: Number bases other than 10 Dear Dr. Rick, The last part of my project is to learn subtraction in other bases. I get confused. What I am supposed to borrow? Can you help some more? Thank you. Agnes Date: 07/29/99 at 11:04:49 From: Doctor Rick Subject: Re: Number bases other than 10 Hi again, Agnes. Subtraction in other bases isn't very hard; I'll just give you some examples. The basic rule is that you always borrow the base. For instance, in base 4: 312 - 133 ----- You can't subtract 3 from 2, so you need to borrow. Take 1 from the 4's column and change it to four 1's; add the four 1's to the two 1's that you have: 3 0 6 - 1 3 3 ------- 3 Now we move to the 4's column. We can't subtract 3 from 0, so we need to borrow again. Take 1 from the 16's column and change it to four 4's; add the four 4's to the zero 4's that you have: 2 4 6 - 1 3 3 ------- 1 3 Finally we move to the 16's column, where we have no problem subtracting 1 from 2: 2 4 6 - 1 3 3 ------- 1 1 3 Let's just check our answer by adding 113 to 133 in base 4. You can check your understanding of adding in base 4: 133 + 113 ----- 6 6 doesn't belong in base 4; we need to break it up into 4 + 2, and carry the four 1's as one 4. 1 133 + 113 ----- 2 1 + 3 + 1 = 5, which again doesn't belong in base 4. Break it into 4 + 1 and carry the four 4's as one 16: 11 133 + 113 ----- 12 Finally we add 1 + 1 + 1 = 3, and no carry is needed: 11 133 + 113 ----- 312 That's the correct answer: we subtracted 133 from 312 to get 113, then we added 133 back on to 113 and got 312 back - all in base 4. Does this make sense? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 07/29/99 at 11:30:36 From: Marc Morrison Subject: Re: Number bases other than 10 Dear Doctor Rick, Thank you a bunch for all your help. You are great! Agnes |
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