Andrew M. answered • 10/31/17
Tutor
New to Wyzant
Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors
base 10 is "based on powers of 10:
10n + 10n-1 + ... + 102 + 101 + 100
and each digit represents how many of the above we have
2312 = 2(103) + 3(102) + 1(101) + 2(100)
The largest digit is 9
base 8: 8n + 8n-1 + ... + 82 + 81 + 80
The largest digit is 7
3428 = 3(82) + 4(81) + 2(80) = 22610
For a negative base system every other digit would
be negative. The number would fluctuate up and down
base -8: (-8)n + (-8)n-1 + ... + (-8)2 + (-8)1 + (-8)0
1342-8 = (-8)3 + 3(-8)2 + 4(-8)1 + 2(-8)0 =
-1536 + 192 - 32 + 2 = -137410
Let's try to turn 134210 to base -8
(-8)4 + 7(-8)3 + . . .
in base 10 so far we have 4096 - 3584 + . . .
At this point we are at 51210 which is 83
How do we get that in base -8?
John H.
ok there it goes, looks like I'll have to break it up into multiple comments...
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11/01/17
John H.
Hmmm... while this appears to answer my question, this answer just doesn't feel right. It places too many inconsistencies in how mathematics works. let's take a look...
1.) in any base system once you reach the largest numeral in the one's place, increasing by one more you place a "1" in the next place and the one's place reverts to a "0". In a positive base system this means you go from (i.e.) 9 to 10 to 11 and the value of 10 is "ten" while 11 is "ten and one (eleven)", but do the same in the negative base version and 9 to 10 means you go from "nine" to "negative ten" and 11 would mean "negative ten and one (negative nine)". This problem then gets compounded when you increase the 2nd place again as you'd jump from 19 "negative one" to 20 "negative twenty". this inconsistency is mathematically illogical.
1.) in any base system once you reach the largest numeral in the one's place, increasing by one more you place a "1" in the next place and the one's place reverts to a "0". In a positive base system this means you go from (i.e.) 9 to 10 to 11 and the value of 10 is "ten" while 11 is "ten and one (eleven)", but do the same in the negative base version and 9 to 10 means you go from "nine" to "negative ten" and 11 would mean "negative ten and one (negative nine)". This problem then gets compounded when you increase the 2nd place again as you'd jump from 19 "negative one" to 20 "negative twenty". this inconsistency is mathematically illogical.
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11/01/17
John H.
2.) continuing on that line... let's look at 5x5. in a positive base system the "5x5" equivalency always equals a total of "25". 1012x1012=110012=2510, 5x5=25... and so on. However in a negative base system this either becomes (a)completely untrue according to your answer, OR (b)mathematics works differently in a negative base system.
(a)5-10x5-10=25-10=-1510 (multiplication is the equvalent of adding the number 'a' to '0' 'b' times... axb with a b=5 becomes ax5 = 0+a+a+a+a+a)
(a)5-10x5-10=25-10=-1510 (multiplication is the equvalent of adding the number 'a' to '0' 'b' times... axb with a b=5 becomes ax5 = 0+a+a+a+a+a)
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11/01/17
John H.
great everything is being shifted around due to copy/paste... and now it won't accept reason (b)
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11/01/17
John H.
in the above 1012 should be 1012 etc. and 5-10 should be 5-10 etc.
(b)but let's assume that it does equal 2510 (as it should). How do you explain multiplication in a base system that does that, without mentioning another base system? In your answer you yourself noted a similar inconsistency when trying to change a base 10 number into base -8
(b)but let's assume that it does equal 2510 (as it should). How do you explain multiplication in a base system that does that, without mentioning another base system? In your answer you yourself noted a similar inconsistency when trying to change a base 10 number into base -8
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11/01/17
John H.
there's a little bit in the above that is missing because this site wouldn't accept it into the comments for some reason.
3.) if (a) is the correct answer [which it is, based upon how multiplication works in a positive base system] and your answer is correct, we end up with a mathematical impossibility (5x5 should always = 5x5 regardless of mathematical base, which is true in positive base mathematics)
If (b) is the correct answer (which it should be as: 5x5 should always = 5x5 regardless...) and your answer is correct, then your answer is incomplete as it does not explain how to compute simple multiplication correctly in a negative base system.
If however your answer is incorrect both (a) and (b) are inconsquential because the negative base system would be improperly defined.
If (b) is the correct answer (which it should be as: 5x5 should always = 5x5 regardless...) and your answer is correct, then your answer is incomplete as it does not explain how to compute simple multiplication correctly in a negative base system.
If however your answer is incorrect both (a) and (b) are inconsquential because the negative base system would be improperly defined.
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11/01/17
John H.
4.) basically my mental process is telling me that your answer is an illogical rote determination based upon a pattern in positive base numbers. if this pattern were to be graphed, in positive bases you'd have a single straight line going up at an angle, but in negative numbers you'd have a "waveform" line. This is logically inconsistent as you are taking the assumption that positive numbers and negative numbers behave the same way. i.e. 5x5=25... two positives multiplied together equal a positive. Whereas -5x-5=25... two negatives multiplied together do NOT equal a negative. if positive base mathematics (as just shown) clearly define a difference in how positive and negative numbers interact, how can you declare that a negative base system would follow the same methodology as a positive base system?
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11/01/17
John H.
still can't figure out why the portion in 2.) (b) is not being allowed in the comments... it's just a formula saying 5 (base -10) multiplied by 5 (base -10) equals 185 (base -10) which is equal to 25 (base 10)
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11/01/17
John H.
oh, and another problem with this answer... positive base systems start with positive numbers... according to your answer a negative base system would start with positive number equivalents as well. One would assume that being a negative base it would have to start with number equivalents that begin in the negatives... why does it not? Just another thing that makes that answer feel... off.
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11/01/17
Andrew M.
John,
my "answer" boils down to a negatively based system is illogical and inconsistent.
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11/02/17
John H.
11/01/17