What Is 0 Times X

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SPECULATIVE SCIENCE

If whatsoever number multiplied by zilch is zero, and any number divided past itself is one, and whatever number divided by cypher is infinity, what is cipher divided past zero?

ZERO DIVIDED by zilch is quite indeterminate. For if A is B times C, then A divided by B is C. But zero is equal to nothing times whatever number whatever. Therefore zero divided by cipher is whatever number whatever.
Ivor Etherington, Easdale, Argyll.

MUST Whatsoever number multiplied past nil e'er equal zero? When only digits are put to paper, it seems so, just when expressed in ordinary language, a different event is possible: if y'all multiply the total value of the change in your pocket by zero, the original amount remains. Some other case of the effect of tangible quantities: taking two circles whose circumferences have been calculated from their identical radii, and dividing the circumference of either into the other volition never get y'all to one.
Jack Belck, USA (nbelck@siue.edu)

TO SAY that "any number divided past nothing is infinity" is not quite correct. Considering normal arithmetic, information technology is not possible to dissever by nix. This is because "dividing by x" is really just a autograph mode of saying "computing the amount which gives the original when multiplied by 10". Since multiplying by naught always gives zero, we really cannot divide anything non-goose egg by zero. In fact, remembering that "dividing by ..." is a shorthand for something involving multiplication, even dividing. The "infinity" notation is, in this sense, a user-friendly shorthand, but not one that is universally valid. In brusque, if we call back that "division by" and "infinity" are convenient notations that are sometimes, simply not always, useful, then information technology shouldn't be too hard to believe that aught divided past zero really is not definable.
Dave Donaghy, Bath (ddonahey@mltsa.united kingdom.lucent.com)

MATHEMATICS is an abstract linguistic communication which describes the behaviour of the things which we call numbers. The fact that it can be related to the real world is fortuitous and convenient. However information technology is far from ever the instance. Pi is a number in whatsoever proper sense (as also is e - the number I talk virtually below). These numbers have an interesting belongings, amongst others - it is impossible to write them downwardly, unless you accept some mistake, because they comprise an infinite "number" of digits. Hence in any "real" experiment one will always get into the mess which Jack Belck describes (above). On the other paw, 0/0 is undefined, not indeterminate. This might sound pedantic just leads to some important consequences. It tin can approach a particular value under given conditions and just when those atmospheric condition are practical, for case if I take a function such equally: f(x)=(1-exp(x))/10 so when ten=0: f(x=0)=-ane, because although when x=0, f(10)=0/0 information technology approaches this value in an ordered fashion. Endeavour information technology out with a piece of graph paper and a pocket computer, or practise the Taylor expansion of exp(x).
(Dr) Simon Evans, London (s.j.evans01@ic.ac.great britain)

The questioner is request u.s.a. to follow through the consequences of some hypotheses in a full general number system. (They exercise non apply to whatsoever number system of any utilize, as explained past previous answers.) His 2d hypothesis gives zero divided by zilch equals one and the third gives zero divided past zero equals infinity. From this nosotros can immediately conclude that in whatsoever number arrangement satisfying his hypotheses, infinity is equal to one.
If we additionally assume that "zero divided past zilch" ways "the unique number which gives zero when muliplied by zippo", we come across that zero is also equal to the common value of one and infinity, and to any other numbers that may exist in the system under consideration.

To sum upward: under the atmospheric condition given, every number is equal to the same value, call it zero, one, infinity or whatever you like.

Pelham Barton, Birmingham, Britain

Must any number multiplied by zero e'er equal aught?If yous multiply the total value of change in your pocket money,the original amount eventually remains.
GAUTHAM, HYDERABAD Republic of india

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