For instance, suppose we did assume that \(0\cdot\infty=1\). Why some people say it's 1: A number divided by itself is 1.

And nowadays again, mathematicians teach that division by zero is impossible, that it is “undefined.” But ever since the mid-1800s, algebraists realized that certain aspects of mathematics are established by convention, by definitions that are established at will and occasionally refined, or redefined. Both addition and multiplication have a (unique) identity element, which leaves any other number alone: Almost every number has an inverse under both addition and multiplication (except zero in the latter case), so that combining a number with its inverse yields the identity: This property is what basically allows us to define subtraction and division in terms of addition and multiplication. 8 divided by 2 equals 4, so 4 times 2 equals 8.

While it’s important to review this concept with them, the best way to help them truly understand is to give them plenty of opportunities to practice their basic multiplication and division  skills.

Thus \(P(-a)=(-x,y)\). We are assuming that we can divide by zero, so 0/0 should work the same as 5/5, which is 1). See if you can spot the error in the problem below: I will attempt to prove that 00=1\frac00 = 1 00​=1.

Ask kids “how many groups of two can go into 8?” When they realize that they can put 4 groups of 2 into 8, explain that 2+2+2+2=8.

Sign up to read all wikis and quizzes in math, science, and engineering topics. Once again, dividing by zero gives us difficulties!

So, we might imagine that at the very top of the circle lives infinity.

Why some people say it's 0: Zero divided by any number is 0. Here are a few common ones: Reply: This is true for any nonzero number, but dividing by 000 is not allowed. Dividing any finite number by \(+\infty\) or \(-\infty\) gives zero. The first paper that rigorously laid out the theory was published by Swedish mathematician Jesper Carlström in 2004. This new structure, where we’ve added a single infinity to the real numbers instead of two, has its own special name — the projectively extended real number line. \[ \begin{align*} 1&=0\cdot\infty\\ &=(2\cdot 0)\cdot\infty\\ &=2\cdot(0\cdot\infty)\\ &=2\cdot 1\\ &=2 \end{align*} \].

Remember that the complex numbers are best understood not as a line of numbers, but as a plane of numbers.

Like zero, this new number seems to be unsigned — neither positive nor negative — so let’s call it unsigned infinity, and let’s write it with a tilde (a little squiggle, like in the Spanish “ñ”) on top. \[\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\], \[m=-\frac{d_i}{d_o}=\frac{h_i}{h_o}=\frac{f}{f-d_o}\].

We just have to remember that while infinity is still a number, it’s not a real number, and it’s not part of any field. But there’s an even more interesting kind of symmetry hidden here as well.

So why not also admit multiple results when multiplying zero by infinity? Addition and multiplication are commutative, meaning that the order of numbers doesn’t matter: Addition and multiplication are also associative, meaning that we can use parentheses to group numbers however we want: \[(a+b)+c=a+(b+c)\qquad (a\times b)\times c=a\times (b\times c)\]. But that means … those answers are approaching infinity again? We can stop right there if we like. More details about wheel theory can be found at the Wikipedia article on the subject. You can spend days trying to explain to kids why dividing by zero isn’t possible. But you can actually see this connection for yourself using nothing but a spoon — or any other concave mirror, the kind that’s sometimes called a “magnifying mirror” in department stores. What we’re going to attempt to do is match points on the number line to points on this unit circle, so that every point on one pairs up perfectly with a point on the other. Suppose the number we’re dividing by isn’t the number of groups, but the size of each group. This is a bit of a hack though, since in the real number system, zero is neither positive nor negative — it’s unsigned, sitting on the boundary but taking neither side. Here's why: Remember that a b \frac{a}{b} b a means … That line is also guaranteed to intersect that circle at some second point — the stereographic projection of that real point onto the circle. What would \(10\div 0\) be based on this situation?

Move your mouse or drag the blue point to see how \(P(a)\) changes as \(a\) changes. So, what happened here? They’ll end up with four cookies in each of the three bags.

Computers use NaN to signal to programmers that some kind of error has occured as a result of a calculation — like those involving the indeterminate expressions we saw earlier.

In some sense, we’ve discovered a new number! This is absurd, so one might imagine that there was something “pre-modern” in Euler’s Algebra, that the history of mathematics includes prolonged periods in which mathematicians had not yet found the right answer to certain problems. which doesn't make any sense for any (finite) choice of 00. We can't share among zero people, and we can't divide by 0. In mathematics, sometimes the impossible becomes possible, often with good reason.

If so, might the result of division by zero change yet again?

Going off on that “tangent” (forgive the pun), graphs can not only “cross through” or “bounce off” zero. \frac00.00​. And if we go all the way, and look at the negative part of the number line in perspective, then we’ve found that again the lines intersect at negative infinity. At one point, a crew member entered a set of data that mistakenly included a zero in one field, causing a Windows NT computer program to divide by zero. Division by zero is an operation for which you cannot find an answer, so it is disallowed. Why some people say it's 1: A number divided by itself is 1.

Reveal the correct answer The expression is undefined \color{#D61F06}{\textbf{undefined}} undefined. Well … as many times as we want!

Now perhaps you’re wondering, “but wait — if we found a good answer to \(1\div 0\), what’s to stop us from finding some other brand-new number as the answer to \(0\cdot\overset\sim\infty\) or \(0\div 0\) and seeing what happens from there?”. Wheel theory is a relatively new branch of mathematics.

How many times could we give away zero apples? If we say that \(10\div 0\) is some number, then that number times \(0\) would have to be \(10\).

That seems plausible, but then remember that \(2\div 0=\overset\sim\infty\) as well, so then we’d have \(0\cdot\overset\sim\infty=2\). However, that would mean that 5 times 0 equals 5 as well, and that’s not the case. We’ve forgotten about something important: If we approach zero from the other side, dividing by smaller and smaller negative numbers, our answer still gets larger and larger, but in the negative direction: \[ \begin{align*} 1\div\frac{-1}{10} &= -10 \\ 1\div\frac{-1}{100} &= -100 \\ 1\div\frac{-1}{1,\!000,\!000} &= -1,\!000,\!000\\&\vdots \end{align*} \]. But this reasoning only makes sense for a nonzero numerator. We and our partners will store and/or access information on your device through the use of cookies and similar technologies, to display personalised ads and content, for ad and content measurement, audience insights and product development.

It seems that we’re at another impasse, but we’ve actually made some progress. So that’s where the true problem is — while zero and infinity are certainly reciprocals of each other, they’re not truly inverses because they don’t multiply to be one. Maybe it just doesn’t even make any sense! Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent, Posted April 12, 2011 More Asia, Blog, Discover, Europe, Science/Medicine/Technology, Writers/Literature, All content © 2010-present NOT EVEN PAST and the authors, unless otherwise noted, Sign up to receive the monthly Not Even Past newsletter, Negative Math: How Mathematical Rules Can Be Positively Bent. The common attitude toward such old notions is that past mathematicians were plainly wrong, confused or “struggling” with division by zero.

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