Robert Lewis University of Tasmania <[email protected]> Maybe you, like me, have always had it instilled in Any number divided into zero gives zero as the you that you do not divide by zero or nasty things answer no matter what the value of a. Similarly, a 0 may happen? Some confusion still surrounds the =1 ; but if this is true, then is equal to 1? a 0 1 number zero, particularly in using it in division. We know that = 1 so let’s try replacing numbers 1 You might never have questioned why you in that equation with numbers that are getting cannot divide by zero, or been asked to explain closer and closer to zero: why it’s a mortal sin. Why would you? It’s fairly obvious, but the fairly obvious is always difficult 0.0001 =1 to explain. How can we explain this concept to 0.0001 children in a method that is easily grasped and 0.00000000001 learnt? The purpose of this short article is to =1 demonstrate a few pedagogy methods to show 0.00000000001 children the trouble that zero can cause. Some 0.0000000000000000000000001 concepts appear trivial at first glance, but look =1 closer and one begins to think deeply about 0.0000000000000000000000001 some of our long held truths about zero. The history of the number zero is an interesting one. In early times, zero was not used as a number As we keep putting in numbers that are getting at all, but instead was used as a place holder to closer and closer to zero the answer is still one. indicate the position of hundreds and tens. For Therefore it is reasonable to say as the numbers example, in representing the number 5048 the tend towards zero the answer is still 1, ergo 0 number of hundreds in this number is nothing, = 1 (in case you are throwing this article down 0 that is, there are no ‘hundreds’ in the hundred in disgust, I know that’s not really true, but stay column. Zero in this case is being used to with me). represent the concept of absence. You might disagree and argue that zero goes 0 Zero first started being used as an actual into zero no times, therefore = 0 and since you 0 number around 458 AD in India by Hindu know that a0=0= 0, I think you may be on some solid astronomer and mathematician Brahmagupta. ground. Let me try and prove it by replacing the He devised the methods by which zero is reached variable ‘a’ with a number that is getting closer and in calculations, for example 6 – 6 = 0, and also closer to zero. how zero is used in equations. Prior to this, zero 0 = 0 was not recognised as a number at all but just 0.0001 a concept representing absence or nothing (Kaplan, 2000). 0 = 0 Let’s look at what appears a fairly intuitive 0.00000000001 calculation: 0 0 = 0 = 0 0.0000000000000000000000001 a 20 amt 71(3) 2015 Just as in the prior example, it is reasonable to Perhaps you want to prove it mathematically. 15 say as the value of a gets closer and closer to zero, Looking at lets replace zero with real numbers 0 0 the answer is still zero, ergo = 0. Wait a minute, getting closer and closer to zero and see what 0 0 I think. Maybe is infinity. happens: 0 15 =15 Let me explain with reference to a simple division: 1 15 15 =5 =150 3 0.1 15 We could ask how many times can we take 3 from =1500 15 before we cannot take any more threes away, 0.01 and of course the answer would be five. Similarly, 15 we could ask “How many times can we take zero =15000000000 away from zero before we cannot take any more 0.000000001 zeroes away?”. Conceptually you may say you can take zero from zero all day—up to an infinite The answer keeps getting larger and larger, amount of times. approaching very large numbers. This seems to So, what is the answer? Is zero divided by support that a = ∞. So is it infinity, or is it zero, 0 zero 1, 0, or infinity? Clearly it cannot be all or does the question not even make sense? three at the same time, therefore the answer is Mathematically, we say that the question is not exactly known or is unable to be determined. unanswerable or that the answer is undefined. Mathematically, the answer to zero divided by zero Calculations involving zero can at first glance is indeterminate. Okay, so zero divided by zero is appear trivial. It is the questions that come from indeterminate but what about: ‘left field’ that have a tendency to challenge the thinking of any teacher. I hope that this article a = undefined has gone some way to challenge the thinking 0 where divisions using zero are used and also Why is any number divided by zero undefined? better prepare teachers to answer student I might argue that the answer should be infinity. questions using critical thinking. After all, you can take an infinite amount of zeroes from any number—keep taking zeros away from References 15 until the end of time and you will still have the number 15 unchanged. Kaplan, R. (2000). The nothing that is: A natural history Using an everyday example that students of zero. United Kingdom:Penguin Books Ltd. understand, I explain that I have a pizza that I want to divide up between six people. How many slices will I need to cut the pizza into—clearly the answer is six slices. Now, what if there are no people demanding a slice of my pizza—how many slices will I need to cut my pizza into? Ignoring the fact that the question doesn’t make sense (how can no people demand a slice of my pizza) I may say ‘I don’t have to cut my pizza into any a slices’, therefore = 0. But this solution also has 0 problems. Look at our previous division problem 15 of = 5. The opposite of division is multiplication, 3 therefore five multiplied by 3 is 15, mathematically 15 you would expect if = 0 then 0 * 0 =15. This 0 obviously is not the case. amt 71(3) 2015 21