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One of the most difficult concepts in mathematics is doing operations involving zero. Contrary to popular opinion zero is far from being nothing, but what exactly it is can be difficult to explain and understand. When we add or subtract using zero we generally think that what we are adding nothing, and in this situation we would be correct in using that thinking. However there are some cases when zero means something or what it means can't be accurately described.
Multiplication by zero does nothing to change the generally held concept of zero being "nothing". When we multiply any number by zero the result is simply zero. While this result is understood as universal, multiplication in more advanced mathematics proves that this is not always so. Using exponents or raising a number to a power is one example where zero doesn't mean nothing.
An exponent represents how many times a number is multiplied by itself. For example 2 x 2 would be 2 raised to the 2nd power or 2^2; 2 x 2 x 2 would be 2 raised to the 3rd power or 2^3, and
2 x 2 x 2 x 2 would be to the 4th power and so on. The use of zero as an exponent however, is defined as being; any number raised to the zero power is equal to 1. So this means that any number with "0" as an exponent is equal to "1". This is the basis of the system of logarithms. Without this definition of "Zero" there could be no way to express numbers that are between "1" and "0" (fractions) in exponential form. The "zero power" allows the use of negative exponents to represent fractions. The use of zero makes negative exponents not only possible but sensible in that fractions can be express in an unlimited or "infinite" range of numbers because of negative exponents. This would not be possible if zero were just nothing.
Another use for zero that means something is zero as a reference point or marking an event. Zero is used to mark the center of the number line and the division between positive and negative numbers. Zero is neither positive or negative, but in the point that divides these two sets of numbers and provides a reference point from which to count. Besides accounting, where zero is the demarcation point for serveral conditions, like profit and loss, you can see it used to mark events. When you think of a space launch, the term t-minus is used during the countdown, this represent the time before launch. The time after launch is t-plus. When the countdown reaches "zero" we have the event (the launch). Once again zero is more than nothing, in fact it is the definition of an event.
One of the more interesting concepts involves the fact that division by zero is undefined. Proof of this statement can yeild some amazing and interesting results. Take for example the fact that the larger the number you divide into another number, the smaller the resulting answer. Conversely the smaller the number you divide into the number the larger the result. This is easy to see if we divide the same number by 5 and then by 3, the answer you get as a result of the division by three would be bigger. Continuing this same thought dividing 2 into the same number would give a larger answer, and dividing by 1/2 would result in a larger answer still. If you continue on this pattern it becomes clear that the smaller the fraction used to divide into a number the larger the answer becomes. As fractions become smaller and smaller they come closer and closer to zero, and at the same time the result of dividing them get larger and larger. 1/1,000,000 goes into 1 , 1 million times! So it would seem that if we divided a number by a fraction so small as to be almost zero the resulting anwser would be infinitely large. So if this is true why can't division by zero be defined? We know that the answer should be infinity, right? Well not quite. Take a look at this simple equation to see the answer.
2X + 6 = X + 3
Solve this equation for X and you get
X = -3
But use the rules of algebra and factor out 2 from 2X + 6 the result is:
2(X + 3) = X + 3
Divide both sides by X + 3 and the result is:
2 = 1
The use of algebra is absolutely correct, so how could this answer be? Well the fact is that the factor (x + 3) that was used to divide into both sides of the equation is actually equal to ZERO, since X = -3 and (-3 + 3)! So what we did in finding a solution is divide both side by Zero. If you tried to define division by Zero you would find that problems like this prove why you can't do it. Division by Zero would make anything equal to anything else!
Zero is a lot of things, but the uses indicated in this paper show that there is a lot more to zero than nothing!
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