Adding and multiplying with zero is straightforward and uncontentious – you can add 0 to 10 to get a hundred – but we shall amean ‘add’ in the less imaginative way of the numerical operation.
Adding 0 to a number leaves that number unchanged while multiplying 0 by any number always gives 0 as the answer. For example, we have 7 + 0 = 7 and 7 × 0 = 0.
Subtraction is a simple operation but can lead to negatives, 7 - 0 = 7 and 0 - 7 = -7, while division involving zero raises difficulties. Let’s imagine a length to be measured with a measuring rod. Suppose the measuring rod is actually 7 units in length. We are interested in how many measuring rods we can lie along our given length. If the length to be measured is actually 28 units the answer is 28 divided by 7 or in symbols 2 8 ÷ 7 = 4. A better notation to express this division is
and then we can ‘cross-multiply’ to write this in terms of multiplication, as 2 8 = 7 × 4. What now can be made of 0 divided by 7? To help suggest an answer in this case let us call the answer a so that
t By cross-multiplication this is equivalent to 0 = 7 × a. If this is the case, the only possible value for a is 0 itself because if the multiplication of two numbers gives 0, one of them must be 0. Clearly it is not 7 so a must be a zero. This is not the main difficulty with zero. The danger point is division by 0. If we attempt to treat 7/0 in the same way as we did with 0/7, we would have the equation
By cross-multiplication, 0 × b = 7 and we wind up with the nonsense that 0 = 7. By admitting the possibility of 7/0 being a number we have the potential for numerical mayhem on a grand scale. The way out of this is to say that 7/0 is undefined. It is not permissible to get any sense from the operation of dividing 7 (or any other nonzero number) by 0 and so we simply do not allow this operation to take place. In a similar way it is not permissible to place a comma in the mid,dle of a word without descending into nonsense. The 12th-century Indian mathematician Bhaskara, following in the footsteps of Brahmagupta, considered division by 0 and suggested that a number divided by 0 was infinite.
This is reasonable because if we divide a number by a very small number the answer is very large. For example, 7 divided by a tenth is 70, and by a hundredth is 700. By making the denominator number smaller and smaller the answer we get is larger and larger. In the ultimate smallness, 0 itself, the answer should be infinity. By adopting this form of reasoning, we are put in the position of explaining an even more bizarre concept – that is, infinity. Wrestling with infinity does not help; infinity (with its standard notation ∞) does not conform to the usual rules of arithmetic and is not a number in the usual sense. If 7/0 presented a problem, what can be done with the even more bizarre 0/0? If 0/0 = c, by cross-multiplication, we arrive at the equation 0 = 0 ×c and the fact that 0 = 0.
This is not particularly illuminating but it is not nonsense either. In fact, c can be any number and we do not arrive at an impossibility. We reach the conclusion that 0/0 can be anything; in polite mathematical circles it is called ‘indeterminate’. All in all, when we consider dividing by zero we arrive at the conclusion that it is best to exclude the operation from the way we do calculations. Arithmetic can be conducted quite happily without it.