by Jeremy Ross

Created on: August 11, 2009   Last Updated: August 13, 2009

Many people have argued over the question, "Can you divide by zero?." Some people believe you can divide by zero, and others firmly believe it is impossible to divide by zero. Division by zero is in fact possible. The reason division by zero is possible is because it allows important results to come to light. There are instances when division by zero is not possible and other circumstances where it is possible and necessary.

During your math education, you were probably told it was not possible. The reason for that was because most of the math that you dealt with during high school and even early college centered around the real number system. The real numbers contain the rational and irrational numbers. Rational numbers are numbers that can be expressed as a fraction and terminate after a decimal point. Irrational numbers are numbers that cannot be expressed as a fraction and never terminate after a decimal point. The real numbers are basically every number about which you were told.

Division is the inverse of multiplication. For instance, if the following is an equation: 5y = 20, one can easily solve for y by dividing both sides by 5 to yield an answer that y=4. This can be checked by multiplying 5 and 4 to get 20. Let's look at a more complicated example: 0y = 0. What is the value of y? Well, y can be any number. It can be any of these numbers: -231, 5, 2/3, . The answers are endless. Because of this fact, y can never be recovered by inverting the process of multiplication. No other number has this property, and it is because of this that division by zero is undefined for the real numbers.

You might be thinking that I am not supporting my side that division by zerp is possible. Well, here comes the supporting evidence. Division by zero is in fact possible. The real numbers are a subset of the complex number systems. In the complex number system, division by zero is defined. If z is any element of the complex number, and z is divided by 0, the result is known as complex infinity. This only holds true if z does not equal 0. This shows this important property: for z not equal to 0, the limit as w goes to 0 of w/z = . This does not mean that z = w. Zero still does not have a multiplicative inverse.

There you have it. Division by zero is not possible in the real number system, but it is possible in the complex number system. The next time someone asks you, "Can you divide by zero?" you will know the answer. It is possible to divide by zero.

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