DIVIDING BY ZERO

What happens when we divide by zero? Several textbooks and math professors disagree on this point. Mathematics seems to be a perfect science. At first. The thought begins with simple addition. This type of operation is done by accountants and is acceptable in court. Unit items, such as doughnuts, are piled up one atop the other so as to represent 'three plus five', for example. Three doughnuts are placed on top of five doughnuts, and the results are counted. The result is eight units. The number 'eight plus zero' involves placing zero doughnuts on top of the eight on the table, and the result is eight.

1. A number plus zero is the same number, which retains the same unit which in this case is a doughnut. Multiplication is a good shortcut, based upon mathematical tables which are learned by rote. The tables themselves are derived by adding units multiple times. For example, the number 'three times five' is found by placing five units onto the table, then piling on five more, then piling on five more. In this example, the units are doughnuts. The result is counted up as fifteen units. The number 'three times zero' involves placing three doughnuts onto the table zero times. As a result, there are zero doughnuts on the table.

2. A number multiplied by zero is zero, which HAS NO UNITS of a doughnut or anything else.

3. Subtraction is shown by taking units off of the table. For example, the number 'fifteen minus eight' is found by placing fifteen units onto the table. Then, eight of the units are taken away. The result is counted up to be seven. This method works well up to a point. The number 'seven minus eight' cannot be found by this method. Starting with seven units on the table, it is impossible to remove eight units. After removing seven doughnuts from the table, there are zero doughnuts left. The physical result of 'seven doughnuts minus eight doughnuts' is less than or equal to zero, and this result is not terribly precise. The physical result of 'seven doughnuts minus one hundred doughnuts' is also less than or equal to zero.

4. A number minus zero is the same number, which retains the same unit which in this case is a doughnut. It has been generally agreed that 'seven minus eight' is 'minus one'. However, in this example using doughnuts, it is impossible to show a negative doughnut. No one has any idea of what a negative one doughnut would look like. In order to be perfectly legal, the mathematician should be limited to using positive numbers only! Otherwise, the results may be somewhat fraudulent. Addition is good, multiplication is okay, and subtraction is somewhat questionable when the result is less than zero. Division, especially long division, involves both multiplication and subtraction. The results of subtraction operations are somewhat questionable. Therefore, the results of division operations are also somewhat questionable.

The results of division must always be checked by multiplying the result with the divisor, in order to recover the quotient. If the recovered quotient is not the same as the original quotient, then the result of that division is fraudulent. While studying in school, we are often taught that the result of a 'divide by zero' operation is infinity. This result is often "proven" by the use of long division. However, we cannot recover the original quotient by checking this division. Several modern texts and computer programs have selected a different value for the sake of convenience, and stated without proof that 'divide by zero' equals zero. Here again, we cannot recover the original quotient by checking this division. This is a puzzling result, as everything else in mathematics is checkable and provable. This 'divide by zero' result is given because the teacher says so, or because this is the answer given by the publisher of the math book, and neither of these results can hold up to rigorous mathematical analysis.

To illustrate the concept of division, try this experiment at home. Start with a bag of doughnuts and a short knife, about half as long as the major diameter of the doughnut. The idea is to form a physical basis for the concept of division. The Doughnut Theory will result. Each doughnut represents a number of arbitrary size, a unit. The short knife represents a smaller number which is used to divide the larger number.

As a physical model of division, this experiment involves cutting the doughnut with the knife. To represent division by two, pick up the knife. Select a new doughnut, and place it on the table. Cut the doughnut once on it's major diameter, then walk halfway around the table and cut the other side of the major diameter. If the cutting is done very carefully, then two equal halves of doughnut result.

The doughnut may be cut any number of times to represent division by any number. Select a new doughnut; cut five times to divide by five, which results in five pieces of a doughnut, or fifths.

Select a new doughnut; cut four times to divide by four, which results in four pieces of a doughnut, or fourths.

Select a new doughnut; cut three times to divide by three, which results in three pieces of a doughnut, or thirds.

Select a new doughnut; cut two times to divide by two, which results in two pieces of a doughnut, or halves.

Select a new doughnut; cut one time to divide by one, which results in one big piece of a doughnut. This is a torus with a gap, similar to the letter 'C'. It is not a legal doughnut anymore, because the dough no longer surrounds the center completely. It is no longer a doughnut! This is the first major point of doughnut theory: Dividing by one causes a mathematical change. Dividing a unit by one changes the unit.

When we divide by one, we are dividing by a unit of what? The result of division by one is dependent on the divisor unit. A doughnut divided by one knife, or divided by one brick is one piece of a doughnut.

Here is the second major point of doughnut theory. Select a new doughnut. Pick up the knife. Divide the doughnut by zero. That is, do not touch the doughnut with the knife. Put the knife down. The doughnut has been divided by zero. Let's look at the results of dividing by zero.
After the doughnut has been divided by zero, are there suddenly an infinite number of them? No. This result is patently absurd. Therefore, most mathematical textbooks are wrong. Dividing a number by zero does not result in any infinity at all.

After the doughnut has been divided by zero, does it disappear? Are there suddenly zero doughnuts to be seen? Once again, no. Several modern math texts and computer programs are wrong. The math chip in the computer on which this is being printed does indeed produce the wrong result when asked to divide by zero.

How many doughnuts result from dividing one doughnut (or any one unit) by zero? One! A unit. This proof of divide by zero is not dependent upon the unit of division. A doughnut divided by zero knife or knives is the same as a doughnut divided by zero brick or bricks.

5. In general, if a number X is not imaginary then X divided by zero (things) is X and no unit of the (thing) can be insinuated. Let's check this result. When dividing by a fractional number, the trick is to invert and multiply. Restate 'X divided by zero' as 'X divided by (zero over one)'. Then, invert and multiply: 'X divided by (zero over one)' is the same as 'X multiplied by (one over zero)' is the same as 'X multiplied by one' is X.

Now, destroy the evidence. Eat the doughnuts!