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Dividing by Zero and Imaginary Numbers

Mathematicians say that division by zero is forbidden. The expression 5/0, for example, is undefined. "Undefined" in this sense means "unable to be determined". Why is this?
When we divide 5 by 5 (5/5) we get 1. Divide 5 by 2 we get 2.5. Each time we make the denominator smaller in the expression 5/x, the expression gets bigger. We make x very small, 5/x gets very big. Make x smaller yet, 5/x gets bigger yet. Why do we get into trouble by making x=0 ? Why can't we say 5/0 equals infinity?

Pick a whole number X. Add three to it. Double what you get. Subtract 4 from this result. Subtract twice your original number. Add 3. The answer must always be 5. Why?

Why do irrational numbers exist? Are they real or imaginary numbers? Is there something unnatural about irrational numbers? We would say that an irrational person is a bit odd. Are irrational numbers an aberration of nature?
If this were so, why do some of the most fundamental things in nature involve irrational numbers?

1. The ratio of the circumference of a circle to its diameter is an irrational number, PI.

2, The diagonal of a square equals one of its sides times the square root of two, an irrational number.

3. In an equilateral triangle, the height is equal to half the base times the square root of three, an irrational number.

Solution Preview

If we say 5รท0 = infinity then that means that infinity * 0 = 5. We can't define any number divided by 0. This is because any number times 0 gives us 0.
Assume y is any nonzero number and y/0 = x. Then x*0 = y. There is no number x for which this can be true unless y = 0. If y = 0, then x can be any real number.
If we allow 0/0 to be a "legal" operation, then 0/0 ...

Solution Summary

This answers some questions about the existence of imaginary numbers and why one can't divide by zero.