Best Answer:
Imaginary numbers doesn't mean the same as your old imaginary friend when you were 2-year-old. That friend didn't really exist and was just in your mind. Imaginary numbers are not fictitious, but they're not "real" in the mathematical sense of the word.

Real numbers are numbers include the set of rational and irrational numbers, so fractions, decimals, pi, e, etc. are all real.

i (being defined as the square root of 1), makes a number imaginary, mathematically speaking.

Two imaginary numbers multiplied together can form a real number (1 + i)(1 - i) = 1² - i² = 1 + 1 = 2.

Now for x/0 being undefined.

Think of something real, like a pie. You can cut it into 12, 8, 6, 4, or even 2 pieces.

A while one is already cut into 1 piece, The more pieces you cut them into, the smaller each piece gets.

The other way around, the less pieces you cut it into, the larger the pieces get. So when you get into dividing by fractions, you end up with larger numbers than which you started with.

As the denominator approaches zero, the quotent gets larger and larger. Take a number like 100 and divide it by 0.00000000001. You get 10,000,000,000,000.

The closer it gets to zero, the larger the answer gets. Add 200 more zeroes between the decimal and the 1, you get 200 more zeroes in your answer, and yet you can still get closer to zero and end up with a larger answer.

You can't cut an object into zero length pieces, if you could, you'd be able to do it an infinate number of times since you're not taking any quantity away from the whole. Cutting a pie in half takes half the pie away. But cutting a "0 sized" piece removes nothing.

That's why you're unable to devide by zero. Some will say it's undefined, some will say it yeilds infinity. Depends on what level of mathematics you're in.

Now for your last item, that's easy.

Take the rule:

a^x * a^y = a^(x + y)

So if you have (x² * x³), it's the same as (xx)(xxx), or or x^5.

Taking it to division, you get:

a^x / a^y = a^(x - y)

x³ / x² = (xxx)/(xx) = x

So now take that formula and have the same exponent:

x² / x² = xx/xx = 1

x^(2 - 2) = x^0 = 1

Why 0^0 = 1 and not 0, I can't answer that, but it also is 1.

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