dividend ÷ divisor = quotient

It is the reverse of multiplication, or course. If you multiplied the quotient by the divisor, you get the dividend, or to be technically correct, multiply the Multiplicand by the Multiplier to get the Product.

Multiplicand x Multiplier = Product

Quotient x Divisor = Dividend

Now, when the Multiplier is zero, no matter value what the Multiplicand is, you will always get a Product of zero.

So, if the Divisor is zero, the Dividend is zero. Correct?

Now let's reverse it with division. The dividend is zero and the divisor is zero. What is the quotient? 1? Yes. 2? Yes. 27? Yes. 1,342,243? Yes. Why all these numbers? It's because when you multiply any of these numbers by zero, you get zero. Therefore, dividing zero by zero is an indeterminate number and is not used in real applications.

Now let's say you want to divide a non-zero number by zero. In multiplication, what number multiplied by zero will give you a non-zero number? Some have argued that you can use an imaginary Product over Zero as the Multiplier, cancel the zero's out, and you have this:

6/0 * 0 = 6?

6 = 6 (cancel out the zeros)

6 ÷ 0 = 6/0?

What is 6/0? What is any nonzero number divided by zero? Let's take a look at this series of division quotients:

24 ÷ 24 = 1

24 ÷ 12 = 2

24 ÷ 8 = 3

24 ÷ 6 = 4

24 ÷ 4 = 6

24 ÷ 3 = 8

24 ÷ 2 = 12

24 ÷ 1 = 24

24 ÷ 1/2 = 48

24 ÷ 1/10 = 240

24 ÷ 10^{-100} = 24 x 10^{100}

So what are we doing here? As the Divisor gets smaller, the Quotient gets bigger!
So what happens when the divisor reaches Zero? The Quotient becomes infinitely big, or infinity,
which is not a real number that can be easily fathomed by any stretch of the imagination.

Now let's make the divisor even smaller! You get a Quotient in the negative zone.

24 ÷ -1/10 = -240

24 ÷ -1 = -24

24 ÷ -24 = -1

24 ÷ -240 = -1/10

The Divisor gets smaller, and the Quotient gets bigger, though it's still in the negative zone. Now what happens when you put infinity as the Divisor?

24 ÷ -infinity = -0

24 ÷ infinity = 0

You have completed a loop of Divisor to Quotient relations. Dividing by infinity is the same as multiplying by zero, so infinity and zero must be reciprocals, right?

What about the equation: n x 0 = 6?

If the "n" is infinity, then you'd get this for division: 6 ÷ 0 = infinity, and infinity x 0 = 6

But the values for the Product and Dividend also work for the Dividend: 27, SQR(-1), pi, 10^{googol}

To summarize this whole mess: If you reverse each division for multiplication, you get the whole series:

anynonzeronumber x infinity = anynonzeronumber ÷ 0 = infinity

anynumber x 0 = anynumber ÷ infinity = 0

anynonzeronumber x anynonzeronumber = anynonzeronumber ÷ anynonzeronumber = determinate

0 x infinity = 0 ÷ 0 = indeterminate

0 x anynumber = 0 ÷ anynumber = 0

0 x 0 = 0 ÷ infinity = 0

infinity x 0 = infinity ÷ infinity = indeterminate

infinity x infinity = infinity ÷ 0 = infinity

infinity x anynonzeronumber = infinity ÷ anynonzeronumber = infinity

This is all in theory. This is why you don't divide by zero, unless you have a legal license to do so.

0^{0}

Why is that undefined? Why isn't the answer 0?

In powers, n^{k} = n multipled k times.

2^{5} = 2 * 2 * 2 * 2 * 2 = 32

1^{5} = 1 * 1 * 1 * 1 * 1 = 1

0^{5} = 0 * 0 * 0 * 0 * 0 = 0

Now what happens when the "k" part becomes zero?

2^{0} = 1

1^{0} = 1

0^{0} = ???

What happens when the "k" part becomes negative?

2^{-4} = 1/16

1^{-4} = 1/1 = 1

0^{-4} = 1/0 = infinity

So the answer for 0^{0} is indeterminate. Correct?

Let's see how you get 2^{5} in more ways than one:

2^{5} = 2^{6} ÷ 2^{1} = (2*2*2*2*2*2)/2 =
2^{5+n}/2^{n}

2^{0} = (2*2*2*2*2*2*2*2*2*2)/(2*2*2*2*2*2*2*2*2*2) =
2^{n}/2^{n}

2^{-4} = (2*2*2*2*2*2)/(2*2*2*2*2*2*2*2*2*2) =
2^{n-4}/2^{n} =
2^{n}/2^{n+4}

So if you try to figure out 0^{0}, you get
0^{n}/0^{n} =
(0*0*0*0*0)/(0*0*0*0*0) =
0 ÷ 0 = indeterminate, which we defined just earlier

So therefore, zero raised to the zero power is undefined and is not used.

n * n = -n * -n = n^{2}

So what happens when you want to get to -n^{2} by squaring a number?
Since all the pairs of numbers are used up, there are no real numbers left for
the square roots of negative numbers. Just as well, since a negative square
such as -16 may look like a square of a number, like +16 is, but both
4 and -4 when squared add up to +16.

If you really insist of exploring the square root of a negative number, let's factor out the 16 part under the square root radical, leaving us with the square root of -1 with the square root of 16. The square root of 16 becomes 4, but it's also negative 4 since it too multiplies out to 16.

But how do you tell what to use? Most real applications use the positive square root, or the principal square root to compute distances and other engineering matters.

This imaginary number is useful in two-dimensional arrays and other stuff that's beyond the scope of this website for now.

-4^{-(1/2)}

That's a bit strange if you ask me. Could it be imaginary?

Let's take a look at these real and imaginary examples:

64^{2} = 4096

64^{1} = 64

64^{1/2} = Square Root of 64 = 8

64^{1/3} = Cube Root of 64 = 4

64^{0} = 1

64^{-1/3} = 1 / 64^{1/3} = 1/4

64^{-1/2} = 1 / 64^{1/2} = 1/8

64^{-1} = 1 / 64^{1} = 1/64

64^{-2} = 1 / 64^{2} = 1/4096

-64^{2} = 4096

-64^{1} = 64

-64^{1/2} = Square Root of -64 = -8*i*

-64^{1/3} = Cube Root of -64 = -4

-64^{0} = 1

-64^{-1/3} = 1 / -64^{1/3} = 1/-4 = -1/4

-64^{-1/2} = 1 / -64^{1/2} = 1/-8*i* = -1/8*i*

-64^{-1} = 1 / -64^{1} = -1/64

-64^{-2} = 1 / -64^{2} = 1/4096

For other negative fractions? Lotsa luck!

In a base eight system, we use ten digits 0-7, and when you add 1 to 7, you roll over the ones place back to 0, and place the first digit after 0 in the tens place, which is a 1.

In a base two system, we use ten digits 0-1, and when you add 1 to 1, you roll over the ones place back to 0, and place the first digit after 0 in the tens place, which is a 1.

All of the above answers are "10".

Now, what happens in a base one system? We don't have a "1" and you use just one digit, "0". So how is one represented if "0" represents zero? "00"? How about 2? "000"?

How can you roll over the "0" and if you do, how can you tell? What's the first digit after the "0"? None. If 0 = 00 = 000 = 0000000000000000000000000000000000000, you can't increment the number since it's always going to be zero!

Now that there's just a zero in a base one system, what use does this system have?

0/0, infinity/infinity, 0

infinity + infinity = infinity, infinity x infinity = infinity, -infinity + -infinity = -infinity, 0

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