Why Do Basic Math Operations and Properties Seem Confusing?

[im learning]

I am trying to learn maths and I don't understand the following:

1. Something multiplied by it's reciprocal equals 1.

2. Something divided by a number is the same as it being multiplied by its reciprocal.

3. A fraction multiplied by another fraction is the same as their numerators multiplied together divided by their denominators multiplied by each other.

4. A number multiplied by a number divided by a number multiplied by a number is the same as one of the numerators divided by the product of the denominators and the total of that multiplied by the other numerator.

5. A number to the power of a negative number is the same as it's reciprocal to the power of the same power with the opposite sign.

6. A number to the power of a fraction is the same as the root of that number to the degree of the denominator (not sure if I wrote that right) and the product of that to the power of the numerator.

Can anyone explain why? Examples aren't helping me. I can see that examples and proofs work, I just don't understand why they work. For example I can see that 3/1 x 1/3 is the same as 1 because 3 x 1 = 3 and 3/3 = 1 but I don't understand why every fraction multiplied by its reciprocal equals 1.

I'm mostly using Khan Academy. I've watched all the playlists in the Arithmetic and Pre-Algebra section.

 

[arildno]

"1. Something multiplied by it's reciprocal equals 1."

It is DEFINED that way, there is no deeper understanding to be had.

It is not anymore to be understood that in a card game, a trump of any value beats a non-trump of every value.

1. is, quite simply, one of the rules of the game we call ordinary maths, by which we play about.
Such rules like 1. are called "axioms" in fancy math lingo.

 

[im learning]

"1. Something multiplied by it's reciprocal equals 1."

It is DEFINED that way, there is no deeper understanding to be had.

No deeper understanding than what?

It is not anymore to be understood that in a card game, a trump of any value beats a non-trump of every value.

I don't know what that means.

1. is, quite simply, one of the rules of the game we call ordinary maths, by which we play about.
Such rules like 1. are called "axioms" in fancy math lingo.

From Wikipedia:

An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

I accept it as true, I just don't understand it.

 

[arildno]

Have you never played card games, where one of the suits is designated trump?

 

[im learning]

Have you never played card games, where one of the suits is designated trump?

I've never played card games.

 

[arildno]

I've never played card games.

Do you know what it is?

 

[im learning]

Do you know what it is?

Do I know what what is?

 

[arildno]

Hereby, I declare you a troll, report you to the moderators, who will henceforth ban you from the forums.

 

[im learning]

Hereby, I declare you a troll, report you to the moderators, who will henceforth ban you from the forums.

Why do you declare me a troll and why do you report me to the moderators? Why will they ban me from the forums?

 

[im learning]

Hereby, I declare you a troll, report you to the moderators, who will henceforth ban you from the forums.

Hereby, I declare you as someone who accuses people who aren't trolls of being trolls, then cries to the moderators.

 

[Norwegian]

Another troll has entered the building. I don't find this one particularly interesting or entertaining, so I advice you to change your act or go troll somewhere else.

 

[im learning]

Another troll has entered the building. I don't find this one particularly interesting or entertaining, so I advice you to change your act or go troll somewhere else.

Who are you talking to?

 

[im learning]

And who about, and why on this thread?

 

[HallsofIvy]

If you are not a troll, and I would prefer to think you are not, then you seem to be asking for some kind of "mystical" answer to basic mathematics statements, and there simply aren't any. Mathematics is a human creation and its properties are whatever we want them to be. If, for some number x, there exist another number, y, such that xy= 1, then we say that "y is the reciprocal of x" and "x is the reciprocal of y" purely as a definition. We can then write y= 1/x and x= 1/y but those are due to the specific notation we choose and no "deeper" properties. That is all that one can say about your questions 1 through 4.

The other questions, 5 and 6, having to do with exponents are slightly more complicated but still largely a matter of the way we have defined elementary arithmetic operations.

". A number to the power of a negative number is the same as it's reciprocal to the power of the same power with the opposite sign."

We can show, just by counting, that for any positive number a and any positive numbers, x and y, $a^{x+ y}= a^xa^y$. a^n means "a multiplied by itself x times" so that $a^{x+y}$ means "a multiplied by itself x+ y times". But we were to write "aaaaa...aaaa" x+ y times, we could break that into (aaaa...aaa)(aaaa...aaa) where there are x "a"s in the first parentheses and y "a"s in the second. The first is the same as a^x, the second is the same as a^y and they are multiplied together: $a^{x+y}= a^xa^y$

That's a very nice, simple, property and we would like it to be true for all x and y. In particular, if it were true for y= 0, we would have $a^xa^0= a^{x+0}= a^x$. Since a^x is never 0 for positive a and x, we can divide both sides by a^x to get $a^0= 1$. Because we want $a^{x+y}= a^xa^y$ to be true, we define $a^0= 1$.

If that formula were true for negative numbers, then we could say, for any positive x, that $a^{x}a^{-x}= a^{x+ (-x)}= a^0= 1$. Again, we can divide both sides by a^x to have $a^{-x}= 1/a^x$. In order to make that nice property true even when the exponent is negative, we define $a^{-x}= 1/a^x$.

6. A number to the power of a fraction is the same as the root of that number to the degree of the denominator (not sure if I wrote that right) and the product of that to the power of the numerator.

We can show, again by simple counting, that [itex](a^x)^y= a^{xy}[/tex]. We can, for example, write a^{xy}= aaaa...a, a total of xy "a"s multiplied together. Place parentheses around each "x" of them so that (aaaa...a) is a^x. But that means there are y parentheses so we are multiplyin a^x by itself y times: $a^{xy}= (a^x)^y$.

Now, we would like that to be true for all x and y, not just integers. In particular, if x is a non-zero integer, and y= 1/x, not an integer, then that property says we must have $(a^{1/x})^x= a^{xy}= a^{x(1/x)}= a^1= a$ and then, taking the x root of both sides, $a^{1/x}= \sqrt[x]{a}$.

But, again, all of those are definitions. We define these things to be true because they make formulas simple. There is no "deeper" (or mystical) reason.