/sci/ - Science & Math

>>5125171
a) 3 x 1/3 = 1 because 3/3 = 1
b) 3/4 = 3 x 1/4 because a quarter of 3 is 3/4.
>CAN SOMEBODY EXPLAIN WHY WITHOUT USING EXAMPLES
Fuck it.

>>5125171
>I just don't understand why they work.
You're dealing with what amounts to different notations of the same basic bits of math. Why do they work? They work because its just saying the same thing in different ways. You're dealing with the math equivalent of synonyms.

>>5125179
Yeah, I'm aware that 3 x 1/3 = 1 because 3/3=1. But why? Why is it that all fractions multiplied by their reciprocal = 1?

Also, I didn't get what you were talking about with b).

>>5125185
Saying the same thing as what? I don't understand what you mean here.

>>5125227
Because 3 in fractional form is 3/1 and 3/1*1/3 is 3*1/1*3 which is 3/3 which is 1. This is true for any 1/x*x/1.

>>5125233
>>5125233
>Because 3 in fractional form is 3/1 and 3/1*1/3 is 3*1/1*3 which is 3/3 which is 1.
Why?

>>5125238
3/1 is 3 divided by 1, which is 3. Thus 3/1 = 3.
3/1*1/3 is the same as 3*1 over 1*3 based on how fractions work.
3*1 is 3, as this multiplication represents 1 set of 3.
3/3 is 3 divided by 3. How many sets of 3 are in 3? 1.

>>5125171
1. definition of reciprocal
2. definition of division
3. follows from commutativity
4. follows from 1. and 3.
5. follows from 1.
6. definition of n-th root

>>5125241
this...
also if OP isn't trolling:
right loudly "n*1/n"
N TIMES 1 OVER N
it's like N times the N-th part of 1.
if you part a pie in n pieces and keep the n pieces, you still get 1 pie.
shit, I think you're trolling after having written that last sentence.

>>5125249
I meant 'read", not "right', brainfart

>*copypasta from about a year ago**or some serious dejavu*
>1. Something multiplied by it's reciprocal equals 1.
If I cut a whole up into some number of pieces, how many time must I glue pieces together to make it whole again?
>Something divided by a number is the same as it being multiplied by its reciprocal.
The amount I cut is always equal to the amount I glue if I want my whole back.
>3. A fraction multiplied by another fraction is the same as their numerators multiplied together divided by their denominators multiplied by each other.
A multiple number of wholes are each cut up the same amount of times. To glue them all back together again, I need to glue each one the same amount of times I cut them
The rest are left for the student as an exercise.

>>5125240
>3/1*1/3 is the same as 3*1 over 1*3 based on how fractions work.
This goes to question 3 in my original post. I can't get the next parts of your post.

Dont be 2 years old and ask why for everything. Explain exactly what you don't understand, otherwise no one can understand what you don't understand and we can't help you.

>>5125241
I'm more confused than ever now.

>>5125249
>right loudly "n*1/n"
Where did we get to n*1/n? I don't get what part you mean. Which part are you explaining and why n*1/n?

>if you part a pie in n pieces and keep the n pieces, you still get 1 pie.
By what do you mean "keep"? In the thing that you are explaining, is the mathematical concept based on "keeping" something?

>>5125252
>If I cut a whole up into some number of pieces, how many time must I glue pieces together to make it whole again?
You need to glue it by the number of pieces you cut it up by.

>The amount I cut is always equal to the amount I glue if I want my whole back.
What do you mean the amount you glue? And glue "back"? Back from what?

>A multiple number of wholes are each cut up the same amount of times.
Wait, I can't go further until I know which parts of the equation are the "wholes".

>>5125259
>Dont be 2 years old and ask why for everything. Explain exactly what you don't understand, otherwise no one can understand what you don't understand and we can't help you.
What exactly I don't understand is the parts that I quote or say I don't understand.

>>5125268
>You need to glue it by the number of pieces you cut it up by.
If you confuse the number of pieces with the number of cuts the student will confuse multiplication with addition and division with subtraction
>What do you mean the amount you glue? And glue "back"? Back from what?
This is why this is an interactive exercise, having the student actually cut and glue either clay blocks or paper.
>Wait, I can't go further until I know which parts of the equation are the "wholes"
This is why some mathematicians make such fucked up teachers...They are more interested in their egos "I'm so smart and your so dumb so just do as I say and call me your highness...look at me mommie, look at me!" Instead of taking the time and creative effort to understand how other people think and tailoring their lessons to the students temperament and ability.
Go drown in a pool of vomit, you arrogant bowl of fail.

>>5125297
>If you confuse the number of pieces with the number of cuts the student will confuse multiplication with addition and division with subtraction
Am I doing that?

>>This is why this is an interactive exercise, having the student actually cut and glue either clay blocks or paper.
I don't understand the whole and pieces in the first place.

>>5125297
>This is why some mathematicians make such fucked up teachers...They are more interested in their egos "I'm so smart and your so dumb so just do as I say and call me your highness...look at me mommie, look at me!" Instead of taking the time and creative effort to understand how other people think and tailoring their lessons to the students temperament and ability.
>Go drown in a pool of vomit, you arrogant bowl of fail.
I'm pretty confused. Are you talking to me or someone else?

>>5125227
>Saying the same thing as what? I don't understand what you mean here.
Its like how "six thousand" can be represented as 6*1000 or just 6000. Six eighths can be represented as (6/8) or 6 * (1/8).

(x/y) * (z/k) = (x*z)/(y*k) <-- this is a goofy one to go into full detail on. More or less, you have a fraction of fractions, so you could look at it as getting z/k, multiplying it by x to get (x*z)/k, then dividing by y to get (x*z)/(k*y) because you have a yth of x (z/k)'s. By inserting 1's and such in that equation, we can get the following:
(1/x) * y = y/x
(1/y) * (y/1) = y/y = 1

Some reasoning with dividing a fraction of something gets this:
(z/x)/y = (z/x) * (1/y) = (z/(x*y))

This can then lead to the 5th question in your post:
A * (x^y) = A * x * x * x ... (A multiplied by x, done y times)
A * (x^-y) = A /x /x /x /x ... (A divided by x, done y times) =A /(x * x * x...) = A / (x^y)

>>5125391
>(x/y) * (z/k) = (x*z)/(y*k) <-- this is a goofy one to go into full detail on. More or less, you have a fraction of fractions, so you could look at it as getting z/k, multiplying it by x to get (x*z)/k, then dividing by y to get (x*z)/(k*y)
so i go (x*z)/k and divide by y. but that's ((x*z)/k)/y

>>5125391
>(z/x)/y = (z/x) * (1/y) = (z/(x*y))
(z/x)/y = ((z/x)*1)/y

>>5125391
>A * (x^y) = A * x * x * x ... (A multiplied by x, done y times)
>A * (x^-y) = A /x /x /x /x ... (A divided by x, done y times) =A /(x * x * x...) = A / (x^y)

Why does the rule work?

What are you having problems with understanding here, OP?

>>5125452
I am pretty sure I understand that example but I don't understand the rule.

>>5125455

What do you mean, you do not understand the "rule"?

>>5125463
I can understand an example like that but I don't understand why all fractions by their reciprocal equal 1.

>>5125467
In his picture, replace 3 with x.

not OP but I don't understand
-*-=+
...how? why?
just an assumption like 0!=1?

also cannot comprehend why after a vector multiplication the resultant vector points to a completely new direction

>>5125468
It's still an example like that because it's using a number (1).

>>5125467

The example of 3 multiplied by its reciprocal 1/3 is simple, you are multiplying 1/3 by three.

Multiplying 1/3 by three is the same as adding three 1/3's together, which make one.

see pic.

>>5125469

Vector multiplication is a "Human defined" Operation. It is not like actual multiplication. it only exists because of it's applications, not because of some underlying mathematical threorum.

For minus times minus, think of it this way. look on a number line, what is 3 times negative two in the opposite direction of negative two? Get it?

>>5125476
Not to mention, even using 1 I still don't quite understand it.

>>5125478
The n example on the right in your picture used the rule in question 3 in the second line.

>>5125479
hmm thinking of the sign as a direction...
that works i guess

>>5125484
What you say makes no sense.

It's not an example. x represents any number you want. 1 is used because that was your question in the first place: "Something multiplied by it's reciprocal equals 1."

>>5125489

I don't know what you are saying here, do you have a problem with my picture?

3 = three = 3/1 = 9/3


8====D = PENIS=COCK=WANGER=FUCK STICK


just different ways of epressing the same thing

Fucking retard.

>>5125501
You explained what I don't understand with what I don't understand either.

>>5125496
I don't get the picture. How does it jump from the first three lines to the fourth?

>>5125511
>You explained what I don't understand with what I don't understand either.

Just so you know, those lines in my picture weren't necessarily related to eachother, they were just various lines of rules and examples that make sense, sorry if you thought they were in an order or something.

You must understand the last example on that picture on the right, surely. that three thirds added together is one whole, i.e 1.

the trick with math is to not look at the numbers theyre confusing and intimidating to stare at. do as much as you can in your head.

just throwin this out there

>have this friend whos a super cool popular guy
>level headed down to earth
>his favorite artist is bomb marley
>be 10:00am he wakes up and shaves his head andeyebrows
>takes a selfie and sends it to his parents
>smashes neighbors glass door
>smashes coffee pot over her head
>chokes her unconcious
>fux her while shes out
>police arrest him outside his apartment building.
>hes refusing a lawyer and to talk to anyone.

Well you're dealing with operations with real numbers. Real numbers are a group with the operation sum and multiplication and thus follow the rules of a group. One of them is the existence of an indentity element for multiplication (an element i which satisfies x . i = x for every x in the group) and an inverse multiplicative for multiplication (for every x=/=0 exists an element y such that x . y = 1). In the case of Real numbers the indentity element is 1 and the inverse element of x is called the reciprocal of x. So by definition an element multiplied by it's reciprocal is 1.
Then regarding division, it's defined as the inverse operation of multiplication ( a.b = c implies that c/b = a and c/a=b) so if lets say x.y =1 (y is the reciprocal of x) and a. x = b, then x.y.b=b which implies that b/x = b.y in other words a number divided by x is the same as the number multiplied by its reciprocal.
I am too lazy to demonstrate the following 4 but they can be demonstrated in the same manner.

>>5125514
>Just so you know, those lines in my picture weren't necessarily related to eachother, they were just various lines of rules and examples that make sense, sorry if you thought they were in an order or something.
I did think this but now I'm corrected

>>5125514
>You must understand the last example on that picture on the right, surely. that three thirds added together is one whole, i.e 1.
Yes, I understand that part, I understand that three 1/3s are 1 and therefore 3/1 multiplied by its reciprocal 1/3 equals 3, but I don't understand why all fractions by their reciprocal equal 1.

>>5125526
>Yes, I understand that part, I understand that three 1/3s are 1 and therefore 3/1 multiplied by its reciprocal 1/3 equals 3, but I don't understand why all fractions by their reciprocal equal 1.

Well, if it works for thirds, then try it with fourths, then fifths.

If this doesn't make sense, then look at the blue N's on my pic.

N is ANY NUMBER, we can see that N multiplied by its reciprocal ( which is 1/n ) is equal to n/n well, anything divided by itself is one.

So we have shown that N, which is also ANY NUMBER, multiplied by its reciprocal is the same as N / N or ANY NUMBER divided by ANY NUMBER, hence divided by itself, which is always 1.

If you don't want to read the long-ass demonstration I posted, just be sure that they work because that how they were defined. It would be like saying why the set of all rocks contains a rock in it. If you take any analysis course they'll go in further detail there.

>>5125529
>Well, if it works for thirds, then try it with fourths, then fifths.
For fifths, 5 wholes of 1/5 = 1. I think I understand 1/3 because I understand all of them that have a 1 in the denominator of one of the fractions and the numerator of the other.


>N is ANY NUMBER, we can see that N multiplied by its reciprocal ( which is 1/n ) is equal to n/n well, anything divided by itself is one.
>
>So we have shown that N, which is also ANY NUMBER, multiplied by its reciprocal is the same as N / N or ANY NUMBER divided by ANY NUMBER, hence divided by itself, which is always 1.
I don't understand the first part of that (question 3) in the first place to understand that as an explanation if I could.

>>5125523
I'll answer this when I'm finished googling all the terms and figuring out what they mean. The first one was "operation sum", the first result of which was "summation" on wikipedia. I hope this is the right place, is it?

>>5125523
Woops, I meant that Real numbers are a field not a group . (a really basic introduction to what a field is: http://en.wikipedia.org/wiki/Field_(mathematics))

>>5125540
>The first one was "operation sum", the first result of which was "summation" on wikipedia
yea you're in the right place, and by the way what you're asking is not that elementary so don't get mad if haven't reached that level of abstraction yet, eventually you'll get there. Everyday we use stuff we don't quite understand but work pretty well.

>>5125540

If a cake is split up into 8 slices, you need 8 of those slices to make one whole cake.

One slice is 1/8 of a whole cake.

So if you multiply that 1/8 by 8, you get a whole cake. A whole cake is 8 lots of 1/8 slices add that up and you get 1.

Now imagine doing it for any number of slices.

>>5125547
So you are saying the understanding of why a number by its reciprocal equals 1 is not elementary?

>>5125540
Well for the reciprocal question:

A reciprocal is just 1 divided by some number. 1 divided by some number, multiplied by the EXACT same number must equal 1.

For your "question 3"

Say you have 4/5 of a pie. You are then ordered to take 6/7 of what is there. 6/7 is the same thing as multiplying by 6, then dividing by 7. Therefore, the new fraction of what you have becomes (4*6)/(5*7), which is 24/35

>>5125552
yea it's not, intuitively it's very obvious that x.1/x =1 but the actual reason for which that holds for any x=/=0 belongs to field theory (i think, any other mathematician thinks I'm wrong please correct me) and much as I love that kind of math , I know it's not taught to everyone (very few people need that kind of math).

>>5125552
I could also ask you why 0 . x=0, it seems obvious but it's impossible to demonstrate with elementary math.

>>5125549
I understand the maths when it's 1 as the numerator of one fraction and 1 as the denominator of the other, but what about 7/9s of a cake split up into 9/7 slices?

>>5125555
>A reciprocal is just 1 divided by some number. 1 divided by some number, multiplied by the EXACT same number must equal 1.
Yeah, that's what I don't understand. Why does the reciprocal of a number multiplied by that EXACT number equal 1?

>>5125555
>Say you have 4/5 of a pie. You are then ordered to take 6/7 of what is there. 6/7 is the same thing as multiplying by 6, then dividing by 7. Therefore, the new fraction of what you have becomes (4*6)/(5*7), which is 24/35
How did you jump from 4/5 x 6 then divided by 7 to (4*6)/(5*7)? Wouldn't 4/5 x 6 and the number divided by 7 be (4/5x6)/7? Or ((4x6)/5)/7?

>>5125568
I get that but I don't get why something like 13/11 x 11/13 = 1.

>>5125572
>I could also ask you why 0 . x=0, it seems obvious but it's impossible to demonstrate with elementary math.
I thought it was because nothing of x equals nothing.

>>5125574
>Why does the reciprocal of a number multiplied by that EXACT number equal 1?

Because you take 1. Then you divide by a number. Then you multiply that same number again. That equals out to the starting number, in this case 1.

>((4x6)/5)/7

Yeah, that's correct. We can simplify that to being (24/5)/7. We can then say this equals to:

(24/7)/5

Now, we can multiply both the numerator and the denominator by 7 to make this look nicer.

(24/7)*7 / (5*7)
=24/35

>>5125586
>13/11 x 11/13 = 1.

put the fractions 13/11 and 11/13 into a calculator.

On a number line, put a point on where 13/11 is and a point where 11/13 is.

1 will be in the middle.

>>5125574
1. Something multiplied by it's reciprocal equals 1.

1/10*10/1 = 1
because one out of ten times ten is one
1/10 is equal to one tenth, 10/1 is equal to 10
ie if you have one tenth of a cake times ten cakes, you have one tenth of each of ten cakes, which you can put together and its the same as one cake, because 10/10 is the same as 1, because ten divided by ten is one, or if there are 10 people and ten cakes they each get one.

>>5125467
Because reciprocals multiplied by each other create a situation where the numerator and the denominator are the same number. Whether it's 6/6 or 36/36, it's basic division that these always equal 1.

Reciprocals are simply a number flipped, which is why you have the same number on both sides of the fraction line. The reciprocal of 23/555 would be 555/23, and both sides are 555*23 or 23*555, which is the same number, which is then divided by itself, which is then 1.

I hope this isn't a troll thread.

I think the thing missing is the understanding of simplification

11/10 is the same as 1.1

>>5125586
>I get that but I don't get why something like 13/11 x 11/13 = 1
well because 13/11 belongs to the field of real numbers and 13/11=/=0 then it has an inverse multiplicative (i.e. reciprocal) it just happens that it's 11/13. The reason why its 11/13 is because lets say 13/11 = k, then 13=k .11 and lets say (q is the reciprocal of k ) then q.13 = q. k .11 but q.k =1 so q.13 = 11 then 11/13=q

>>5125599
>Yeah, that's correct. We can simplify that to being (24/5)/7. We can then say this equals to:
>
>(24/7)/5
I did not know this. Is this another rule?

>>5125599
>Now, we can multiply both the numerator and the denominator by 7 to make this look nicer.
>
>(24/7)*7 / (5*7)
>=24/35

So is this the logic behind 4/5 x 6/7 = 4x6 / 5x7?

>>5125603
What difference does that make?

>>5125638
It follows from other rules. If you have 6/8 of a pie, and you take a third of it, then you have (6/3)/8 of a pie, or 2/8 of a pie.

>So is this the logic behind 4/5 x 6/7 = 4x6 / 5x7?
It's one of the ways to look at it.

>>5125638
>So is this the logic behind 4/5 x 6/7 = 4x6 / 5x7?
the logic behind that would be lets say a/b =n and c/d = m then a=bn and c=dm, if u multiply both u bet ab=bdmn which implies that ab/bd = mn but m = c/d and n a/b so ab/bd = (a/b)(c/d)

>>5125611
>Reciprocals are simply a number flipped, which is why you have the same number on both sides of the fraction line. The reciprocal of 23/555 would be 555/23, and both sides are 555*23 or 23*555, which is the same number, which is then divided by itself, which is then 1.
A lot have provided this explanation, but I'm still in the process of understanding why 23/555 x 555/23 = 23x555/555x23.

>>5125665
read my demonstration >>5125621

>>5125665
and the other one too >>5125664

ITT a derp didn't pay attention when he was supposed to learn that multiplication is serial addition and division is serial subtraction

>>5125621
>The reason why its 11/13 is because lets say 13/11 = k, then 13=k .11
so 13 = 13/11x11 ? 13 = 13/121?

>>5125672
(not OP) but what do you by serial addition/substraction

>>5125675
>so 13 = 13/11x11 ? 13 = 13/121?
i didnt meant 13/11x11 i meant (13/11)x11

>>5125171
>1.
The real numbers are a field so every number with the exception of 0 has a multiplicative inverse
that means, that for every number a there exists exactly one number b, such that <span class="math">a*b=1 [/spoiler]
>2.
don't think of division as an own operation. it is just multiplying with the multiplicative inverse.
so if c is the multiplicative inverse of b, than <span class="math"> a/b=a*c [/spoiler]
>3.
let b' be the multiplicative inverse of b and d' be the multiplicative inverse of d
than <span class="math">(a/b)*(c/d)=(a*b')*(c*d') [/spoiler]. since multiplication is a commutative and distributive operation on the real numbers we can do the following step:
<span class="math">(a*b')*(c*d')=(a*c)*(b'*d')=(a*c)/(b*d)[/spoiler]
note that I used, that b'*d' is the multiplicative inverse of b*d since <span class="math">(b*d)*(b'*d')=(b*b')*(d*d')=1*1=1 [/spoiler]
>4.
reading this gave me a headache, sorry
>5.
we defined, that
<span class="math">x^{a+b}=x^a*x^b[/spoiler]
for any real numbers a,b,x
and <span class="math">x^0=1[/spoiler] for every x=/=0
now lets take
<span class="math">1=x^0=x^{a-a}=x^{a+(-a)}=x^a*x^{-a}[/spoiler]
that implies, that x^(-a) is the multiplicative inverse of x^a
>6
the root is just the inverse function of the "power" function (whatever you call that in english)
since <span class="math"> (x^a)^b=x^(a*b) [/spoiler]it's pretty clear, why <span class="math"> \sqrt[n]{x}=x^{1/n} [/spoiler]

>>5125686
that's what I've been trying to say, OP just carefully read that post and try to understand it.

>>5125664
I'm not getting this exactly. If a/b=n, then a=bn and b=a/n, so ab=bn(a/n).

>>5125678
multiplication is serial addition
2*7 = 2+2+2+2+2+2+2 which is a series of 2s added to itself 7 times
or 7+7 which is a series of 7s added to itself twice

division is serial subtraction: asking "what is 20/5" is the same as asking "how many times can 5 be subtracted from 20 until you reach zero?" the answer is 4, 20-5=15 15-5=10 10-5=5 5-5=0, we subtracted 5 from 20 a total of 4 times to reach zero.

>>5125672
I did pay attention to that.

>>5125701
>I did pay attention to that
how about 0.7 x 3 ? what would be your "serial expansion" for that

>>5125684
OK sorry, I got the numbers mixed up.

>>5125621
>and lets say (q is the reciprocal of k ) then q.13 = q. k .11 but q.k =1
Are you saying that what you said before q.k=1 showed why q.k=1, or is that just a part of what you are showing?

>so q.13 = 11
What are the implications of this?

>>5125710
are you serious here? .7+.7+.7=2.1

>>5125713
>What are the implications of this?
it implies that q=11/13

>>5125713
>Are you saying that what you said before q.k=1 showed why q.k=1, or is that just a part of what you are showing?

remember q is the reciprocal of k (i mentioned that in a parenthesis like this one) so q.k=1

>>5125716
>are you serious here? .7+.7+.7=2.1
then 0.3 x 0.7 ?

>>5125716
the point I was trying to make is that the "serial expansion" is more of a consecuence of the properties of multiplication rather than its definition.

>>5125719
Not him, but it's 0.21.
Are you trying to make a point or something?

>>5125686
>let b' be the multiplicative inverse of b and d' be the multiplicative inverse of d
>than (a/b)*(c/d)=(a*b')*(c*d') . since multiplication is a commutative and distributive operation on the real numbers we can do the following step:
>(a*b')*(c*d')=(a*c)*(b'*d')=(a*c)/(b*d)
It's taking me a long time to get this but I think I'm getting there.

>reading this gave me a headache, sorry
I mean like a*b/c*d = (a/c*d) x b. I didn't get a video on gravity because of this.

>>5125724
no... serial addition IS what multiplication is...

>>5125686
>1=x^0=x^{a-a}=x^{a+(-a)}=x^a*x^{-a}
>that implies, that x^(-a) is the multiplicative inverse of x^a
That makes sense. But why does that lead to x*-a = 1/x^a?

>>5125730
lets say, <span class="math"> c' [/spoiler]is the multiplicative inverse of <span class="math"> c [/spoiler]
<span class="math"> a*b/c*d = a*(b*c')*d=(a*c'*d)*b=(a/c*d)*b [/spoiler]
we can do this, because multiplication on the real numbers is commutative <span class="math">(a*b=b*a)[/spoiler] and associative <span class="math"> (a*b)*c=a*(b*c) [/spoiler]

>It's taking me a long time to get this but I think I'm getting there.
I know. It seems to be a bit of an unintuitive approach to tackle the problem. But it's way better than seeing multiplication as repeated addition and all that crap (because that doesn't work well on irrational numbers).
Also it prepares you for higher math

>>5125686
>since (x^a)^b=x^(a*b) it's pretty clear, why \sqrt[n]{x}=x^{1/n}
I don't get how (x^a)^b=x^(a*b) leads to \sqrt[n]{x}=x^{1/n}

>haven't done math in years
>can slowly feel the basic principles I learned leaking out of my head

This is scary

>>5125748
if a is the multiplicative inverse of b, than <span class="math">1/a=b [/spoiler]and <span class="math">1/b=a [/spoiler]
so if <span class="math">x^{-a} [/spoiler]is the is the multiplicative inverse of <span class="math">x^a[/spoiler], then <span class="math">x^{-a}=1/(x^a)[/spoiler]

>>5125753
lets say <span class="math"> \sqrt[n]{x}=x^a [/spoiler]
we know, that <span class="math">(\sqrt[n]{x})^n=x^1[/spoiler]
so <span class="math">(x^a)^n=x^1[/spoiler]
so <span class="math">x^(a*n)=x^1[/spoiler]
so <span class="math">a*n=1[/spoiler]
that means, that a is the multiplicative inverse of n
that means, that <span class="math">a=1/n[/spoiler]
that means, that <span class="math">\sqrt[n]{x}=x^a=x^{1/n}[/spoiler]

>>5125771
sorry. it should read
<span class="math"> x^{a*n}=x^1 [/spoiler]
in the 4th row

>>5125750
>lets say, c' is the multiplicative inverse of c
>a*b/c*d = a*(b*c')*d=(a*c'*d)*b=(a/c*d)*b
>we can do this, because multiplication on the real numbers is commutative (a*b=b*a) and associative (a*b)*c=a*(b*c)
It took me 10 minutes but I finally got this. I had to take the b/c = b*c' part at face value or for granted (however you say that) because I still don't fully understand that and the 'associative' part confuses me a little but I got it.

>>5125750
>I know. It seems to be a bit of an unintuitive approach to tackle the problem. But it's way better than seeing multiplication as repeated addition and all that crap (because that doesn't work well on irrational numbers).
>Also it prepares you for higher math
Yeah, I really appreciate your time and effort. I got the one I said I was getting there to based on the principles above. I'm now trying to get >>5125762

>>5125786
>I had to take the b/c = b*c' part at face value or for granted (however you say that) because I still don't fully understand that
that's a very important part.
dividing a number x by another number y is the same as multiplying x and the multiplicative inverse of y
<span class="math"> x/y=x*y' [/spoiler]
I don't really like the concept of division, because it complicates the whole thing.
People tend to think, that they need to remember new rules when it comes to division, but those can be derived from the rules we have for multiplication

>>5125762
So that is right because 1/(x^a) = (1/x)^a? What is that called?

physicsforums.com/showthread.php?t=641649

Why can't our mods be like their mods? :(

>>5125816
>>>5125762
>So that is right because 1/(x^a) = (1/x)^a? What is that called?
it's because <span class="math"> p^a*q^a=(p*q)^a [/spoiler]
so if p=1 and q is the inverse of x (lets call it <span class="math">x' [/spoiler]again), than
<span class="math"> 1/(x^a)=(1^a)/(x^a)=(1^a)*(x'^a)(1*x')^a=(1/x)^a [/spoiler]
(I used here, that <span class="math">1^a=1[/spoiler])

>>5125846
damn. it should be
<span class="math"> 1/(x^a)=(1^a)/(x^a)=(1^a)*(x'^a)=(1*x')^a=(1/x)^a [/spoiler]

>>5125821
I guess because they aren't so weak that they are scared of the regulars and do exactly as they say, and also fold under questioning?

>>5125771
What is x^a? I've been reading this for over 20 minutes and I don't get it.

>>5125849
>1/(x^a)=(1^a)/(x^a)
How does that work?

this is the finest maths thread on /sci/ in a while

>>5125855
>What is x^a? I've been reading this for over 20 minutes and I don't get it.
I assumed, that the nth root of x is some power of x.
I didn't assume, that it was 1/n so I just called it a.
I proved, that a must equal 1/n, so we can write the nth root of x as x^(1/n)

>>5125855
>How does that work?
like i said,
I used, that 1=1^a for all real numbers a

>>5125846
OK so 1=1^a so 1/x^a is (1^a)/(x^a)

and that is the same as 1^a by the inverse of x^a, which is x'^a

Multiplying those two is just x'^a, which is (1/x)^a

How did you do all those steps so quickly to get from 1/(x^a) to (1/x)^a?

>>5125887
>How did you do all those steps so quickly to get from 1/(x^a) to (1/x)^a?
it's mostly experience.
also
>inverse of x^a, which is x'^a
I almost forgot to prove this, since it's not that trivial, so:
<span class="math"> x^a*x'^a=(x*x')^a=(1)^a=1 [/spoiler]
therefore <span class="math">x'^a[/spoiler] is the multiplicative inverse of <span class="math">x^a[/spoiler]
(i used, that <span class="math">x*x'=1[/spoiler])

>>5125171
> Can anyone explain why instead of just using examples

Not really.

It's the examples that convince everyone that the rules make sense.

The rules are created in a way that keeps the patterns as consistent as possible. Without examples, you can't express what the patterns are.

Like negative powers for example. You just start with positive powers, and work the pattern down:

2^3 = 8 -- divide both sides by 2 to get:
2^2 = 4 -- divide both sides by 2 to get:
2^1 = 2 -- divide both sides by 2 to get:
2^0 = 1 -- divide both sides by 2 to get:
2^(-1) = 0.5 -- divide both sides by 2 to get:
2^(-2) = 0.25 -- etc.

The pattern of the exponents is perfectly regular. This regularity makes it extremely compelling that 2^0 should be 1, and 2^(-1) should be 0.5. If anyone claimed that 2^0 or 2^(-1) are something else, then they'd have a shitload of difficult explaining to do as to why they aren't conforming to the pattern.

Notice that it's this EXAMPLE that makes the negative power rule compelling and understandable. If I just had a bunch of dry theory or explanation without any examples, then it just wouldn't have the impact that this example does.

I've had a number of math instructors who just don't get this. They think that the theory and the bla bla bla explanations are what makes it understandable. Well, that's usually not the case. Instead, it's usually a well-constructed example that gets through to the students the best.

>>5125906
OK I still haven't gotten why a number to the power of a fraction is the same as the denominator root of that fraction by it's numerator.
I just can't get why

>since (x^a)^b=x^(a*b) it's pretty clear, why \sqrt[n]{x}=x^{1/n}

>>5125994
If b is a number such that b^n = a, meaning it's the n-th root of a, then b = a^(1/n). Then b^m = (a^(1/n))^m = a^(m/n) = (a^m)^(1/n), which is the n-th root of the m-th power of a.

>>5125994
have you read
>>5125771
and the first half of
>>5125866
?

>>5126022
Yes, I went through the proof and could see that each step worked out. But I don't get it.

>>5126059
>But I don't get it.
Have you considered trying harder?

>>5125171
>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.

>>Field axioms
>there exist an 'number' '1'≠'0' such that for all 'numbers' x 1*x=x*1=x
>for all 'numbers' x there exist a multiplicative inverse x⁻¹ such that x*x⁻¹= x⁻¹*x=1 usually written as x⁻¹=1/x
>if a, b, c are 'numbers' then (a*b)*c=a*(b*c) and we then just write abc to stand for either
>if a and b are 'numbers', then a*b=b*a

>proposition <span class="math">[/spoiler] the multiplicative inverse is unique.
Proof: assume a and b are inverses for x, then a=a*1=a(xb)=(ax)b=1*b=b so the inverses are the same.

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

(ba)*(1/a)*(1/b)=(b)*(a*(1/a))*(1/b)=(b)*(1)*(1/b)=b*1/b=1
thus (1/a)*(1/b) is the multiplicative inverse for ab and thus by the earlier proposition (1/a)*(1/b)=(ab)=1/(ab)

>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.

notation: let a/b stand for a*1/b
let a,b,c,d be 'numbers'

then (a/b)(c/d)=a*1/b*c*1/d=a*c*1/b*1/d=ac*1/b*1/d by 3 above we have 1/d*1/b=1/(bd) and so ac*1/b*1/d=ac*1/(bd)=(ac)/(bd)

>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.

notation let aⁿ=a*a*a...*a n times
notation let a⁻ⁿ=(aⁿ)⁻¹
>proposition <span class="math">[/spoiler] (a⁻¹)ⁿ=a⁻ⁿ
proof: assume it holds for n then for n+1 (aⁿ⁺¹)*(a⁻¹)ⁿ⁺¹=a*(aⁿ)(a⁻¹)ⁿ*a⁻¹ by the hypotheses (aⁿ)(a⁻¹)ⁿ=(aⁿ)(a⁻ⁿ)=1 so a*(aⁿ)(a⁻¹)ⁿ*a⁻¹=a*1*a⁻¹=1 thus (a⁻¹)ⁿ⁺¹=a⁻ⁿ⁻¹

>>5126093
I'm trying as hard as I can. I focus on what I'm reading and try to make sense of it. I mean I read the proof and I can see that it proves that for x^a to be the same as \sqrt[n]{x}, a must be 1/n, and therefore \sqrt[n]{x} = x^1/n, but I don't get how "since (x^a)^b=x^(a*b) it's pretty clear, why \sqrt[n]{x}=x^{1/n}".

>>5126138
>(1/a)*(1/b)=(ab)=1/(ab)
(1/a)*(1/b)=(ab)⁻¹=1/(ab)

>>5126145
> I don't get how "since (x^a)^b=x^(a*b) it's pretty clear, why \sqrt[n]{x}=x^{1/n}".
okay forget about that.
I was too lazy to write that out.
sorry for confusing you. the first part should be all you need

>>5126145
You've had more than one explanation posted already. If you can't grasp elementary multiplication then you're truly dim.

>>5126146
>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.

Define your field. If it's ℚ there might not be a such number. Anyway by definition let b=√a represent the 'number' if it exist such that b²=a and b>0 and write b=<span class="math"> \rm a^{1/2} [/spoiler]. Definition: <span class="math">\ rm \sqrt{n}{a}=a^{1/n}[/spoiler]

proposition: n roots are unique
proposition: <span class="math">\ rm {a^{1/n}}^{n}=a[/spoiler]

>>5126138
Still reading through this, about half-way through.

>>5126155
What do you mean the first part?