SCHOOL, A New Adventure

Speedie: I have to hand it to you for persisting to understand what you got wrong and why. You clearly are going to do great!

I have been known to be a wee bit stubborn. ... from time to time. ;)

Ha. Good work! I'm glad you were able to clear your head of it.

Goooood morning!!

...Well, I think it's good... except for... now there's another thing I can't seem to clear my head of. Paul, would you mind helping me out here, please?

What's getting me is the WHY and the HOW of the rules of negative exponents.

WHY is it that Aˉ² = 1/A² ?? HOW did the Aˉ² become positive?? What is the logic, and what are the math steps that turned it from Negative to Positive?? I mean, aside from the rules.
I'm not entirely good with MEMORIZING rules without understanding the reasoning behind them -- (at least, in math), and this is making me NUTS!! I've done all sorts of searches (mathisfun.com, khanacademy, purplemath, several .edu websites), and they all say the same thing: THE RULE SAYS DO THIS. But not one of 'em says WHY and HOW.
Can you explain it, please? Thank you! =)

P.S. Yes, I was one of "THOSE" kids (but, WHY?? but, HOW??)... I never got any answers back then, which is why I'm stuck in this boat NOW at 49 years old! < =/

Meanwhile, if I can get this to stick in my head, then I'll be ready for the next quiz, and then the next Progress Exam on Tuesday! That'll be Progress Exam #6 out of 8, WHEEEE!!!!! =)

Speedie,


Try this on.
http://answers.yahoo.com/question/index?qid=20101028140033AA7FRnO
I googled negative exponent proof.

"So, you accept the definition for positive integer. What about 0? Do you accept that:

a^0 = 1

for any a not equal to 0? Why do we define a^0 to be this? Well, we do so, more or less, because it fits in with our established laws of exponents. So:

a^(m + n) = a^m a^n
a^(m - n) = a^m / a^n
a^(mn) = (a^m)^n

If we substitute n = 0 into the first two, we see that they still hold:

a^(m + 0) = a^m = a^m * 1 = a^m * a^0
a^(m - 0) = a^m = a^m / 1 = a^m / a^0

If we do the same thing to the third rule, we see that it works too. Also, consider:

a^0 = a^(n - n) = a^n / a^n = 1

Basically, it fits everything we expect from exponents, so we add it in, as a kind of continuation of the exponents. We justify negative exponents the same way. Consider:

a^(-m) = a^(0 - m) = a^0 / a^m = 1 / a^m = (1/a)^m"


P

This message was edited Sep 26, 2013 6:00 AM

Duplicate post**

This message was edited Sep 26, 2013 11:34 AM

HUH????????????????????????????

"I understand your proof and I know it can't be explained mathematically just like multiplying can't be proved. It is an idea and derived on cumulative logic."

Ah, "cumulative logic" now there's a fine kettle of fish! So, if it walks like an exponent and follows the rules of an exponent we use it as an exponent til the collective wisdom recollects itself into some thing better....

God bless all of the whys alive and well in the world. Magic is afoot.

Heehee my eyes just glazed over, just like when DH starts talking Calc or physics. Sorry.

LOL, you guys crack me up; Gita with your huh???? and Jan with your glazed (donut) over eyes, Hahahahahaahaaa!!!!!!! Heeheeheee Judy; DH keeps telling me "It's not magic, it's math!" =)

OK, to the nitty gritty here. Paul, thank you, I think I get it! Sooo... basically, we know (intuitively) that: with a positive exponent, we multiply using that exponent's value. (eg: 8^5 would be 8 X's itself 5 times)... Logic then tells us, with a negative exponent, we are doing the opposite, ie: Divide. To my Polish, laymen's brain, positive exponents mean we want a bigger number, thus MULTIPLY. Negative ones mean we're going "down the number scale" -- aka: "getting smaller", so we DIVIDE. But no matter how you slice it, we do the action BY that VALUE of the exponent... not by the sign. Which is why the - in A^ -2 becomes a +A, thus it = 1/A^2...Because the 2 is the value of the exponent.

If that's it, then I GET IT!!! HALLELUJAH!!!! That makes PERFECT sense to me!!!
Ooooh, I just KNEW you'd be the man for the job, thank you ever so much!!! YOU DA MAN!!!!!!

Oh, and by the way... A^0=1 makes perfect sense to me as well, including all the mathematical reasoning behind it. That's the easy part! =)

If it works for you that is great. Me, it is best understood deriving it from the definition. Everybody is different!

hooray for Paul...and speedie!

Heeheeheee, Hooray for Paul, to be sure! =) Thank you for confirming that, regardless of how I got there, I still got there! (not the first time I've been called "different" either, by the way) ;P

I had actually asked my "professor" (and I use that term very very loosely!) at school this same question yesterday during class time... she had no idea, nor did she understand it when I asked her "Can you 'show me the work' to show me how you got there?" She said (I kid you not!) "What do you mean show you the work?" She told me that she guesses 'we are not meant to know' the "how" of how we got there. WHAT!?!?! Really!?!?! This is a very basic idiot-math class, and you can't explain it!?!? ... deep sigh...

Paul, are you familiar with 'ratemyprofessor.com'? Would/could you recommend that as a helpful tool in choosing my next math "teacher"?

I am familiar with it. Your best bet is just to talk to your academic advisor. Be open and honest, and communicate what you're looking for.

This message was edited Sep 27, 2013 7:43 AM

I agree with Terp.

Can't say it better than Calvin and Hobbs!

First frame
Calvin: You know, I don’t think math is a science. I think it’s a religion.
Hobbes: A religion?

Second frame
Calvin: Yeah. All these equations are like miracles. You take two numbers and when you add them, they magically become one new number! No one can say how it happens. You either believe it or you don’t.

Third frame
Calvin: This whole book is full of things that have to be accepted on faith! It’s a religion!

Fourth frame
Hobbes: And in the public schools no less. Call a lawyer.
Calvin: As a math atheist, I should be excused from this.

Haahahaahaaaahaaa Judy! Where was this when I was in High School!?! < =D

Paul, that is a much better idea, I agree with you.

Been reviewing and reviewing the last 3 chapters since Thursday... Got the next Quiz tomorrow, and then Progress Exam #6 TOMORROW!! If any of you have a spare second for a quick prayer please, I sure would appreciate it! Thank you guys!

After this, there will be one more Progress Exam covering the next 3 chapters (preceded by their quizzes, of course), and then the Final! WHEEEE!!!

Speedie, it sounds like you are speeding through this course! :-) Good luck with the exams!!

Aaawwww, God Bless your warm fuzzy heart!! < =D

YEAH BUDDY!!! I SMOKED those tests today!!!!!!!!! THANK YOU LORD!!!!!! And Thank You Paul for your help; there were LOADS of those awful types of problems on the Progress Exam, and were it not for your help in understanding how they work, I would have been a flounder! But, thanks to you, I just looked at 'em and said "Ah, I know what this is, I can do this!". WOOHOOO!!!!!! NOT ONE OF 'EM WRONG!!!!! < =D

Speedie---

I have NEVER doubted that you are a 100 in anything....
You are such a smartypants.....:o)
G.

That's terrific, Speedie! Congrats!! :-)

That's awesome! Well congratulations to you, credit goes to your curiousity and hard work.

Hip-hip...

HURRAAAAYYY!!!!!!!!!!!!!!!!!!!!!!!!!! < =D

I do take full credit for being stubborn and nosy and one of "those" pain in the butt students! "But teacher, why? Why?? WHYYY!?!?!" < =D LOL!!!!!!!!

Speedie--
And I bet the teachers love you for it--as you are NOT a boring lump on a log.
Asking "WHY" is a sign of intelligence--you want to know MORE!!!
You are NOT satisfied with partial knowledge.

You are a mature (??) adult and have learned the value of understanding
WHAT you are learning...not just sitting in class cause you have to.

Don't ever change!!! Gita

I think I may have annoyed this instructor when she couldn't answer that last query; she had to go out to her car to get a book to try to find an answer. Granted, there is no algebraic formula or "work" to show how to get from one point to the next, but there IS a pattern to follow and intuition/logic to follow... I may have frustrated her because she just simply did not know anything but the "rule". "Don't make me think about it, I can just follow a rule" type of thing. Sorry, this girl don't play like that. =P I don't think she enjoyed being challenged that way. **shrug**

When I took those last 2 tests the other day, I did it on "my own time", so I used one of the open math labs, didn't do it on class time in her class, so I've not seen her for a while. Not going in today either 'cause class time is only 50 minutes, and I am NOT gonna drive 40 minutes each way just to sit in class for 50 minutes... just to do what I can do uninterruptedly at home!! That's just a stupid waste of gas and time! =)

Ah Grasshopper you calculate well.

I think college instructors should not be annoyed when a student asks a challenging question--they should be happy that someone is showing that much interest and depth of thought, and that they themselves are being challenged and stretched! Hopefully she did feel that way, despite the inconvenience of getting the book from the car.

Book? Did someone say book? My favorite mobile device!

Me three!! No such thing as too many! =)

amen, Speedie--especially when it comes to books! :-)

My love for the written word began back when I was about... oh, about 4 years old I think, when I started trying to teach myself to read using my Dad's newspaper. I've always been a bookaphile... readaphile..... libraryaphile..... one of those things! < =D

Speedie--
What key do you use to draw a line through text--like you did?
I tried when I was editing my haves list a while back--and nothing worked....

What you type is < s > (with no spaces in between) right before the word(s), and then < / s > (with no spaces) right at the end of the word. If you click on the "Quote" link, to try to quote me, you'll see it in action on your forehead here. =)

ReALLY!
I thought that using "---" may be involved.....

OK! Gonna try it here.

I am tired of all this rain we are having.!!! OOOPS!

There is NO "quote" next to the reply box!!!! So--if I send it--
then I can no longer type all the .....in it--unless I edit.

Maybe you missed telling me I need to edit???

Sending it now...