Then, on top of that, you get derivatives and integrals, which are to exponents what exponents are to multiplication.
No, that would be incorrect. Integrals and derivatives are not numerical operations. You'd want this:
http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
None.
I'm confused at what you're saying...you mean that you can't multiply x.5 and x3 without simplifying? You're right...and it doesn't get any simpler. You keep the base, add the exponents (x.5 + 3, and get x3.5.[/color]
I didn't quote the post I was responding to (It was Roy's).. but read it and then re-read mine. Context
is helpful.
if you still don't understand
Show an equivalent of 4.5*3.5 using only addition without any form of multiplication... I am only allowing you to expand things, not simplify them (so you can't just assume or do 3.5*.5 and write the results down). You'd get 4.5 + 4.5 + 4.5 + something. But what?
I mean that, but with exponents and multiplcation (4.5^3.5).
Post has been edited 6 time(s), last time on Apr 9 2011, 1:59 am by FaRTy1billion.
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We can't explain the universe, just describe it; and we don't know whether our theories are true, we just know they're not wrong. >Harald Lesch
Multiplication is division
48 * 0.5 * (9 + 3)
That's it baby. Almost a whole page of pointless discussion while the truth being glaringly obvious, once proof is found. Way to be a boss guys.
Please get to sqrt(2) with just a finite series of multiplications and additions.
You cannot get an
exact representation of sqrt(2) with a finite number of ANY operations.
The exponent should not be included as an elementary operator because it can be computed by using only multiplications and additions.
If I give you just a pen and paper and tell you to calculate 3^pi, how would you do it? There's only one way to do this and it involves an infinite number of multiplications and additions (the same goes for any irrational exponent).
Computers do it the same way. Whether it's log, exponent, the gamma function or whatever you want, they're all computed as multiplications and additions, albeit in finite numbers to obtain a good approximation but if an infinite number was used, you'd get the exact representation.
For example, exp(x)=sum(x^k/k!) for k=0 to infinity.
None.
The exponent should not be included as an elementary operator because it can be computed by using only multiplications and additions.
And multiplication can be computed only using addition. So how is it different?
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And multiplication can be computed only using addition. So how is it different?
Well, for one, it would complexify computations drastically.
I don't know whether or not it's impossible to completely exclude multiplication as an elementary operator for real numbers but here's an argument that should be more or less satisfactory:
How would you compute the product of, say, 0.5*4.5 in terms of additions? Sure, you could argue that it's 1+1.25, 3.25-2, ... but there exists a x+y representation for ALL real numbers; you'd have no "good" manner(i.e. algorithm) of picking out which number you'd need to use in your addition.
Wait... maybe it's just late and I'm forgetting something basic here.
None.
Multiplication and exponentiation can be reduced to a lower-order operation
if and only if the operands are integers. 4 * 4 = 4 + 4 + 4 + 4, but 4 * 3.5 cannot be expressed in terms of its operands using only addition; you must perform the multiplication 4 * 0.5 at some point. Similarly, you cannot reduce fractional exponents like 4
3.5 to just multiplications; you
must perform an exponentiation at some point with a fractional part.
Obv. we can use Taylor series to express everything, but that is irrelevant as it is an infinite series.
None.
How would you compute the product of, say, 0.5*4.5 in terms of additions? Sure, you could argue that it's 1+1.25, 3.25-2, ... but there exists a x+y representation for ALL real numbers; you'd have no "good" manner(i.e. algorithm) of picking out which number you'd need to use in your addition.
That was the same argument I was using. xD
if you still don't understand
Show an equivalent of 4.5*3.5 using only addition without any form of multiplication... I am only allowing you to expand things, not simplify them (so you can't just assume or do 3.5*.5 and write the results down). You'd get 4.5 + 4.5 + 4.5 + something. But what?
I mean that, but with exponents and multiplcation (4.5^3.5).
Except I'm trying to extend it to exponents instead of multiplication... I'm claiming exponents are to multiplication as multiplication is to addition. How would you do 0.5^4.5 with only multiplication?
... So why are exponents excluded from the "addition and multiplication are the basic operators!"?
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Multiplication and exponentiation can be reduced to a lower-order operation if and only if the operands are integers. 4 * 4 = 4 + 4 + 4 + 4, but 4 * 3.5 cannot be expressed in terms of its operands using only addition; you must perform the multiplication 4 * 0.5 at some point. Similarly, you cannot reduce fractional exponents like 43.5 to just multiplications; you must perform an exponentiation at some point with a fractional part.
Obv. we can use Taylor series to express everything, but that is irrelevant as it is an infinite series.
Read my first post in this thread. The ONLY way to compute x^i where i is an irrational number is by using an infinite number of multiplications and additions.
Except I'm trying to extend it to exponents... I'm claiming exponents are to multiplication as multiplication is to addition.
... So why are exponents excluded from the "addition and multiplication are the basic operators!"?
Because all exponents can be computed as additions and multiplications (i.e. it's the ONLY way) whereas not all multiplications can be expressed as additions.
Think of it fundamentally: what's an operation in the first place? Let us consider a binary representation for the sake of simplicity. For addition, you'd need to "memorise" the following "associations":
1+0 makes 1
1+1 makes 10
0+0 makes 0
You can then repeat that process for numbers that are as large as you like.
For multiplication it would be:
1*0 makes 0
0*0 makes 0
1*1 makes 1
However, you do not need to "memorise" any associations for exponents or anything else as they are simply extensions of those two operations.
Note, however, that if you were to "memorise" more such associations, you could also considerably reduce computation time.
None.
Multiplication and exponentiation can be reduced to a lower-order operation if and only if the operands are integers. 4 * 4 = 4 + 4 + 4 + 4, but 4 * 3.5 cannot be expressed in terms of its operands using only addition; you must perform the multiplication 4 * 0.5 at some point. Similarly, you cannot reduce fractional exponents like 43.5 to just multiplications; you must perform an exponentiation at some point with a fractional part.
Obv. we can use Taylor series to express everything, but that is irrelevant as it is an infinite series.
Read my first post in this thread. The ONLY way to compute x^i where i is not a whole number is by using an infinite number of multiplications and additions.
You clearly did not read
my post since you stipulated that I read your post when in fact I have actually responded to your post within the body of mine.
The short answer: No, sir, Taylor series are horribly inefficient for fast computation requirements and are definitely not the only way to compute those things. This is why we use alternatives such as variations of Newton's method to do things like
Fast invSqrt. Please do some research before claiming these things.
None.
Where did I ever say that a taylor series was the only way to compute it? I said that you needed to use additions and multiplications. You could also memorise the numbers but that's not an operation.
You don't need to get pissy, it certainly won't get us anywhere.
None.
Just here for the activity... well not really
48÷2(9+3)
48/2*9+2*3
48/(18+6)
48/(24)
= 2
Distributive property, guys.
trollface.jpg
guy lifting weight (animated smiley):
O-IC
OI-C
"Oh, I see it"
I feel stupid for voting 2...
I just realized.
None.
An artist's depiction of an Extended Unit Death
48/2(9+3)
48/2*9+2*3
48/(18+6)
48/(24)
= 2
Distributive property, guys.
trollface.jpg
Actually, you would be distributing 48/2, because it is a fraction and cannot be separated in the manner you've shown.
48/2(9+3)
(48/2)*9 + (48/2)*3
24*9 + 24*3
216 + 72
= 288
Distributive property, guys.
Not reading the whole mess, I assume 288 was agreed upon because anything else is just making an associative assumption.
Saw this thread on TeamLiquid too, was astounded how many people don't understand the reasoning behind rules and just blindly memorize them.
None.
Saw this thread on TeamLiquid too, was astounded how many people don't understand the reasoning behind rules and just blindly memorize them.
The education system isn't doing a good job.
None.