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I've begun a new thread on this matter because I believe I had gone about phrasing my question in a convoluted manner in my previous related thread.
I believe this question warrants a distinct and succinct answer; I feel I only have myself to blame for not asking the right question.
This thread does justice to a question put forth online several times and, as far as I can tell, is only answered in part. What I'm finding online is summarized below and as one can see... there is something missing.
To reiterate, I'd been thinking about the terms relation, function and operation as in their primary similarities and differences.
For instance, a function is always a relation, but a relation is not necessarily a function. A relation is not necessarily a function because a relation,
unlike a function, may involve more than one output.
In the same vein it can also be said that an operation is always a function, but a function is not necessarily an operation. A function is not necessarily an operation because a function,
unlike an operation, ______________________________.
When is an operation not a function?
____________________________________________________________________________________________________________________________________________________________
I just want to make sure my understanding of certain fundamental terms will not ultimately be contradictorial. I apologize for my misinterpretation.
I met up with my math tutor today and he wasn't sure, but he suggested that perhaps factorization was the inverse operation of exponentiation.
That is, if exponentiation is repeated multiplication, then factorization would be repeated division.
For example 2x2x2= 8 as 8/2/2= 2.
He also pointed out that as ShivamS suggested, in a way, nth root and logarithm are inverse operations of eachother.
That is, the index of the nth root is the solution (or exponent) of logarithm, and the base of logarithm is the solution (or root) of nth root.
For example, the 3rd root of 8= 2 as the log base 2 of 8= 3.
If this is a practical understanding, I wonder now why there is little on the internet to support it. So what do you all think about this? Can factorization be considered an operation?
How could you define nth root and logarithm in relation to exponentiation and factorization, as they are in relation to multiplication and division, and as they are in relation to addition and subtraction?
It all seems very fitting so far, though of course I'm not so far along as you all. Would this understanding become a problem later on as my mathematical education advances?
I have a couple questions in regards to the relationship and evidently non-relationship between exponentiation, logarithms and nth roots.
For starters let me just start of at the beginning...
...Addition is the operation of combining quantities
Subtraction is the operation of inverting addition...
...Multiplication is the operation of repeating addition
Division is the operation of inverting multiplication...
Exponentiation is the operation of repeating multiplication...
...but what is the inverse operation of exponentiation? I read in wiki that the inverse of exponents are logarithms and if this is true, then what about nth roots?
All three of these terms seem to be directly related, even though nth root is given different terms for its inputs and output.
It's true that...
Base is to exponentiation (and logarithm) as root is to nth root
Exponent is to exponentiation (and logarithm) as index is to nth root
Power is to exponentiation (and logarithm) as radicand is to nth root
E.G.
2^3= 8 as the LOG base 2 of 8= 3 as the cubed root of 8= 2
So if logarithm is in fact the inverse of exponentiation, then why? And what is the inverse operation of nth root?
Thanks for the reading suggestions. As for the definitions, I believe it's best for me to learn a little more and then come back to it.
An operand is an object upon which a mathematical operation is performed. Sometimes you can use them interchangeably, but they have different meanings.
An operand is an object upon which a mathematical operation is performed... which is not always an input? Can you give an example when an operand is not an input.
Before I go, I feel compelled to ask just one more question. Is there any difference what-so-ever between the operand and the input? Every time I've used the term thus far I've noticed that the term "input" would have
sufficed. It seems to make things a little more clear to use "input", especially because the antonym to input is output and the antonym to operand seems to be output as well. Any thoughts?
An ordered pair is just a set of numbers (or rather, "mathematical objects"). They don't have to be "input and output", but that does not make a difference in the definition of a function.
A function is just a subset of the cartesian product X x (multiplication symbol) Y.
I was under the impression that a function required an input and an output.
Though I was curious about(assuming that a function did in fact require an input, relation, and an output) how unary functions could be explained.
For example, -2,
input, - (?)
relation, 2 (??)
output, -2 (???)
I got the idea that a function has an input, relation, and output from the link provided by Bob Bundy above.... and now directly below:
http://www.mathsisfun.com/sets/function.htm
...Your definition for an operation looks ok to me.
Relation.
Not quite....
Perhaps this is an improvement:
Operation is a function of an output from one or more inputs
Function is a relation of an output to one or more inputs
Relation is a collection of one or more outputs and one or more inputs within a set
In other words, a function f is a set of ordered pairs (x, y) where x is an element of a set X, y is an element of a set Y and no two pairs in f have the same first element.
Correct me if I'm wrong, but ordered pairs do not necessarily have input and output, but rather are simply an arranged pair.
Is this a proper interpretation of these four terms? Am I understanding them correctly?
Operation is a function which produces an output from one or more operands
Function is a relation that exists between an input and output
Relation (i.e. predicate) is the output value of an operation
Operand (i.e. argument) is the input value of an operation
OTHER RELATED DEFINITIONS:
Value is a known numerical amount
Quantity is a known or unknown numerical amount
I'd appreciate any input. Thank you.
I don't completely understand, what is the visual indicator of the exponential operation if it is not font size?
When using a keyboard and the superscript option is not available, the caret (i.e. circumflex) is used. (e.g. 3^2=9)
Also, if I'm understanding the dual usage of LOG, then I believe this is true, correct?
[LOG] is to operator (of logarithms), and something else (i.e. a function) as
[-] is to operator (of subtractions), and something else (i.e. a negative number)
In other words, LOG is the operator of logarithms, and, when in doubt, the circumflex is used as the operator for exponentiation (otherwise the superscript is the visual indicator of the operation to be performed).
[it must be or we would have, e.g., 32 = 9, and obviously thirty-two does not equal nine]
I believe it's true to assume that all operations require an operator of some sort in order to indicate what operation is to be performed. Am I misunderstanding? In a manner of speaking, how do you know which way to go without a direction? I'm not trying to be difficult, I'm just looking for clarity.
Defining an operator as an indicator of an operation raises some questions, such as in regards to exponentiation. Correct me if I'm mistaken, but I believe it can be said that...
addition is to the plus sign as
subtraction is to the minus sign as
multiplication is to the times sign as
division is to the over (or obelus) sign as
exponentiation is to the (?superscript?) sign as
nth root is to the radical sign as
logarithms are to the (?subscript?) sign?
Are the sizes of the exponent's and base's font in exponentiation and logaritms respectively, considered the indicator of the operation? Seems like a silly question, but if it's true it certainly is overlooked.
Also, why write LOG at all? Why not EXPO?
Hello, obviously I'm studying mathematics. I have plenty of technical questions that would help improve my understanding. I appreciate any and all help. Thank you.
MATHEMATICS
Mathematics is the science of quantities
Number is the symbol for a known quantity
Variable is the symbol for an unknown quantity
Amount is the extent of something
Value is a known numerical amount
Quantity is a known or unknown numerical amount
In other words, a quantity is always a value, but a value is not necessarily a quantity. Take for example the variable, which is a quantity that can have different values.
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