The main point is that operation in maths is defined as a function, and a function is defined
as a relation that doesn't give two results for one input.
So for root function new definitions that build on these and complex set allow root to have two results.
I think this has something to do with this:
Equation a+b=c defines two operations + and - that are closed in integer set.
Equation a*b=c defines two operations * and / that are closed in rational set.
But equation a^b=c defines three operations: exponentiation, root and logarithm so it's not a field or ring...
(Another question: are ^, root, log closed in complex set?)
Hello there,
This kind of question comes up a lot and i see this really a lot on the web all over the place.
The problem here stems from the way they teach math in school, or rather the way they have to teach it.
When we learn something new we usually learn the basics first, then start to expand on that with later years of study. That's because we cant learn every single thing about something right away sometimes because it is too far reaching to do that. First we learn the alphabet, then words, then pronunciation etc., but we dont learn every single word on the first day. It takes small steps to learn anything, so what happens is that we are usually taught one small step at a time, and unfortunately sometimes these smaller steps are a bit misleading.
And then there is the matter of how we speak. Sometimes we speak in a more causal way which has a way of encompassing things in groups that dont belong in groups or glancing over the little details. What is happening here is in the past you have glanced over the details due to the manner of teaching that was presented to you and now you are questioning that because you came across a contraction. This is typical so no reason for real alarm.
In this example you were told that a function can only have one y value for any one x value. Strictly speaking, this is 100 percent true. However, there are times when we stretch this definition to include some of the more common functions that we encounter that actually have two values for one x. Even though strictly speaking this is not a function, we might still think of it as a function because even though it has more than one value the number is still limited, and in the case here it is limited to two.
Another example is the circle. We often define the circle as:
R^2=x^2+y^2
but if this is solved for y we find we have to use two different functions to make a circle because the circle has two values for each x except at two distinct points. So we have:
y^2=R^2-x^2
and in order to solve this we have to take the square root of both sides and get:
y=sqrt(R^2-x^2)
but clearly this cant be the end of it because this means y would have to be positive always and that means we cant construct the part of the circle that is under the x axis, so we have to modify it to:
y=+/- sqrt(R^2-x^2)
and so we end up with two values of y, one positive and one negative, for each value of x.
This is really the learning procedure in action. We first learn some things, then later we learn how to modify them to fit a wider class of applications. Now that you are learning that a square root can possibly have two different values, you are starting to modify your old definitions to encompass a wider range of applications.
This isnt the last time this kind of thing is going to happen. You'll next want to look into how to find the square root of a negative number. Doing that you'll get into more theory and that will expand your understanding of what you can do with math. Before this you could only do positive square roots, now you'll be able to do much more as well as enter into the world of complex analysis where some more complicated problems can be simplified simply by knowing how to use complex numbers.
You'll run into the number which is called "j" or in some texts "i", which is the square root of minus 1.
Without going into this too deep right now, you can often think of 'j' as a number which can not actually be calculated but acts as a placeholder for the algebraic expression such as:
y=1+2*j
Note that if we look at this as a regular algebraic expression, we can not solve it directly because we dont know what j is really. If j was 3, we would have the answer: 7, and if j was 4 we would have the answer: 9, but j is not a number which is directly calculable so we have to leave that expression as is. When we do this we call the '1' the "real part" and the '2' the "imaginary part". I dont want to get too much into this right now as you have a lot to think about already.