Why Does the Proof of \(y^n = x\) Consider Specific Cases for \(t\)?

In summary, the theorem states that for every positive x and integer n, there is only one positive real number y that when raised to the power of n equals x. The book proves this by dividing into two cases, t= x/(1+x) and t>1+x, and showing that t exists in a set E and that 1+x is an upper bound for E. The question is why these specific cases are chosen instead of t> x/(1+x) and t=1+x. However, the original statement is incorrect as there are two values of y, one positive and one negative, when n is even. The statement has been corrected to specify that y must be positive.
  • #1
jwqwerty
43
0
the theorem goes likes this:
For every x>0 and every integer n>0 there is one and only one real y>0 such that y^n=x

The book starts the proof by stating E as set consisting of all positive real numbers t such that t<x^n. Then it states that:

If t= x/(1+x) then 0<t<1. Hence t^n<t<x. Thus t exists in E and E is not empty
If t>1+x then t^n>t>x so that t does not exist in E. Thus 1+x is an upper bound of E.

My questions is this:
Why does it divide into two cases, t= x/(1+x) and t>1+x? And instead, why can't we divide into t> x/(1+x) and t=1+x? Don't they still have the same meaning as the previous one in the way that 0<t<1 and t>1?
 
Last edited:
Physics news on Phys.org
  • #2
The original statement is wrong. If n is even there are two values of y, one positive and the other negative.
 
  • #3
mathman said:
The original statement is wrong. If n is even there are two values of y, one positive and the other negative.

oh sorry i changed the statement from all real y to all real y>0
 

FAQ: Why Does the Proof of \(y^n = x\) Consider Specific Cases for \(t\)?

1. What is the meaning of "proof of a theorem (y^n=x)"?

The phrase "proof of a theorem (y^n=x)" refers to the process of providing evidence or logical reasoning to demonstrate that a given statement, known as a theorem, is true. In this case, the statement is in the form of an equation where the left side is y raised to the power of n, and the right side is the variable x.

2. How do I prove a theorem (y^n=x)?

To prove a theorem (y^n=x), you must follow a logical and systematic approach. Start by understanding the theorem and its components, such as the variables and the given conditions. Then, use mathematical principles, such as algebra and calculus, to manipulate the equation and show that both sides are equal. This process may also involve using definitions, properties, and previously proven theorems.

3. Can I use different methods to prove a theorem (y^n=x)?

Yes, there are various methods or techniques that you can use to prove a theorem (y^n=x). Some common approaches include direct proof, proof by contradiction, and mathematical induction. The choice of method may depend on the complexity of the theorem and your personal preference.

4. Why is it important to prove a theorem (y^n=x)?

Proving a theorem (y^n=x) is essential because it provides a solid foundation for mathematical knowledge and understanding. It allows us to verify the accuracy and validity of mathematical statements and concepts. Additionally, the process of proving theorems helps develop critical thinking and problem-solving skills.

5. Are there any tips for successfully proving a theorem (y^n=x)?

Yes, here are a few tips to help you successfully prove a theorem (y^n=x):

  • Start by carefully reading and understanding the theorem and its components.
  • Break down the theorem into smaller, more manageable parts.
  • Use definitions, properties, and previously proven theorems to guide your reasoning.
  • Be patient and persistent, as the process of proving a theorem may require several attempts.
  • Check your work and make sure to provide clear and concise explanations for each step.

Similar threads

I Multivariable fundamental calculus theorem in Wald
Replies
2
Views
2K
B Why do I not understand what seems to be basic Calculus?
Replies
11
Views
2K
I On convolution theorem of Laplace transform: Schiff
Replies
4
Views
2K
B Proving the Existence and Value of a Derivative at a Point
Replies
11
Views
1K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
Replies
9
Views
2K
I What is the difference between these two power series theorems?
Replies
5
Views
750
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
Replies
19
Views
3K
How to conclude a mathematical expression with the provided background
Replies
17
Views
2K
B Why is the growth of exps & logs so fast & slow, respectively?
Replies
5
Views
2K
I Question about uniform convergence in a proof
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top