Real Analysis: Proving mn=1 Implies m & n = ±1

Since m cannot equal 0, we are left with the options m=1 and m=-1.In summary, if m and n are integers such that mn=1, then either m=1 and n=1, or m=-1 and n=-1. This is because for mn to equal 1, m must be either 1 or -1, with n being the corresponding inverse. Any other value for m would result in a non-integer value for n.
  • #1
johnjuwax
2
0
Prove or disprove that if m and n are integers such that mn = 1 then either m= 1 & n = 1 or else m = -1 & n = -1.
 
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  • #2
johnjuwax said:
Prove or disprove that if m and n are integers such that mn = 1 then either m= 1 & n = 1 or else m = -1 & n = -1.

If mn=1, then m/1=1/n. But for n an integrer, then m is a fraction. The only way both are integers is if n=1 so that m=1, or if n=-1 so that m=-1.
 
  • #3
It follows from the fact that, if m>1, then 0<n<1, and if m<-1, then -1<n<0. Therefore we have three options left to check, m=-1, 0, 1.
 

1. What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and their properties. It focuses on understanding the behavior and properties of real-valued functions, sequences, and series.

2. What does the statement "mn=1 Implies m & n = ±1" mean?

This statement is commonly known as the "inverse property of multiplication" and it means that if the product of two real numbers is equal to 1, then those two numbers must be either 1 or -1.

3. How is this statement related to Real Analysis?

The statement is related to Real Analysis because it involves the manipulation and understanding of real numbers and their properties. It is often used in proofs and demonstrations within the field of Real Analysis.

4. How can we prove this statement?

To prove this statement, we can use the axioms and properties of real numbers, such as the commutative, associative, and distributive properties of multiplication, along with the definition of the multiplicative inverse. We can also use logical reasoning and algebraic manipulation to show that the statement holds true.

5. What are some real-world applications of the inverse property of multiplication?

The inverse property of multiplication has many practical applications, such as in engineering, physics, and economics. For example, it is used in calculating the resistance of electrical circuits, determining the velocity and acceleration of objects in motion, and in calculating the exchange rates of currencies.

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