Math Problem Statement
prove that if x is ration and x does not equal 0, then 1/x is rational
Solution
To prove that if is a rational number and , then is also rational, we start by recalling the definition of a rational number.
Proof:
-
By definition, a rational number can be expressed as a fraction: where and are integers, , and and have no common divisors other than 1 (i.e., is in its simplest form).
-
Since , .
-
Consider the reciprocal of , which is :
-
Since and are integers and , is a fraction of two integers with a non-zero denominator.
-
Therefore, is a rational number, as it is the ratio of two integers with a non-zero denominator.
Conclusion:
Since is a rational number, we have proven that if is rational and , then is also rational.
Would you like further details, or do you have any questions?
Additional Questions
- What happens if is irrational? Is rational or irrational?
- Can the proof be extended to complex numbers?
- How would you define rational numbers in terms of decimal representations?
- Is the sum of two rational numbers always rational? Why?
- If is a rational number between 0 and 1, is an integer?
Tip
Remember that a non-zero rational number has a reciprocal that is also rational because multiplying or dividing two integers (with no remainder) always yields another rational number.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rational Numbers
Reciprocal Properties
Fractions
Formulas
x = a/b where a and b are integers and b ≠ 0
1/x = b/a
Theorems
Definition of Rational Numbers
Reciprocal of a Rational Number
Suitable Grade Level
Grades 9-10