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 xx is a rational number and x0x \neq 0, then 1x\frac{1}{x} is also rational, we start by recalling the definition of a rational number.

Proof:

  1. By definition, a rational number xx can be expressed as a fraction: x=abx = \frac{a}{b} where aa and bb are integers, b0b \neq 0, and aa and bb have no common divisors other than 1 (i.e., ab\frac{a}{b} is in its simplest form).

  2. Since x0x \neq 0, a0a \neq 0.

  3. Consider the reciprocal of xx, which is 1x\frac{1}{x}: 1x=1ab=ba\frac{1}{x} = \frac{1}{\frac{a}{b}} = \frac{b}{a}

  4. Since aa and bb are integers and a0a \neq 0, ba\frac{b}{a} is a fraction of two integers with a non-zero denominator.

  5. Therefore, ba\frac{b}{a} is a rational number, as it is the ratio of two integers with a non-zero denominator.

Conclusion:

Since 1x=ba\frac{1}{x} = \frac{b}{a} is a rational number, we have proven that if xx is rational and x0x \neq 0, then 1x\frac{1}{x} is also rational.

Would you like further details, or do you have any questions?

Additional Questions

  1. What happens if xx is irrational? Is 1x\frac{1}{x} rational or irrational?
  2. Can the proof be extended to complex numbers?
  3. How would you define rational numbers in terms of decimal representations?
  4. Is the sum of two rational numbers always rational? Why?
  5. If xx is a rational number between 0 and 1, is 1x\frac{1}{x} 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.

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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