Question

Reasoning  The numbers 1.333 . . . and 1.444 . . . are both rational numbers. Which of the following is an irrational number that is between these two rational​ numbers? Use pencil and paper. Show why there has to be an irrational number between any two rational numbers.
2.334333 . . .
1.443444 . . .
1.334
1.334333 . . .
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Part 1
Which of the following is an irrational number that is between the two rational​ numbers? Select all that apply.
A.1.334
1.334
B.
2.334333 . . .
C.
1.443444 . . .
D.
1.334333 . . .

Answer 1

Pick C only: 1.443444 . . .

Reasoning about the choices
- A. 1.334 is a finite decimal (hence rational), so not acceptable.
- B. 2.334333 . . . is >2, so it is not between 1.333… and 1.444….
- D. 1.334333 . . . = 1.334333333… has eventually repeating 3's, so it is rational.

C. 1.443444 . . . is between the two given rationals because
1.333333… < 1.443444… < 1.444444….
It is irrational because its decimal expansion is nonterminating and nonrepeating: the pattern is 4,4,3,4,4,4,3,4,4,4,4,3,… (between successive 3’s the block of 4’s keeps getting longer), so no fixed finite block repeats forever.

Why there must always be an irrational between any two rationals
Let a and b be rationals with a < b. Take any fixed irrational t in (0,1) (for example t = 1/√2 ≈ 0.7071…). Then x = a + t(b − a) satisfies a < x < b. If x were rational then t = (x − a)/(b − a) would be a ratio of rationals and hence rational, a contradiction. Therefore x is irrational. This shows there is always an irrational between any two rationals.