Is -6.743743… a rational or irrational number? Your

Rational because .743 repeats

Rational because the 6 is negative

Irrational because the number never ends

Irrational because the 6 is negative

It's rational because it contains a repeating digit pattern, 743 (first option).

In particular, if

x = -6.743743…,

then

1000x = -6,743.743743… .

Subtracting, we can eliminate the trailing end and solve for x exactly:

1000x - x = -6,743.743743… - (-6.743743…)

999x = -6,737

x = -6,737/999