Math

QuestionDetermine if 5.787787778 ... is rational or irrational. Provide your reasoning.

Studdy Solution

STEP 1

Assumptions1. A rational number can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. . An irrational number cannot be expressed as a simple fraction - it has non-repeating, non-terminating decimal parts.

STEP 2

The number given is5.787787778... . We can see that the digits after the decimal point are repeating. This is a characteristic of a rational number.

STEP 3

To further confirm, we can express this number as a fraction. Let's denote the number as xx.
x=5.787787778...x =5.787787778...

STEP 4

We can express xx in terms of a fraction by multiplying it by a power of10 that moves the decimal point to just after the repeating part. Since the repeating part is787, we multiply by 103=100010^3 =1000.
1000x=5787.78778778...1000x =5787.78778778...

STEP 5

Subtract the original number xx from this result to create an equation where the right side is a number with no decimal part.
1000xx=5787.78778778...5.78778778...1000x - x =5787.78778778... -5.78778778...

STEP 6

implify the left side and the right side of the equation.
999x=5782999x =5782

STEP 7

olve for xx by dividing both sides of the equation by999.
x=5782999x = \frac{5782}{999}

STEP 8

Therefore, the number5.787787778... can be expressed as a fraction, 5782999\frac{5782}{999}, where both the numerator and the denominator are integers.
So,5.787787778... is a rational number.

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