Show that 3.142678 is a rational number. In other words, express 3.142678 in the form p/q, where p and q are integers and q ≠ 0.

To show that 3.142678 is a rational number, we need to express it in the form of a fraction \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers, and \(q \neq 0\).

Step 1: Write the given number as a decimal.

The given number is 3.142678. We can separate it into its integer part and its decimal part:

3.142678 = 3 + 0.142678.

Step 2: Express the decimal part as a fraction.

To convert the decimal part (0.142678) into a fraction, count the number of digits after the decimal point. There are 6 digits after the decimal point.

Thus, we can multiply the decimal part by \(10^6 = 1000000\) to eliminate the decimal point:

0.142678 × 1000000 = 142678.

Now we have:

0.142678 = \(\dfrac{142678}{1000000}\).

Step 3: Combine the integer and fractional parts.

The number 3.142678 can now be written as:

3.142678 = 3 + \(\dfrac{142678}{1000000}\).

To combine these, write 3 as a fraction with the same denominator (1000000):

3 = \(\dfrac{3 \times 1000000}{1000000} = \dfrac{3000000}{1000000}\).

So now:

3.142678 = \(\dfrac{3000000}{1000000} + \dfrac{142678}{1000000} = \dfrac{3000000 + 142678}{1000000} = \dfrac{3142678}{1000000}\).

Step 4: Simplify the fraction if possible.

Check if the numerator and denominator have any common factors. The greatest common divisor (GCD) of 3142678 and 1000000 is 2.

So, divide both the numerator and denominator by 2:

\(\dfrac{3142678}{1000000} = \dfrac{3142678 \div 2}{1000000 \div 2} = \dfrac{1571339}{500000}\).

Step 5: Conclusion.

Thus, 3.142678 can be expressed as the rational number \(\dfrac{1571339}{500000}\), where 1571339 and 500000 are integers, and the denominator is not zero. Therefore, 3.142678 is a rational number.