To express the quantity of programming errors for every programmer as a rational number, you can set up the ratio of programming errors to programmers.
You have 14 programming errors for 56 programmers. To express this as a rational number, you can write it as:
\[ \frac{14 \text{ errors}}{56 \text{ programmers}} \]
Next, you can simplify this fraction. Both the numerator and the denominator can be divided by 14:
\[ \frac{14 \div 14}{56 \div 14} = \frac{1}{4} \]
Thus, the rational number that represents 14 programming errors for every 56 programmers is \( \frac{1}{4} \).