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Rational Approximation — Definition, Formula & Examples

Rational approximation is replacing a number that has no exact fraction form, such as π\pi or 2\sqrt{2}, with a fraction that comes very close to it. The fraction is easier to calculate with, but it is never exactly equal to the irrational number.

A rational approximation of a real number α\alpha is a rational number pq\frac{p}{q}, where pp and qq are integers and q>0q > 0, chosen so that the error ∣α−pq∣\left|\alpha - \frac{p}{q}\right| is small. Because the rational numbers are dense in the real numbers, every real number can be approximated by fractions to any accuracy you want. An approximation is considered better when it gives a smaller error for a given size of denominator qq.

Key Formula

E=∣α−pq∣E = \left|\alpha - \frac{p}{q}\right|
Where:
  • EE = the error, meaning how far the fraction is from the true value
  • α\alpha = the number being approximated, often irrational
  • pp = an integer numerator
  • qq = a positive integer denominator

Worked Example

Problem: Which is the better rational approximation of π≈3.14159265\pi \approx 3.14159265: the decimal 3.14 or the fraction 227\frac{22}{7}?
Step 1: Write both as fractions so you can compare them fairly. The decimal 3.14 is itself a fraction.
3.14=314100=15750,227≈3.1428573.14 = \frac{314}{100} = \frac{157}{50}, \qquad \frac{22}{7} \approx 3.142857
Step 2: Find the error of each approximation by subtracting from π\pi and taking the absolute value.
∣π−15750∣≈0.00159,∣π−227∣≈0.00126\left|\pi - \frac{157}{50}\right| \approx 0.00159, \qquad \left|\pi - \frac{22}{7}\right| \approx 0.00126
Step 3: Compare the errors. The smaller error wins. Note that 227\frac{22}{7} also uses a much smaller denominator.
0.00126<0.001590.00126 < 0.00159
Answer: 227\frac{22}{7} is the better approximation. Its error is about 0.00126, while the error of 3.14 is about 0.00159.

Why It Matters

Calculators and computers store only finitely many digits, so every value of π\pi or 2\sqrt{2} they use is really a rational approximation. Choosing a good one matters in geometry and engineering. For example, 355113\frac{355}{113} is within one millionth of π\pi, which is accurate enough for most real-world circle calculations.

Common Mistakes

Mistake: Writing π=227\pi = \frac{22}{7} or π=3.14\pi = 3.14 as if they were exact.
Correction: An irrational number can never equal a fraction. Use the symbol ≈\approx, as in π≈227\pi \approx \frac{22}{7}, and keep the exact symbol π\pi when an exact answer is required.

Related Terms

  • Pi Approximations — a closely related concept
  • Irrational Numbers — The numbers that rational approximations try to estimate
  • Integers — Numerators and denominators of approximating fractions
  • Natural Numbers — Possible values of the positive denominator q
  • Digit — Truncating decimal digits gives a rational approximation
  • Aleph Null — Rationals are countable yet approximate every real