Introduction to Exact Calculations

In most of your math lessons, you’ve probably been used to rounding your answers to 2 decimal places or 3 significant figures. While that’s great for real-life measurements, mathematicians love being exact. 1/3 is much more precise than 0.33!

In this chapter, we will learn how to provide answers that are 100% accurate by using fractions, multiples of \(\pi\), and surds. No more rounding, no more "approximate" squiggles (\(\approx\))—just perfectly exact results.

Quick Review: An exact value is a number that has not been rounded or truncated. For example, \(\frac{1}{3}\) is exact, but \(0.333\) is an approximation.


1. Calculations with Fractions and \(\pi\)

The simplest way to keep a calculation exact is to avoid your calculator's "S-D" button. If a question asks for an exact answer, follow these two rules:

Working with Fractions

If you are adding, subtracting, multiplying, or dividing, keep your numbers as proper or improper fractions. Example: If you calculate \( \frac{2}{3} \times \frac{5}{7} \), your exact answer is \( \frac{10}{21} \). Do not turn this into \(0.476...\)

Working with \(\pi\) (Pi)

When dealing with circles, cylinders, or spheres, you will often have \(\pi\) in your calculation. To keep it exact, simply treat \(\pi\) like a letter in algebra (like \(x\)).

The Analogy: Think of \(\pi\) as a "name tag." If you have 5 of them, you just write \(5\pi\).

  • Example: Find the area of a circle with a radius of \(3cm\).
  • Area = \( \pi \times r^2 \)
  • Area = \( \pi \times 3^2 = 9\pi \)
  • The exact answer is \(9\pi\).

Key Takeaway: If the question says "leave your answer in terms of \(\pi\)" or "give an exact answer," don't press the \(\pi\) button on your calculator to get a decimal!


2. Understanding Surds

A surd is an expression containing a square root (or cube root, etc.) that results in an irrational number. This means if you typed it into a calculator, the decimals would go on forever without repeating.

Did you know? \(\sqrt{9}\) is not a surd because it equals \(3\). However, \(\sqrt{2}\) is a surd because its decimal form is \(1.41421356...\)

The Golden Rules of Surds

To calculate with surds, you need to know these three rules:

  1. Multiplication: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
  2. Division: \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)
  3. Adding/Subtracting: You can only add or subtract "like" surds (just like \(2x + 3x = 5x\)).
    Example: \( 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3} \).
    Warning: \( \sqrt{a} + \sqrt{b} \) is NOT \( \sqrt{a+b} \)! For example, \( \sqrt{9} + \sqrt{16} \) is \(3 + 4 = 7\), but \( \sqrt{25} \) is \(5\).

3. Simplifying Surds

Simplifying a surd makes it easier to work with. We do this by finding the largest square number that is a factor of the number under the root.

Square Numbers Cheat Sheet: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Step-by-Step: How to Simplify \(\sqrt{50}\)

  1. Find the factors of 50: (1, 50), (2, 25), (5, 10).
  2. Identify the largest square number factor: It is 25.
  3. Rewrite the root: \( \sqrt{50} = \sqrt{25 \times 2} \)
  4. Split the root using the multiplication rule: \( \sqrt{25} \times \sqrt{2} \)
  5. Calculate the square root of the square number: \( 5\sqrt{2} \)

Don't worry if this seems tricky at first! Just keep dividing your number by the square numbers (4, 9, 16, 25...) until you find one that fits perfectly.

Key Takeaway: Always look for square number factors to "pull out" of the square root.


4. Rationalising the Denominator (Higher Tier)

In mathematics, having a surd on the bottom of a fraction (the denominator) is considered "untidy." Rationalising is the process of moving the root to the top.

Type 1: Simple Denominator

If the fraction is \( \frac{1}{\sqrt{a}} \), multiply the top and bottom by \( \sqrt{a} \).

Example: Rationalise \( \frac{3}{\sqrt{5}} \)

  1. Multiply top and bottom by \( \sqrt{5} \): \( \frac{3 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \)
  2. Since \( \sqrt{5} \times \sqrt{5} = 5 \), the answer is: \( \frac{3\sqrt{5}}{5} \)

Type 2: Complex Denominator (The Conjugate)

If the denominator is something like \( 2 + \sqrt{3} \), we use a trick called the conjugate. You multiply the top and bottom by the same expression but change the sign in the middle.

Example: Rationalise \( \frac{1}{\sqrt{3} + 1} \)

  1. The conjugate of \( \sqrt{3} + 1 \) is \( \sqrt{3} - 1 \).
  2. Multiply top and bottom: \( \frac{1(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \)
  3. Expand the bottom (using FOIL/Double brackets): \( \sqrt{3}\sqrt{3} - \sqrt{3} + \sqrt{3} - 1 = 3 - 1 = 2 \).
  4. The simplified answer is: \( \frac{\sqrt{3} - 1}{2} \)

Common Mistake: Forgetting to change the sign. If you multiply \( \sqrt{3} + 1 \) by \( \sqrt{3} + 1 \), the middle surd terms won't cancel out, and you'll still have a root on the bottom!

Key Takeaway: Rationalising is just a fancy way of saying "get the root off the bottom."


Summary Checklist

Quick Review Box:

  • Exact means no decimals (unless they terminate) and no rounding.
  • Leave circle answers in terms of \(\pi\).
  • To simplify a surd, find the square factor (e.g., \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)).
  • \( \sqrt{x} \times \sqrt{x} = x \).
  • Rationalise by multiplying the top and bottom to remove the root from the denominator.