Mathematical Fundamentals · MS01-001-G
By the end of this session, you should be able to:
Key Rule — PEMDAS
Why it matters: The order of operations is a fundamental rule that lets us evaluate any numerical expression unambiguously. Changing the order changes the result.
Warning
Calculators do not always apply priorities correctly without parentheses. When in doubt, add parentheses.
The absolute value of a real number \(a\), written \(|a|\), is its distance from zero on the number line.
\[|a| = a \text{ if } a \geq 0, \qquad |a| = -a \text{ if } a < 0\]
Key Properties
Application: Measuring deviations from a target — quality control, temperature comfort zones, forecast errors.
A power \(a^{n}\) is the product of \(a\) by itself \(n\) times.
Key Properties
| Property | Rule |
|---|---|
| Product | \(a^{n} \times a^{m} = a^{n+m}\) |
| Quotient | \(\dfrac{a^{n}}{a^{m}} = a^{n-m}\) |
| Zero exponent | \(a^{0} = 1\) |
| Negative exponent | \(a^{-n} = \dfrac{1}{a^{n}}\) |
| Power of a power | \((a^{n})^{m} = a^{nm}\) |
| Square root | \(a^{1/2} = \sqrt{a}\) |
Warning
\(a^{n} + a^{m} \neq a^{n+m}\) — the product rule does not apply to sums.
The square root of \(a \geq 0\) is the unique non-negative number \(b\) such that \(b^{2} = a\).
\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \; (b > 0)\]
Warning
\(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\) — roots do not distribute over addition.
Connection to powers: \(\sqrt{a} = a^{1/2}\), so all power rules apply.
A fraction \(\dfrac{a}{b}\) (\(b \neq 0\)) represents the division of \(a\) by \(b\).
Fundamental Rules
Addition / Subtraction — find a common denominator, then add numerators: \[\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\]
Multiplication — multiply numerators, multiply denominators: \[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]
Division — multiply by the reciprocal of the divisor: \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]
A percentage is a fraction with denominator 100.
Percentage Change Formula
\[\Delta\% = \frac{V_{\text{final}} - V_{\text{initial}}}{V_{\text{initial}}} \times 100\]
Applying a rate \(t\) multiplies the value by the factor \((1 + t)\) (with \(t\) as a decimal). Two successive rates \(t_1\), \(t_2\) give factor \((1 + t_1)(1 + t_2)\) — they do not simply add.
Index numbers: \(\text{Index} = \dfrac{\text{current value}}{\text{reference value}} \times 100\)
Warning
Always divide by the initial value, not the final value.
If two ratios are equal, the quantities are proportional.
Rule of Three
\[\frac{a}{b} = \frac{c}{x} \;\Rightarrow\; x = \frac{b \times c}{a}\]
Everyday uses: scaling recipes, converting prices, comparing unit costs, splitting quantities proportionally, currency exchange.
Example: If 3 kg of coffee cost €27, then 5 kg cost \(\dfrac{27 \times 5}{3} = €45\).
A coat costs €200. It receives a 10 % sale discount, a 5 % loyalty discount, and a 15 % birthday discount.
Step 1 — Final price
Each discount multiplies the running price by its factor:
\[200 \times 0.90 \times 0.95 \times 0.85 = \mathbf{€145.35}\]
Step 2 — Overall discount rate
\[T = 1 - 0.90 \times 0.95 \times 0.85 = 1 - 0.72675 \approx \mathbf{27.33\%}\]
Not 30 % (the naive sum 10 + 5 + 15), because each rate applies to an already-reduced base.
Step 3 — Index number
\[\text{Index} = \frac{145.35}{200} \times 100 = 72.675\]
The index fell below 100, confirming a price decrease of about 27.3 %.
Order of operations governs every calculation — resolve parentheses first, then exponents, then multiplication/division, then addition/subtraction.
Successive percentage changes multiply, they do not add — applying \(+t_1\%\) then \(-t_2\%\) gives an overall factor of \((1 + t_1)(1 - t_2)\), which almost never equals \(1 + t_1 - t_2\).
Always divide by the initial value for a percentage change — the reference base is where you started, not where you ended up.
These slides summarise the key rules for lecture use.
For complete theory, step-by-step proofs, guided exercises, interactive simulations, and collapsible solutions, see the full module:
→ Module 1: Essential Mathematical Calculations