1️⃣ Essential Mathematical Calculations for Economics and Management
Master the fundamental operations and properties of real numbers, with a focus on applications in economics and management: order of operations, absolute value, powers, roots, fractions, percentages, index numbers, and proportionality.
- Apply the correct order of operations in numerical and business calculations.
- Use absolute value, powers, and square roots and know their properties.
- Manipulate fractions, percentages, and index numbers.
- Solve everyday problems with the rule of proportionality.
Mathematics is an indispensable universal language in economics and management. Behind every decision about production, pricing, investment, or financing lie calculations — sometimes simple, sometimes more involved, but always based on the fundamental rules of real numbers. Mastering these basics is therefore not merely an academic exercise: it is a daily tool for reasoning and professional effectiveness.
In this module we revisit the essential operations: order of operations, absolute value, powers, square roots, fractions, percentages, index numbers, and proportionality. The goal is not only to know the theoretical rules, but above all to understand their usefulness in concrete situations of economic life and business management.
1. Order of Operations
The order of operations is a fundamental rule of mathematics that lets us evaluate a numerical expression correctly. The order in which we perform the calculations can completely change the result: we cannot add, multiply, or raise to a power in just any order.
To avoid mistakes we apply a strict hierarchy: first parentheses, then powers and roots, then multiplications and divisions (from left to right), and finally additions and subtractions (from left to right). This rule, often summarised by the acronym PEMDAS, is essential both for manual calculation and for using a calculator or spreadsheet.
- Parentheses: \((\ )\), \([\ ]\), \(\{\ \}\)
- Exponents (powers and roots)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
A calculator does not always apply priorities correctly, especially without parentheses. When in doubt, add parentheses.
Worked example — net income
You worked \(20\) hours paid at \(30\) €/h and \(40\) hours paid at \(55\) €/h. The gross-to-net factor is \(0.8\). Compute your total net income in euros.
\[ 0.8 \times (20 \times 30 + 40 \times 55) = 0.8 \times (600 + 2200) = 0.8 \times 2800 = 2240 \text{ €} \]
Exercises
- \(8 \div 2 \times (2 + 2)\) → \(16\)
- \(4 \times (5 - (9 - 2))\) → \(-8\)
- \(10 + 2 \times (-1) \times 2^{3}\) → \(-6\)
- \(\big(10 + 2 \times (-1) \times 2^{3} - 4 \times (5 - (9 - 2))\big)\) → \(2\)
- \(\big[12 - (3 + 2^{3} \times (4 - 1)) \div 3\big] \times 2 - 5^{2}\) → \(-19\)
- \(3 + 5 \times 2 = 16\) → \((3 + 5) \times 2 = 16\)
- \(10 - 2 + 4 = 4\) → \(10 - (2 + 4) = 4\)
TODO: interactive order-of-operations quiz (deferred — see plan Complexity Tracking). For now the exercises above provide the same practice statically.
2. Absolute Value
The absolute value of a real number \(a\), written \(|a|\), is its distance from zero:
- \(|a| = a\) if \(a \geq 0\);
- \(|a| = -a\) if \(a < 0\).
- \(|a| \geq 0\)
- \(|a| = |{-a}|\)
- \(|a \times b| = |a| \times |b|\)
- \(\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}\) if \(b \neq 0\)
- \(|a| = 0 \iff a = 0\)
- \(|a| = |b| \iff a = b \text{ or } a = -b\)
In general \(|a + b| \neq |a| + |b|\) (this is the triangle inequality: \(|a + b| \leq |a| + |b|\)).
Applications
Thermal comfort — the gap between the actual and the target temperature:
- Morning: \(|T^{\text{actual}}_{\text{morning}} - T^{\text{target}}_{\text{morning}}|\)
- Evening: \(|T^{\text{actual}}_{\text{evening}} - T^{\text{target}}_{\text{evening}}|\)
- Balance index: \(\big|\text{morning gap} - \text{evening gap}\big|\)
Quality control (metal plate) — with measured length \(L_r\), target \(L_c\), measured width \(l_r\), target \(l_c\):
- Overall defect: \(|L_r - L_c| + |l_r - l_c|\)
- Imbalance: \(\big||L_r - L_c| - |l_r - l_c|\big|\)
Synthesis exercise
\[ -2 \times \big|-4 - |1 - 5|\big| - \big|(-2)^{2}\big| \times |4 - 7| + |-3|^{2} \]
Step by step:
- \(|1 - 5| = 4\), so \(\big|-4 - 4\big| = 8\) and \(-2 \times 8 = -16\);
- \(|(-2)^2| = 4\) and \(|4 - 7| = 3\), so \(-4 \times 3 = -12\);
- \(|-3|^2 = 9\).
Total: \(-16 - 12 + 9 = \mathbf{-19}\).
3. Powers
A power \(a^{n}\) is the product of \(a\) by itself \(n\) times.
- \(a^{n} \times a^{m} = a^{n+m}\)
- \(\dfrac{a^{n}}{a^{m}} = a^{n-m}\)
- \(a^{0} = 1\)
- \(a^{-n} = \dfrac{1}{a^{n}}\)
- \((a^{n})^{m} = a^{nm}\)
- \(a^{1/2} = \sqrt{a}\)
\(a^{n} + a^{m} \neq a^{n+m}\).
Economic application — compound (exponential) growth: each increase applies to an already-increased base, e.g. compound interest and capitalisation (developed in Modules 4 and 5).
\[ \left( \frac{2^{3} \times 4^{-2} \times 8}{2^{-1} \times \sqrt{64} \times 8} \right)^{2} \]
Rewrite in base \(2\): numerator \(2^3 \times 2^{-4} \times 2^3 = 2^{2}\); denominator \(2^{-1} \times 2^{3} \times 2^{3} = 2^{5}\). Ratio \(2^{2-5} = 2^{-3}\), squared \(2^{-6} = \tfrac{1}{64}\).
Result: \(\dfrac{1}{64}\).
4. Square Root
The square root of \(a \geq 0\) is the positive number \(b\) such that \(b^{2} = a\).
- \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
- \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\) (for \(b > 0\))
\(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\).
5. Fractions
A fraction is a division of two real numbers, written \(\dfrac{a}{b}\) with \(b \neq 0\).
- Addition / subtraction: with the same denominator, add the numerators; otherwise reduce to a common denominator (the LCM is convenient).
- Multiplication: multiply numerators together and denominators together.
- Division: multiply by the reciprocal of the second fraction.
\[ \left( \frac{3}{4} + \frac{2}{3} \times \left(1 - \frac{5}{6}\right) \right) \div \left( \frac{7}{8} - \frac{1}{2} \right) \]
Inner bracket \(1 - \tfrac56 = \tfrac16\), so \(\tfrac23 \times \tfrac16 = \tfrac{2}{18} = \tfrac19\). Then \(\tfrac34 + \tfrac19 = \tfrac{27}{36} + \tfrac{4}{36} = \tfrac{31}{36}\). Divisor \(\tfrac78 - \tfrac12 = \tfrac38\). Finally \(\tfrac{31}{36} \div \tfrac38 = \tfrac{31}{36} \times \tfrac83 = \tfrac{248}{108} = \tfrac{62}{27}\).
Result: \(\dfrac{62}{27}\).
6. Percentages and Index Numbers
A percentage is a fraction with denominator \(100\). For example, \(25\%\) of \(80\) is \(\dfrac{25}{100} \times 80 = 20\).
Applying a discount of rate \(t\) multiplies the price by the factor \((1 - t)\). Applying two successive discounts \(t_1\) then \(t_2\) multiplies by \((1 - t_1)\) then by \((1 - t_2)\), hence by the product \((1 - t_1)(1 - t_2)\). The overall discount rate \(T\) satisfies \(1 - T = (1 - t_1)(1 - t_2)\), so
\[ T = 1 - (1 - t_1)(1 - t_2). \]
This is not \(t_1 + t_2\): successive discounts do not simply add, because each rate applies to a base already reduced by the previous one.
Worked example — successive discounts
A coat costs \(200\) €. It receives a \(10\%\) sale discount, a \(5\%\) loyalty discount, and a \(15\%\) birthday discount. The final price is
\[ 200 \times 0.90 \times 0.95 \times 0.85 = 145.35 \text{ €}. \]
The overall discount is \(1 - 0.90 \times 0.95 \times 0.85 = 1 - 0.72675 = 0.27325\), i.e. about \(\mathbf{27.33\%}\) (not \(30\%\)).
Index numbers
\[ \text{Index} = \left( \frac{\text{current value}}{\text{reference value}} \right) \times 100 \]
- Index \(= 100\): stable;
- Index \(> 100\): increase;
- Index \(< 100\): decrease.
Example: a price index of \(\dfrac{145.35}{200} \times 100 = 72.675\) shows the price fell to about \(72.7\%\) of its reference level.
7. Proportionality Rule
If two ratios are equal, the quantities are proportional:
\[ \frac{a}{b} = \frac{c}{x} \;\Rightarrow\; x = \frac{b \times c}{a}. \]
This “rule of three” is the everyday tool for scaling recipes, converting prices, comparing unit costs, or splitting quantities proportionally.
If \(3\) kg of coffee cost \(27\) €, what do \(5\) kg cost at the same rate? \[ x = \frac{27 \times 5}{3} = 45 \text{ €}. \]