Study Notes: Number – The Four Operations (0607 International Mathematics)
Welcome to the foundational chapter of IGCSE Mathematics! Before we tackle complex algebra or geometry, we need to ensure our basics are rock-solid. This chapter is all about mastering the four operations: Addition, Subtraction, Multiplication, and Division, across all types of numbers you will encounter (integers, fractions, and decimals).
Think of these four operations as the essential tools in your mathematical toolbox. If you use the tools incorrectly, your results will be wrong, no matter how clever the problem is! Let’s dive in.
1. The Golden Rule: Order of Operations
When a calculation involves more than one operation (like addition AND multiplication), we must follow a strict order to ensure everyone gets the same answer. This rule is often remembered by the acronyms BODMAS or PEMDAS.
What is BODMAS/PEMDAS?
This mnemonic tells you which operation takes priority:
- Brackets (or Parentheses)
- Order (Powers, Indices, or Square Roots)
- Division and Multiplication (Done from left to right)
- Addition and Subtraction (Done from left to right)
Analogy: Imagine BODMAS is like a checklist you MUST follow when cooking. If you mix up the steps, the recipe won't work!
Step-by-Step Guide to Using BODMAS
Let's calculate \(4 \times (5 - 2)^2 + 8 \div 4\):
- B - Brackets: Solve everything inside the brackets first.
\(4 \times (\mathbf{3})^2 + 8 \div 4\) - O - Order (Indices): Calculate any powers or roots.
\(4 \times \mathbf{9} + 8 \div 4\) - DM - Division and Multiplication: Work from left to right.
\(\mathbf{36} + \mathbf{2}\) - AS - Addition and Subtraction: Work from left to right.
\(\mathbf{38}\)
Common Mistake to Avoid: Remember that Division and Multiplication have equal priority. If multiplication comes before division in the expression (reading left to right), you do multiplication first. The same applies to Addition and Subtraction.
Quick Review: Order of Operations
Priority Level 1: Brackets ($()$)
Priority Level 2: Powers/Roots ($a^2, \sqrt{a}$)
Priority Level 3: Multiplication/Division ($\times, \div$) (Left to Right)
Priority Level 4: Addition/Subtraction ($+, -$) (Left to Right)
2. Calculations with Integers
Integers are whole numbers, including positive numbers, negative numbers, and zero. Calculations involving negative numbers are frequent in IGCSE questions, especially in practical situations like temperature or finance.
Addition and Subtraction of Negative Numbers
The key here is understanding what happens when two signs meet:
- Two adjacent positive signs or two adjacent negative signs become Positive (+). (e.g., \(5 + (-3)\) becomes \(5 - 3\); \(5 - (-3)\) becomes \(5 + 3\)).
- A positive and a negative sign become Negative (-).
Memory Aid (The Money Analogy):
- If you have $5 but owe $8: \(5 - 8 = -3\) (You still owe $3).
- If the temperature is $-2^\circ\text{C}$ and it drops by $3^\circ\text{C}$: \(-2 - 3 = -5^\circ\text{C}\).
Step-by-Step Example: Calculate \(-10 + 5 - (-2)\)
- Simplify the signs: \(-10 + 5 + 2\)
- Solve left to right: \((-10 + 5) = -5\)
- Finish: \(-5 + 2 = -3\)
Multiplication and Division of Negative Numbers
The rule is simple and applies to both multiplication and division:
- Same Signs: The answer is Positive.
Example: \((-5) \times (-3) = 15\); \(-10 \div (-2) = 5\) - Different Signs: The answer is Negative.
Example: \((-5) \times 3 = -15\); \(10 \div (-2) = -5\)
Did you know? Dividing or multiplying *any* number by zero is undefined. However, $0 \div (\text{any non-zero number})$ is always 0. E.g., \(0 \div 5 = 0\).
Key Takeaway: Integers
Mastering sign changes is essential: A minus sign followed immediately by another minus sign means you add!
3. Calculations with Fractions
Fractions often trip students up, but if you follow the specific rules for each operation, they become much easier.
Prerequisite: Improper Fractions
Before you start adding, subtracting, multiplying, or dividing, you must convert all mixed numbers (like \(2 \frac{1}{3}\)) into improper fractions (like \(\frac{7}{3}\)).
Example: \(3 \frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}\).
3.1. Adding and Subtracting Fractions
You can only add or subtract fractions if they have the same denominator (the bottom number). You must find the Lowest Common Denominator (LCD).
Step-by-Step Example: Calculate \(\frac{1}{2} + \frac{2}{5}\)
- Find the LCD for 2 and 5. The lowest common multiple is 10.
- Convert both fractions to have a denominator of 10.
\(\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}\)
\(\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}\) - Add the numerators (the top numbers). The denominator stays the same.
\(\frac{5}{10} + \frac{4}{10} = \frac{9}{10}\) - Simplify the final answer (if possible). \(\frac{9}{10}\) is in its simplest form.
3.2. Multiplying Fractions
Multiplication is the easiest! You do not need a common denominator.
Rule: Multiply the numerators together, and multiply the denominators together.
Example: \(\frac{3}{4} \times \frac{2}{7}\)
Numerator: \(3 \times 2 = 6\)
Denominator: \(4 \times 7 = 28\)
Result: \(\frac{6}{28}\)
Remember to simplify your final answer: \(\frac{6 \div 2}{28 \div 2} = \frac{3}{14}\).
Trick: Before multiplying, you can often "cross-cancel" to make the numbers smaller and the final simplification easier. In \(\frac{3}{4} \times \frac{2}{7}\), you can divide the 4 (bottom left) and the 2 (top right) by 2: \(\frac{3}{\mathbf{2}} \times \frac{\mathbf{1}}{7} = \frac{3}{14}\).
3.3. Dividing Fractions
Division is often remembered using the popular mnemonic KFC (Keep, Flip, Change) or KCF (Keep, Change, Flip).
Rule: Keep the first fraction, Change the division sign to multiplication, and Flip the second fraction (find its reciprocal).
Example: \(\frac{1}{3} \div \frac{2}{5}\)
- Keep the first fraction: \(\frac{1}{3}\)
- Change the sign: \(\times\)
- Flip (take the reciprocal of) the second fraction: \(\frac{5}{2}\)
Now multiply: \(\frac{1}{3} \times \frac{5}{2} = \frac{5}{6}\)
Key Takeaway: Fractions
Convert mixed numbers first! Addition/Subtraction needs LCD. Division requires the KCF rule.
4. Calculations with Decimals
Calculations with decimals are used frequently in real-world contexts like calculating money (C1.15) or measures (C1.11). While your Graphic Display Calculator (GDC) handles complex decimals, you must know the rules for non-calculator papers (Paper 1 or 2).
4.1. Adding and Subtracting Decimals
The most important rule is to keep the decimal points aligned vertically.
Example: Calculate \(15.3 + 2.78\)
Step 1: Write the numbers lining up the decimal points. (You can add zeros as placeholders.)
\(15.30\)
\(+ \ 2.78\)
\(----- \)
\(18.08\)
4.2. Multiplying Decimals
Ignore the decimal points until the very end!
Step-by-Step Example: Calculate \(1.2 \times 0.05\)
- Ignore the decimals and multiply the integers: \(12 \times 5 = 60\).
- Count the total number of decimal places (d.p.) in the original numbers.
1.2 has 1 d.p.
0.05 has 2 d.p.
Total d.p. needed: \(1 + 2 = 3\) d.p. - Move the decimal point 3 places to the left in your answer (60).
$60 \to 0.060$ (or simply 0.06)
4.3. Dividing Decimals
To divide, convert the divisor (the number you are dividing by) into a whole number by multiplying both numbers by a power of 10.
Example: Calculate \(4.2 \div 0.6\)
- Make the divisor (0.6) a whole number by multiplying by 10. You must do the same to 4.2.
\(4.2 \times 10 = 42\)
\(0.6 \times 10 = 6\) - The calculation is now $42 \div 6$.
- Answer: 7.
Key Takeaway: Decimals
Align decimal points for addition/subtraction. Count the total number of decimal places for multiplication.
5. Working in Context: Practical Situations (C1.6)
The syllabus requires you to apply these operations to real-life problems. These often involve integers (like temperature or bank balances) or decimals (money and time).
Example: Temperature Changes
The temperature in Moscow is \(-8^\circ\text{C}\). Over the course of the day, it rises by $5^\circ\text{C}$, and then drops by $10^\circ\text{C}$. What is the final temperature?
Calculation:
Start: \(-8\)
Rises: \(-8 + 5 = -3\)
Drops: \(-3 - 10 = -13\)
Answer: \(-13^\circ\text{C}\)
Example: Combined Operations with Money
A builder buys $2 \frac{1}{2}$ kg of sand at $1.50 per kg and 3 bags of cement costing $5.25 each. Calculate the total cost.
Note: We need to use multiplication and addition, ensuring we deal with the mixed number first.
- Sand cost: Convert $2 \frac{1}{2}$ to $2.5$ (or \(\frac{5}{2}\)).
Sand cost = \(2.5 \times 1.50 = \$3.75\) - Cement cost: \(3 \times 5.25 = \$15.75\)
- Total cost: \(3.75 + 15.75 = \$19.50\)
Don't worry if this seems tricky at first! The key to success is practising the mechanical rules repeatedly, especially when mixing number types and signs. Keep reviewing your BODMAS/PEMDAS order to ensure you tackle problems the right way, every time!