International Mathematics (0607) | Ordering

🔢 International Mathematics (0607) Study Notes: Number – Ordering (C1.5 / E1.5)

Hello future mathematician! This chapter is all about comparing different values and putting them in the correct order, either from smallest to largest (ascending) or largest to smallest (descending).

Ordering is a fundamental skill. Whether you’re comparing prices in a shop, checking who ran the fastest race time, or handling scientific data, you need to know which quantity is truly bigger. Don't worry if ordering fractions or negative numbers seems challenging—we'll break it down into easy, manageable steps!

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1. The Language of Ordering: Inequality Symbols

In mathematics, we use special symbols to show the relationship between two quantities. You must be completely familiar with these six symbols:

• \(=\) : Equals (The values are exactly the same.)
• \(\neq\) : Not Equal to (The values are different.)
• \(>\) : Greater than (The number on the left is bigger.)
• \(<\) : Less than (The number on the left is smaller.)
• \(\ge\) : Greater than or Equal to (The number is bigger than or possibly the same as.)
• \(\le\) : Less than or Equal to (The number is smaller than or possibly the same as.)

🧠 Memory Aid: The Crocodile Rule

Imagine the inequality sign is an alligator or crocodile mouth! The mouth always opens wide to eat the BIGGER quantity.
Example: \(10 > 3\) (10 is bigger, so the mouth points toward 10).

Key Takeaway: Understand what each symbol means. Pay close attention to whether the inequality is strict (\(<\) or \(>\)) or inclusive (\(\le\) or \(\ge\)).

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2. Ordering Integers and Decimals

Ordering whole numbers (integers) and decimals relies on understanding the number line.

2.1 Ordering Integers (Including Negative Numbers)

The core rule is simple: Numbers become larger as you move to the right on the number line, and smaller as you move to the left.

The trickiest part for many students is ordering negative numbers.

Analogy: Temperature
If it is \(-10^\circ C\), it is much colder (smaller value) than \(-2^\circ C\).

If you are asked to order these numbers in ascending order (smallest to largest):
\(5, -12, 0, 8, -4\)

Step 1: Find the biggest negative number (the one furthest left on the number line). This is \(-12\).
Step 2: Order the remaining negatives: \(-4\).
Step 3: Place zero: \(0\).
Step 4: Order the positives: \(5, 8\).
Ascending Order: \(-12, -4, 0, 5, 8\)

2.2 Ordering Decimals

When comparing decimals, it is crucial to compare the digits based on their place value, starting from the left.

Step-by-Step Method:

1. Line up the decimal points. This makes comparison easier.
2. Add trailing zeros so all numbers have the same number of decimal places. This doesn't change the value, but makes it look fairer.
3. Compare column by column, starting with the largest place value (the left).

Example: Order \(0.5, 0.49, 0.505, 0.4\)

• \(0.500\) (Added two zeros)
• \(0.490\) (Added one zero)
• \(0.505\) (Original)
• \(0.400\) (Added two zeros)

Now compare:
The smallest tenths digit is 4 (\(0.490\) and \(0.400\)). Between these two, \(0.400\) is clearly smaller than \(0.490\).
The larger tenths digit is 5 (\(0.500\) and \(0.505\)). \(0.500\) is smaller than \(0.505\).

Ascending Order: \(0.4, 0.49, 0.5, 0.505\)

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3. Ordering Fractions

Comparing fractions is often the trickiest task in ordering, because they look so different! You cannot compare them directly unless they share a common feature (either the numerator or the denominator).

Method 1: Using a Common Denominator

This is the mathematically purest way to compare fractions.

Example: Order \(\frac{2}{3}, \frac{5}{6}, \frac{3}{4}\)

1. Find the Lowest Common Multiple (LCM) of the denominators (3, 6, 4).
   The LCM of 3, 6, and 4 is 12. This will be your new common denominator.
2. Convert each fraction to have a denominator of 12.
   • \(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)
   • \(\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\)
   • \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
3. Compare the numerators.
   Since \(8 < 9 < 10\), the order of the fractions is clear.

Ascending Order: \(\frac{2}{3}, \frac{3}{4}, \frac{5}{6}\)

Method 2: Converting to Decimals or Percentages

If you are allowed to use a calculator (Paper 3, 4, 5, or 6), converting fractions to decimals is often the fastest way to order them.

Example: Order \(\frac{2}{3}, \frac{5}{6}, \frac{3}{4}\)

• \(\frac{2}{3} \approx 0.667\)
• \(\frac{5}{6} \approx 0.833\)
• \(\frac{3}{4} = 0.75\)

Comparing the decimals gives: \(0.667 < 0.75 < 0.833\).

Important Tip: When converting fractions to decimals, use enough decimal places (usually 3 or 4) to ensure you can distinguish between the values before ordering.

Key Takeaway: You cannot order fractions effectively until they share a common basis (common denominator or decimal form).

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4. Ordering Quantities in Different Forms (Mixed Comparisons)

In International Mathematics, you are often required to order a mixture of numbers: fractions, decimals, percentages, mixed numbers, and even numbers in standard form or with indices.

The Golden Rule: Standardise!

You must convert all quantities into a single, easy-to-compare format. The easiest format is usually a decimal.

Step-by-Step Conversion Strategy:

1. Convert everything to Decimal Form.
   • Percentages: Divide by 100. (e.g., \(45\% = 0.45\))
   • Fractions: Divide the numerator by the denominator. (e.g., \(\frac{3}{8} = 0.375\))
   • Mixed Numbers: Convert to an improper fraction first, or just keep the whole number and convert the fraction part. (e.g., \(2\frac{1}{4} = 2.25\))
   • Standard Form (\(A \times 10^n\)): Convert to an ordinary number. (e.g., \(3.1 \times 10^{-2} = 0.031\))
2. Line up the decimal points and add zeros (as practiced in Section 2.2).
3. Order the decimals from smallest to largest magnitude.
4. Write your final answer using the original forms of the numbers.

Example of Mixed Ordering:

Order these in descending order (largest to smallest):
\(0.82, \frac{4}{5}, 85\%, 8.1 \times 10^{-1}\)

Conversion to Decimals:

• \(0.82\) (Stays as \(0.82\))
• \(\frac{4}{5} = 4 \div 5 = 0.80\)
• \(85\% = 85 \div 100 = 0.85\)
• \(8.1 \times 10^{-1}\) = \(0.81\)

Comparing (Adding zeros for clarity):
\(0.82\)
\(0.80\)
\(0.85\)
\(0.81\)

Descending Decimal Order: \(0.85, 0.82, 0.81, 0.80\)

Final Answer (using original form): \(85\%, 0.82, 8.1 \times 10^{-1}, \frac{4}{5}\)

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5. Ordering Rational and Irrational Numbers

The syllabus requires you to understand and order both rational numbers (which can be written as a simple fraction, like \(\frac{1}{2}\) or \(-3\)) and irrational numbers (which cannot be written as a simple fraction, like \(\sqrt{2}\) or \(\pi\)).

The Strategy: Approximating Irrational Numbers

To order an irrational number like \(\sqrt{5}\) alongside fractions or decimals, you must find its approximate decimal value.

Example: Order \(\frac{7}{3}, 2.3, \sqrt{5}\)

1. Convert to Decimals (Approximation):
   • \(\frac{7}{3} = 2.3333...\)
   • \(2.3\) (Stays as \(2.3\))
   • \(\sqrt{5}\)
     (Did you know? Since \(2^2 = 4\) and \(3^2 = 9\), \(\sqrt{5}\) must be between 2 and 3. Using a calculator: \(\sqrt{5} \approx 2.236\))
2. Compare the Approximations:
   • \(2.236\) (\(\sqrt{5}\))
   • \(2.300\) (\(2.3\))
   • \(2.333\) (\(\frac{7}{3}\))

Ascending Order: \(\sqrt{5}, 2.3, \frac{7}{3}\)

⚠️ Common Mistake to Avoid

When you are asked to show your working for ordering fractions, do not rely solely on decimals if the numbers are very close (e.g., \(\frac{11}{20}\) vs \(0.551\)). Showing the calculation using a common denominator proves your understanding of magnitude comparison, especially in non-calculator papers.