Ordering

1. Overview

Ordering in IGCSE Mathematics (0580) involves arranging numbers by their magnitude (size). This includes integers, fractions, decimals, percentages, and, for Extended students, negative numbers, surds, and numbers in standard form. You must be fluent with inequality symbols (=, ≠, >, <, ≥, ≤) to compare and order these values accurately. Mastering this topic is crucial for solving inequalities, interpreting data, and understanding relative scales.

Key Definitions

• Magnitude: The absolute size or value of a quantity, disregarding its sign.
• Ascending Order: Arranging numbers from the smallest to the largest (e.g., -3, 0, 1, 2.5).
• Descending Order: Arranging numbers from the largest to the smallest (e.g., 10, 5, 1, 0, -1).
• Inequality: A mathematical statement that compares two quantities, indicating that they are not necessarily equal.
• Integer: A whole number (can be positive, negative, or zero); no fractions or decimals.

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Core Content

The Inequality Symbols

You must be able to recognize and use the following symbols:

• $= \quad$ Equal to: The values on both sides are identical.
• $\neq \quad$ Not equal to: The values are different.
• $> \quad$ Greater than: The value on the left is larger.
• $< \quad$ Less than: The value on the left is smaller.
• $\ge \quad$ Greater than or equal to: The value on the left is at least as big as the one on the right.
• $\le \quad$ Less than or equal to: The value on the left is at most as big as the one on the right.

Method: Ordering Different Formats

To order a list containing fractions, decimals, and percentages:

1. Convert all values into the same format (usually decimals is easiest).
2. Align the decimal points to compare place value columns (tenths, hundredths, etc.).
3. Rank them according to the question (ascending or descending).
4. Rewrite the final answer using the original numbers given in the question.

Worked example 1 — Ordering decimals, fractions, and percentages

Place the following numbers in ascending order: $0.7, \frac{3}{4}, 72%, 0.705$

• Step 1: Convert all to decimals.
  • $0.7 = 0.700$
    • Reason: Add zeros for easy comparison.
  • $\frac{3}{4} = 3 \div 4 = 0.750$
    • Reason: Divide numerator by denominator.
  • $72% = 0.720$
    • Reason: Divide by 100.
  • $0.705 = 0.705$
    • Reason: Already in decimal form.
• Step 2: Compare the values.
  • $0.700 < 0.705 < 0.720 < 0.750$
    • Reason: Compare tenths, then hundredths, then thousandths.
• Step 3: Write in original format.
  • $0.7, 0.705, 72%, \frac{3}{4}$
    • Reason: The question requires the answer in the original format.

Final Answer: $0.7, 0.705, 72%, \frac{3}{4}$

Worked example 2 — Ordering decimals and fractions

Arrange the following numbers in descending order: $\frac{2}{5}, 0.3, \frac{1}{3}, 0.25$

• Step 1: Convert all to decimals.
  • $\frac{2}{5} = 2 \div 5 = 0.4$
    • Reason: Divide numerator by denominator.
  • $0.3 = 0.3$
    • Reason: Already in decimal form.
  • $\frac{1}{3} = 1 \div 3 = 0.333...$
    • Reason: Divide numerator by denominator.
  • $0.25 = 0.25$
    • Reason: Already in decimal form.
• Step 2: Compare the values.
  • $0.4 > 0.333... > 0.3 > 0.25$
    • Reason: Compare tenths, then hundredths, then thousandths.
• Step 3: Write in original format.
  • $\frac{2}{5}, \frac{1}{3}, 0.3, 0.25$
    • Reason: The question requires the answer in the original format.

Final Answer: $\frac{2}{5}, \frac{1}{3}, 0.3, 0.25$

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Extended Content (Extended Curriculum Only)

Extended students are expected to order more complex values, including negative numbers, surds (roots), and numbers in standard form.

Comparing Negative Numbers Remember: The further a negative number is from zero, the smaller it is. On a number line, numbers decrease as you move to the left. Therefore, $-10 < -2$.

Comparing Powers and Roots Without a calculator, you may need to estimate roots or square numbers to compare them. Knowing common squares (e.g., $1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25$) is very helpful. For roots, think about the perfect squares that the number falls between. For example, to estimate $\sqrt{20}$, note that 20 is between 16 ($4^2$) and 25 ($5^2$), so $\sqrt{20}$ will be between 4 and 5.

Comparing Numbers in Standard Form Numbers in standard form are written as $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer. To compare numbers in standard form, first compare the powers of 10 ($n$). The number with the larger power of 10 is larger. If the powers of 10 are the same, then compare the values of $a$.

Worked example 3 — Ordering negative numbers, surds, and $\pi$

Arrange in descending order: $-2^2, \sqrt{10}, 3.1, \pi$

• Step 1: Calculate/Estimate values.
  • $-2^2 = -(2 \times 2) = -4$
    • Reason: Exponent applies only to 2, not the negative sign.
  • $\sqrt{10} \approx 3.16$
    • Reason: Since $\sqrt{9}=3$ and $\sqrt{16}=4$, $\sqrt{10}$ is slightly greater than 3. $\sqrt{10}$ is closer to $\sqrt{9}$ than $\sqrt{16}$, so it's closer to 3 than 4.
  • $3.1 = 3.10$
    • Reason: Add zero for easy comparison.
  • $\pi \approx 3.14$
    • Reason: $\pi$ is approximately 3.14159...
• Step 2: Compare magnitudes.
  • $3.16 > 3.14 > 3.10 > -4$
    • Reason: Positive numbers are greater than negative numbers. Compare the decimal places.
• Step 3: Final Answer.
  • $\sqrt{10}, \pi, 3.1, -2^2$
    • Reason: Write the answer in the original format.

Final Answer: $\sqrt{10}, \pi, 3.1, -2^2$

Worked example 4 — Ordering numbers in standard form

Place the following numbers in ascending order: $3.2 \times 10^5, 5.1 \times 10^4, 8.9 \times 10^3, 1.0 \times 10^6$

• Step 1: Compare the powers of 10.
  • $10^3 < 10^4 < 10^5 < 10^6$
    • Reason: The larger the exponent, the larger the number.
• Step 2: Order based on powers of 10.
  • $8.9 \times 10^3 < 5.1 \times 10^4 < 3.2 \times 10^5 < 1.0 \times 10^6$
    • Reason: Numbers with smaller powers of 10 are smaller.

Final Answer: $8.9 \times 10^3, 5.1 \times 10^4, 3.2 \times 10^5, 1.0 \times 10^6$

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Key Equations

While there are no formulas for ordering, the following symbol logic is essential:

Symbol	Meaning	Example
$>$	Strictly greater than	$5 > 2$
$<$	Strictly less than	$-5 < -2$
$\ge$	Greater than or equal to	$x \ge 4$ (includes 4)
$\le$	Less than or equal to	$x \le 10$ (includes 10)

These symbols are NOT provided on the IGCSE formula sheet; they must be memorized.

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Common Mistakes to Avoid

• ❌ Wrong: Thinking $-5$ is greater than $-2$ because $5 > 2$.
  • ✓ Right: On a number line, $-5$ is to the left of $-2$, so $-5 < -2$. Remember that negative numbers become smaller as their absolute value increases.
• ❌ Wrong: Forgetting to convert all numbers to the same format before comparing. For example, trying to compare $\frac{1}{4}$ directly with $0.2$ without converting one of them.
  • ✓ Right: Convert $\frac{1}{4}$ to $0.25$ or $0.2$ to $\frac{1}{5}$ before comparing.
• ❌ Wrong: Writing the decimal versions in your final answer instead of the numbers provided in the question.
  • ✓ Right: Always rewrite your final answer using the original numbers given in the question. Note: You will lose marks if the final list does not use the original fractions/percentages.
• ❌ Wrong: Confusing the $<$ and $>$ symbols.
  • ✓ Right: Tip: The "mouth" always eats the bigger number! Alternatively, read the inequality from left to right. For example, $3 < 5$ reads "3 is less than 5".
• ❌ Wrong: Incorrectly evaluating $-2^2$ as $(-2)^2 = 4$.
  • ✓ Right: Remember that $-2^2 = -(2 \times 2) = -4$, while $(-2)^2 = (-2) \times (-2) = 4$. Pay close attention to parentheses.

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Exam Tips

• Command Words: Look for "Ascending" (Smallest $\rightarrow$ Largest) or "Descending" (Largest $\rightarrow$ Smallest). Mixing these up is a common way to lose all marks for the question.
• Calculator Tip: Use the $S \Leftrightarrow D$ button on your scientific calculator to quickly convert fractions and $\pi$ into decimals for easy comparison.
• Show Working: Even if the question is only worth 2 marks, write down your decimal conversions. If you make one slip in the final order but your conversions are correct, you may still earn "Method Marks" (M1).
• Standard Form: If numbers are in standard form (e.g., $3 \times 10^4$), compare the powers of 10 first. The higher the power, the larger the number. If powers are equal, compare the decimal parts.
• Typical Values: It is helpful to memorize common decimal equivalents:
  • $\frac{1}{8} = 0.125$
  • $\frac{3}{8} = 0.375$
  • $\frac{5}{8} = 0.625$
  • $\frac{7}{8} = 0.875$
• Double Check: After ordering, quickly scan your answer to ensure it logically follows the ascending or descending order requested. A quick visual check can catch simple mistakes.