POP

E1.5 Ordering Quantities

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01 — Explanation

How do I compare numbers given in different forms?

Ordering asks you to arrange quantities using $=, \neq, >, <, \geqslant, \leqslant$ . At Extended level the values being compared are rarely all the same type — a question might mix surds, negative or fractional indices, and recurring decimals in one list. The reliable strategy is to convert every value to a decimal (around $4$ d.p. is enough), keeping track of which decimal came from which original expression.

02 — Explanation

Why is \sqrt{a} bigger than a when 0 < a < 1?

It feels natural to assume square-rooting always shrinks a number, but that's only true for numbers greater than $1$ . For a proper fraction, you're asking "what number times itself gives this small value?" — and that number has to be closer to $1$ , so it's larger than the original.

\sqrt{0.25} = 0.5, \qquad 0.5 > 0.25

Examiner Tip: This is a genuine exam favourite. Whenever a comparison list includes $\sqrt{a}$ for some $a<1$ , expect it to sit higher in the order than you might instinctively guess.

03 — Explanation

How do I lay out an ordering question so I don't lose track?

Write the original expression next to its decimal approximation on the same line as you go, so that once you've sorted the decimals, you can translate straight back into the notation the question used — never report an answer purely in decimal form.

Worked Example

Arrange in order of size, smallest first:

3^{-2}, \quad \sqrt{0.1}, \quad 0.\dot{1}\dot{1}, \quad \left(\frac{1}{3}\right)^2

$3^{-2} = \dfrac{1}{9} \approx 0.1111$
$\sqrt{0.1} \approx 0.3162$
$0.\dot{1}\dot{1} = 0.111111\ldots \approx 0.1111$
$\left(\frac{1}{3}\right)^2 = \dfrac{1}{9} \approx 0.1111$
Comparing: three of the four values are equal ( $\approx 0.1111$ ), and $\sqrt{0.1}\approx0.3162$ is clearly the largest.

3^{-2} = 0.\dot{1}\dot{1} = \left(\frac{1}{3}\right)^2, \ \sqrt{0.1}

04 — Explanation

Does the same "convert to decimal" strategy work when negative numbers are involved?

Yes, but remember that on the number line, a negative number is always smaller than any positive number, and among negatives, the one closer to zero is the larger value: $-2 > -8$ , even though $8 > 2$ . Convert every value to a decimal exactly as before, but plot them mentally on a number line running left (small) to right (large) rather than just comparing sizes of the digits.

Worked Example

Arrange in order of size, smallest first: $-\sqrt{9}, \ -2.5, \ (-3)^2 \div 10, \ -\dfrac{7}{3}$

$-\sqrt{9} = -3$
$-2.5$ stays as $-2.5$
$(-3)^2 \div 10 = 9 \div 10 = 0.9$ (squaring a negative gives a positive result)
$-\dfrac{7}{3} \approx -2.333$
Ordering on the number line from most negative to most positive: $-3 < -2.5 < -2.333 < 0.9$ .

-\sqrt{9}, \ -2.5, \ -\frac{7}{3}, \ (-3)^2\div10

Examiner Tip: A squared negative number becomes positive, which often means it jumps to the opposite end of the ordering from where students instinctively place it. Always evaluate powers fully before comparing signs.

05 — Explanation

How much decimal accuracy should I keep while comparing values mid-question?

Keep at least $3$ – $4$ decimal places for every converted value until the final ordering is decided, especially when two values are close in size — rounding too early can make two genuinely different values look identical, or worse, swap their apparent order entirely.