Rounding Numerical Answers and Distinguishing Exact Values

1. The Core IB Rule: Exact or Three Significant Figures

The IB has a very specific instruction for the accuracy of numerical answers that appears on the cover of every exam paper. It's the first and most important rule to learn.

This rule gives you two options for your final numerical answers:

1. Give the exact value.
2. Give the value correct to three significant figures (3 s.f.).

You must use one of these two forms unless the question explicitly asks for a different level of accuracy (e.g., "give your answer to two decimal places"). Understanding when to use which form, and how to do so correctly, is our goal for today.

2. Distinguishing Exact Answers from Approximations

An exact answer is a value that has not been rounded or truncated. It perfectly represents the mathematical quantity. An approximation is a value that is close to the exact value but not precisely the same, typically because it has been rounded.

On Paper 1 (non-calculator), your answers should almost always be exact. On Paper 2, you will often need to decide between giving an exact value or a 3 s.f. approximation from your calculator.

So, what constitutes an exact answer?

• Integers and terminating decimals: e.g., $42$, $-5$, $0.125$
• Fractions: e.g., $\frac{1}{3}$, $\frac{22}{7}$. Note that $\frac{1}{3}$ is exact, while its decimal form, $0.333$, is an approximation.
• Expressions involving irrational numbers: $\pi$, $e$, $\sqrt{2}$, $\ln (5)$. The symbols themselves represent the exact values. For example, the exact area of a circle with radius 1 is $\pi$. The number $3.14$ is an approximation of $\pi$.

The following guide explains the relationship between fractions and decimals and discusses irrational numbers, reinforcing this distinction.

3. The Mechanics of Rounding

When an exact answer isn't practical or required, you'll need to round. The default is 3 significant figures, but questions can also specify a number of decimal places. Given your background in Physics HL, you are likely already familiar with significant figures, but the rules in a math exam are applied with particular strictness.

Rounding to Significant Figures (s.f.)

Significant figures are counted from the first non-zero digit on the left. The general rule is to look at the digit to the right of the one you're rounding to:

• If it's 4 or less, you round down (keep the last digit as is).
• If it's 5 or more, you round up (increase the last digit by 1).

This video provides an excellent and thorough review of rounding to significant figures, covering many of the tricky cases you might encounter.

Example: Round $0.078552$ to 3 significant figures.

1. Identify the first three significant figures. Counting starts at the '7': $0.0\underline{7}\underline{8}\underline{5}$. The '5' is the third significant figure.
2. Look at the next digit, which is another '5'.
3. Since this digit is 5 or more, we round up the last significant figure. The '5' becomes a '6'.
4. The answer is $0.0786$.

Rounding to Decimal Places (d.p.)

Sometimes, a question will specify a number of decimal places instead. The rounding rule is the same, but you count digits from the decimal point.

A common point of confusion is when you need to round up a '9'. The video below clearly explains this scenario.

Example: Round $17.3695$ to three decimal places.

1. The third decimal place is the '9'. The next digit is '5', so we must round the '9' up.
2. Rounding '9' up gives '10'. We write down the '0' and carry the '1' over to the previous digit.
3. The '6' becomes a '7'.
4. The answer is $17.370$. Note that you must write the trailing zero to show that you have rounded correctly to three decimal places.

4. IB Exam Strategy and Common Pitfalls

Knowing the rules is one thing; applying them under exam pressure is another. Here are the most important strategic points for your mock exam.

• Never Round Intermediate Steps: This is the single most common and avoidable error related to accuracy. When you perform a multi-step calculation on your TI-84, use the full, unrounded number from the calculator display for the next step.

  • Bad Practice: Calculate $x=\sqrt{5}\approx 2.24$. Then calculate $x^2$ as $2.24^2=5.0176$. This is incorrect.
  • Good Practice: Calculate $x=\sqrt{5}$. Keep the full value $\mathrm{2.236067977...}$ on your calculator. Then calculate $x^2$, which will give you exactly $5$. You can also use the STO→ feature on your TI-84 to store the unrounded value in a variable like A for later use.

• "Show that..." Questions: If a question asks you to "show that" a value is, for example, $12.5$, you must work with unrounded intermediate values. Your final calculated result should be very close to $12.5$ (e.g., $\mathrm{12.5001...}$ or $\mathrm{12.4999...}$). You can then state that this is equal to $12.5$ as required. If you round too early, your final answer might be $12.4$ or $12.6$, and you won't get the marks.

• Accuracy Penalties (AP): In IB marking, if you make an accuracy error (e.g., rounding to 2 s.f. instead of 3 s.f., or rounding incorrectly), an Accuracy Penalty (AP) is applied. This means you lose the final answer mark. This penalty is typically applied only once per paper. Getting your rounding right from the start is an easy way to protect your marks.

Conclusion

In this lesson, we've focused on a crucial aspect of IB exam technique: numerical accuracy. Mastering this will ensure you don't lose preventable marks and can communicate your mathematical results like a professional.

Key Takeaways:

• The default IB rule is to give numerical answers exactly or correct to three significant figures, unless otherwise specified.
• Exact answers include integers, fractions, and expressions with symbols like $\pi$, $e$, and $\sqrt{2}$. They are not rounded.
• Approximations are typically rounded decimals.
• Crucially, never round values used in intermediate steps of a calculation. Use your calculator's full display or memory.
• Incorrect rounding typically results in an Accuracy Penalty (AP), costing you the final answer mark.

In our next lesson, we will continue building your IB exam skills by exploring common command terms. Words like "write down," "calculate," "find," and "determine" have very specific meanings in an IB context, and understanding them is key to providing the right type of answer (e.g., exact vs. approximate, with or without working).