Standard Form & Approximation (AI HL)
Standard form gives a compact, exact way to write very large or very small numbers, while approximation is about how honestly a rounded number represents the true value behind it. This topic covers multiplying, dividing, adding and subtracting numbers in standard form, rounding to significant figures or decimal places, finding the upper and lower bounds of a rounded value, and calculating percentage error.
What the syllabus says
This topic maps onto two points in the official IB Applications & Interpretation syllabus, both common to SL and HL.
| Code | Syllabus content |
|---|---|
| SL1.1 | Operations with numbers in the form \(a\times10^k\), where \(1\le a<10\) and \(k\) is an integer. Calculator or computer notation (e.g. 5.2E30) is not acceptable and must be written as \(5.2\times10^{30}\). |
| SL1.6 | Approximation: decimal places, significant figures - choosing an appropriate degree of accuracy for given data. Upper and lower bounds of rounded numbers: if \(x=4.1\) to one decimal place, then \(4.05\le x<4.15\). Percentage errors, including measurement and rounding errors. Estimation, and recognising whether a calculated result is reasonable. |
SL1.1 to SL1.5 are shared content with Analysis & Approaches, so standard form works identically across both courses.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is standard form?
Standard form writes a number as \(a\times10^k\), where \(1\le a<10\) and \(k\) is an integer - it keeps very large or very small numbers compact and lets you compare their sizes at a glance from the power of 10 alone.
e.g. \(45\,000 = 4.5\times10^4.\)
What are significant figures?
Significant figures count the meaningful digits in a number, starting from the first non-zero digit - unlike decimal places, they work the same way for numbers of very different sizes.
e.g. \(0.048351\) rounded to 3 significant figures is \(0.0484\), since the fourth significant figure is 5 and rounds the third up.
What is an upper or lower bound?
The upper and lower bounds of a rounded number are the largest and smallest values that could have rounded to it - found by adding or subtracting half of the smallest place value it was rounded to.
e.g. \(8.4\) m to 2 significant figures has bounds \(8.35 \le L < 8.45\), since the half-width is \(0.5\times0.1=0.05\).
What is percentage error?
Percentage error measures how far an approximate value is from the true (exact) value, as a percentage of the true value - it's always calculated by dividing by the exact value, never the approximate one.
e.g. measuring \(1.60\times10^{-19}\) C against a true value of \(1.602\times10^{-19}\) C gives \(\varepsilon = \dfrac{|1.60-1.602|}{1.602}\times100\% \approx 0.12\%.\)
What is premature rounding error?
Premature rounding error is the inaccuracy that creeps in when you round a value in the middle of a calculation instead of carrying the full value through to the end - it can shift your final answer's last significant figure.
e.g. squaring the rounded value \(1.85\) gives \(3.4225\), but squaring the full value \(1.847562\) gives \(3.413\) (4 s.f.) - a small but real difference.
Key formulas
This topic has one genuine booklet formula - percentage error - plus a handful of conventions and definitions the tables and cards below cover in full.
Formula reference
Percentage error is explicitly in the formula booklet; standard form and bounds are conventions and definitions you're expected to know from prior learning.
| Formula | Used for | Booklet? |
|---|---|---|
| \(a\times10^k,\ 1\le a<10\) | Standard form notation | Not in booklet - notation convention |
| Bound \(= \text{value} \pm \dfrac{1}{2}\times\text{smallest place value}\) | Upper and lower bounds of a rounded number | Not in booklet - definition |
| \(\varepsilon = \left|\dfrac{v_A-v_E}{v_E}\right|\times100\%\) | Percentage error between an approximate and exact value | ✓ Yes |
Decimal places vs significant figures
Both are ways of rounding, but they count differently and suit different kinds of numbers.
| Feature | Decimal places (d.p.) | Significant figures (s.f.) |
|---|---|---|
| Counts from | A fixed position after the decimal point | The first non-zero digit, wherever it falls |
| Best suited to | Numbers of similar, everyday size | Numbers of very different magnitudes |
| Example | \(13.96528 \to 13.97\) (2 d.p.) | \(0.0072649 \to 0.00726\) (3 s.f.) |
Standard form arithmetic
Multiplying and dividing in standard form is quick because the powers of 10 combine on their own - adding and subtracting need one extra step first.
Multiplying and dividing
Multiply (or divide) the leading numbers, then add (or subtract) the powers of 10. Adjust back into standard form if the leading number is no longer between 1 and 10.
Not in the formula booklet - index lawsAdding and subtracting
Rewrite both numbers with the same power of 10 first, then add or subtract the leading numbers - you can't combine them directly if the powers don't match.
Not in the formula booklet - requires matching indicesConverting to standard form
Count how many places the decimal point moves to put exactly one non-zero digit before it - that count (with the correct sign) is the power of 10.
Not in the formula booklet - definitionBounds and error
These ideas connect a rounded or measured value back to the range of true values it could represent.
Upper and lower bounds
Add or subtract half of the smallest place value the number was rounded to. A value rounded to the nearest 0.1 has bounds \(\pm0.05\); to 2 s.f., the half-width depends on the digit's place value.
Not in the formula booklet - definitionPercentage error
\[\varepsilon = \left|\dfrac{v_A-v_E}{v_E}\right|\times100\%\]
Always divide by the exact (true) value \(v_E\), never the approximate value \(v_A\).
✓ In the formula bookletCompounding error
When a measured quantity is raised to a power in a formula (e.g. \(V\propto r^3\)), the percentage error roughly multiplies by that power - a small error in radius becomes a larger error in volume.
Not in the formula booklet - consequence of the formulaWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\((2.5\times10^{6})\times(8.0\times10^{-2}).\)
(a) Evaluate it, giving your answer in standard form.
(b) Evaluate \((4.0\times10^{6})\times(2.0\times10^{-3}).\)
Worked solution
(a) \(2.5\times8.0=20\) and \(10^{6}\times10^{-2}=10^{4}.\) M1
\(20\times10^{4}=2.0\times10^{5}.\) A1
(b) \(4.0\times2.0=8.0\) and \(10^{6}\times10^{-3}=10^{3}.\) M1
\(8.0\times10^{3}\) is already in standard form. A1
A component must have length \(50\) mm with a percentage error of at most \(1.5\%\).
(a)(i) Find the lower bound of the acceptable range of lengths.
(a)(ii) Find the upper bound.
(b) A batch is measured at \(50.9\) mm. State whether it is within tolerance, justifying your answer.
Worked solution
(a)(i) \(1.5\%\) of \(50=0.75\) mm. M1
Lower bound \(=50-0.75=49.25\) mm. A1
(a)(ii) Upper bound \(=50+0.75=50.75\) mm. A1
(b) \(\varepsilon=\left|\dfrac{50.9-50}{50}\right|\times100\%.\) M1
\(\varepsilon=1.8\%.\) A1
Since \(1.8\%>1.5\%,\) R1
the batch is not within tolerance. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Adding or subtracting standard form numbers without matching the powers of 10 first. \((5.6\times10^5)+(8.0\times10^4)\) can't be combined directly - rewrite \(8.0\times10^4\) as \(0.80\times10^5\) first, then add the leading numbers.
- Using the full place value instead of half of it when finding bounds. A number rounded to the nearest \(0.1\) has bounds \(\pm0.05\), not \(\pm0.1\) - the bound is always half the smallest unit of rounding.
- Dividing by the wrong value in the percentage error formula. The denominator must always be the exact (true) value, never the approximate one - swapping them gives a different, incorrect percentage.
- Rounding a value too early in a multi-step calculation. Carry extra decimal places (or the exact value) through every intermediate step, and only round the final answer - early rounding compounds into a wrong final digit.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
For very large or very small numbers - avoids typing long strings of zeros and prevents rounding errors.
- Scientific notation means \(a\times10^n\), e.g. \(3.2\times10^{8}\) or \(4.5\times10^{-3}\).
- Use 2nd → , (EE) to enter the ×10 part: type 3.2 2nd , 8 to enter \(3.2\times10^{8}\). Do NOT type ×10^ separately.TI-84
- Use the EE key (or type ×10^ from the keyboard template) to enter scientific notation. Or just type \(3.2\times10^{8}\) using the ^ key.Nspire
- Use the ×10ˣ key (EXP key) - type 3.2 then EXP then 8. Do NOT type ×10^ manually.Casio
- To display answers in scientific notation: on TI-84 press MODE and choose SCI; on Casio set the display mode in SET UP.
Tip: A common mistake is typing ×10^ instead of using the EE/EXP key - this gives ×10×... (multiplication, then a power) rather than proper scientific notation.
Bounds and percentage-error questions don't need a special calculator routine - they're ordinary arithmetic once you've set up the expression. Type the working directly on the home screen (or its equivalent), and switch the display mode to Sci or Norm using the steps above if the question specifically asks for standard form.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Standard form & approximation questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What counts as correct standard form?
A number written as a times 10 to the power k, where 1 is less than or equal to a, which is less than 10, and k is an integer. Calculator notation like 5.2E30 is not accepted - it must be written as 5.2 times 10 to the power 30.
How do I find the upper and lower bound of a rounded number?
Take half of the smallest place value the number was rounded to, then subtract it for the lower bound and add it for the upper bound. If x = 4.1 to one decimal place, the bounds are 4.05 is less than or equal to x, which is less than 4.15.
What's the formula for percentage error?
Percentage error equals the absolute value of (approximate value minus exact value) divided by the exact value, all times 100 percent. It's in the formula booklet, and you must divide by the exact (true) value, not the approximate one.
Why do I need standard form if my calculator handles big numbers anyway?
Standard form is examined as a written skill in its own right, including questions with no calculator allowed. It also keeps very large or very small numbers readable and avoids rounding errors from typing out long strings of zeros by hand. See the GDC guide for calculator-specific entry steps.
Sub-topics
Standard Form & Approximation broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AI HL syllabus unit, in case you want to keep going.