Standard Form (AA SL)
Standard form (also called scientific notation) writes any number as a single digit times a power of ten, so astronomical distances and sub-atomic masses can be handled without a string of zeros. This topic covers converting numbers into and out of standard form, and multiplying, dividing, adding and subtracting numbers written this way - all without losing precision or breaking the \(1\le a<10\) rule.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| 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 is not acceptable - for example \(5.2\text{E}30\) is not acceptable and should be written as \(5.2\times10^{30}\). |
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 any number as \(a\times10^{k}\), where \(1\le a<10\) and \(k\) is an integer. It lets you write very large or very small numbers compactly, without a long string of zeros.
e.g. \(52\,000 = 5.2\times10^{4}\).
What is the coefficient in standard form?
The coefficient is the number \(a\) in front of the power of ten - it must always be at least \(1\) and strictly less than \(10\). If your working gives a coefficient outside this range, you haven't finished simplifying.
e.g. In \(3.4\times10^{-7}\), the coefficient is \(3.4\).
What is the index in standard form?
The index (or exponent) \(k\) tells you how many places to move the decimal point, and in which direction. A positive index means a large number; a negative index means a small number less than 1.
e.g. In \(6.7\times10^{5}\), the index is \(5\), meaning multiply by \(10^{5}=100\,000\).
How do you multiply numbers in standard form?
Multiply the coefficients together and add the indices, then adjust back into standard form if the new coefficient is \(10\) or more. This works because multiplying powers of the same base adds their exponents.
e.g. \((2\times10^{3})(3\times10^{5})=6\times10^{8}\).
How do you add numbers in standard form?
You can only add the coefficients directly if the powers of ten match. If they don't, rewrite one number so both share the same power first, then add and convert back to standard form.
e.g. \(3\times10^{5}+2\times10^{4} = 300\,000+20\,000=320\,000=3.2\times10^{5}\).
Key formulas
These aren't formula-booklet entries - they're index-law rules you're expected to already know and apply to numbers written as \(a\times10^{k}\).
Formula reference
Standard form is built entirely from prior-knowledge index laws - none of these rules appear as separate formula-booklet entries.
| Rule | Used for | Booklet? |
|---|---|---|
| \((a\times10^{m})(b\times10^{n}) = (ab)\times10^{m+n}\) | Multiplying | Not in booklet - prior knowledge |
| \(\dfrac{a\times10^{m}}{b\times10^{n}} = \dfrac{a}{b}\times10^{m-n}\) | Dividing | Not in booklet |
| \((a\times10^{m})^{n} = a^{n}\times10^{mn}\) | Powers of a standard-form number | Not in booklet |
| \(1\le a<10\) | Valid range for the coefficient | Not in booklet - definition |
Ordinary form vs standard form
The same number, written two ways - practise converting confidently in both directions.
| Ordinary number | Standard form |
|---|---|
| \(45\,000\) | \(4.5\times10^{4}\) |
| \(0.0032\) | \(3.2\times10^{-3}\) |
| \(670\,000\,000\) | \(6.7\times10^{8}\) |
| \(0.000\,000\,091\) | \(9.1\times10^{-8}\) |
Converting to and from standard form
The direction the decimal point moves - and the sign of the index - depends on whether the original number is large or small.
Large numbers
Move the decimal point left until only one non-zero digit remains before it. The number of places moved is the positive index \(k\).
Small numbers
Move the decimal point right until only one non-zero digit remains before it. The number of places moved is the negative index \(k\).
Back to ordinary form
Move the decimal point \(k\) places right (positive index) or left (negative index), filling any gaps with zeros.
Operations in standard form
Multiplying and dividing only need the index laws; adding and subtracting need matching powers first.
Multiplying
Multiply the coefficients, add the indices, then re-adjust if the coefficient is no longer between \(1\) and \(10\).
Dividing
Divide the coefficients, subtract the indices, then re-adjust into standard form.
Adding & subtracting
Rewrite so both numbers share the same power of ten (or convert to ordinary numbers), combine, then convert the result back into standard form.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\(45\,000.\)
(a) Write it in the form \(a\times10^{k}\).
(b) Write \(670\,000\) in the form \(a\times10^{k}\).
Worked solution
(a) \(45\,000=4.5\times10^{4}.\) A1
(b) \(670\,000=6.7\times10^{5}.\) A1
\((2\times10^{3})^{3}.\)
(a) Evaluate it, giving your answer in standard form.
(b) Evaluate \((3\times10^{2})^{3}\), giving your answer in standard form.
Worked solution
(a) \((2\times10^{3})^{3}=2^{3}\times10^{3\times3}.\) M1
\(=8\times10^{9}.\) A1
(b) \((3\times10^{2})^{3}=3^{3}\times10^{2\times3}.\) M1
\(=27\times10^{6}=2.7\times10^{7}.\) 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.
- Leaving the coefficient outside \(1\le a<10\). \(45\times10^{3}\) is correct arithmetic but not standard form - it must be rewritten as \(4.5\times10^{4}\).
- Adding or subtracting without matching the powers of 10 first. \(3\times10^{5}+4\times10^{4}\) is not \(7\times10^{5}\) or \(7\times10^{4}\) - convert to a common power (or to ordinary numbers) before combining.
- Losing the sign of a negative index. \(2.1\times10^{-4}\) is a small decimal, not a large one - a sign slip on the index turns a tiny number into a huge one.
- Writing calculator E notation as a final answer. \(5.2\text{E}30\) is explicitly not acceptable on the IB syllabus - it must be written out as \(5.2\times10^{30}\).
Using your GDC
The 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 3.2×10^8 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.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Standard form questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the rule for the coefficient in standard form?
It must satisfy \(1\le a<10\) - a single non-zero digit before the decimal point, then the rest of the digits after it. \(45\times10^{3}\) is correct maths but not standard form; \(4.5\times10^{4}\) is.
How do I add two numbers in standard form with different powers of 10?
You can't add the coefficients directly unless the powers match. Rewrite one number so both share the same power of 10 (or convert both to ordinary numbers), add, then convert the result back into standard form.
Can I write my answer using calculator E notation?
No. The IB syllabus explicitly states that calculator or computer notation such as \(5.2\text{E}30\) is not acceptable - you must write it out as \(5.2\times10^{30}\) in your final answer.
Can I use my GDC for standard form questions?
Yes, for entering and computing with very large or small numbers - but you still need to convert the calculator's output into proper \(a\times10^{k}\) notation by hand, since calculator notation isn't accepted as a final answer. See the GDC guide for model-specific instructions.
Sub-topics
Standard Form broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA SL syllabus unit, in case you want to keep going.