Converting Standard Form (AA SL)
This page is about switching a number between its ordinary decimal form and standard form \(a\times10^{k}\), in both directions - including tidying up a calculator's \(\text{E}\) notation into something an examiner will accept. It's part of the broader Standard Form topic.
19 questions on this sub-topic.
Key rules
Covered under IB syllabus reference 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 such as \(5.2\text{E}30\) is never acceptable as a final answer - it must be written \(5.2\times10^{30}\).
Into standard form
Shift the decimal point until exactly one non-zero digit sits before it, giving \(a\) with \(1\le a<10\). The exponent \(k\) is the number of places you shifted - positive if the digits moved left (the original number was large), negative if they moved right (the original number was small).
Back to ordinary form
Move the decimal point \(k\) places right (positive index) or left (negative index), filling any gaps with zeros.
Need the full syllabus wording and formula-booklet reference table? See Standard Form.
Worked examples
Write in the form \(a\times10^{k}\), \(1\le a<10\), \(k\in\mathbb{Z}\).
(a) \(384\,000\)
(b) \(0.000\,56\)
Worked solution
(a) \(384\,000 \to 3.84\): the digits moved \(5\) places left, so \(384\,000 = 3.84\times10^{5}.\) A1
(b) \(0.000\,56 \to 5.6\): the digits moved \(4\) places right, so \(0.000\,56 = 5.6\times10^{-4}.\) A1
Write as ordinary numbers.
(a) \(7.5\times10^{3}\)
(b) \(2.04\times10^{-2}\)
Worked solution
(a) \(7.5\times10^{3}\): move the digits \(3\) places right \(\Rightarrow 7500.\) A1
(b) \(2.04\times10^{-2}\): move the digits \(2\) places left \(\Rightarrow 0.0204.\) A1
Evaluate \((2\times10^{3})^{4}\), in standard form.
Worked solution
\((2\times 10^{3})^4 = 2^4\times (10^{3})^4.\) M1
\(2^4 = 16\) and \((10^3)^4 = 10^{12}\), so \(16\times 10^{12}.\) A1
\(16 = 1.6\times 10^{1}\), so the answer is \(1.6\times 10^{13}.\) A1
Evaluate \((2\times 10^{3})^{4}\), giving your answer in standard form.
Worked solution
\((2\times 10^{3})^4 = 2^4\times (10^{3})^4.\) M1
\(2^4 = 16\), \((10^3)^4 = 10^{12}\), so \(16\times 10^{12}.\) A1
\(16 = 1.6\times 10^{1}\), giving \(1.6\times 10^{13}.\) A1
A calculator displays \(2.8\text{E}7\) and \(6.5\text{E}{-}4\).
(a) Write each in standard form.
(b) Write \(2.8\text{E}7\) as an ordinary number.
Worked solution
(a) \(2.8\text{E}7 = 2.8\times10^{7}.\) A1
\(6.5\text{E}{-}4 = 6.5\times10^{-4}.\) A1
(b) \(2.8\times10^{7}\): move the digits \(7\) places right \(\Rightarrow 28\,000\,000.\) A1
Common mistakes
- Leaving the coefficient outside \(1\le a<10\). \(45\times10^{3}\) is correct arithmetic but not standard form - it must be rewritten as \(4.5\times10^{4}\).
- Counting the shift the wrong way. Moving the decimal point left to reach \(a\) means the exponent is positive (the number was large); moving right means the exponent is negative (the number was small) - swapping the sign is a very common slip.
- Leaving calculator notation in a final answer. A display of \(5.2\text{E}30\) or \(1.2^{9}\) must be rewritten as \(5.2\times10^{30}\) before it is submitted as an answer, even though the calculator's own screen never shows the \(\times\) sign.
Ready to practise properly?
19 standard-form conversion questions, marked instantly like the real exam.
Quick answers
What counts as standard form?
A number written as \(a\times10^{k}\) where \(1\le a<10\) and \(k\) is an integer. Calculator notation such as \(5.2\text{E}30\) is never acceptable as a final answer.
How do I convert a standard form number back to an ordinary number?
Move the decimal point \(k\) places right for a positive index or \(k\) places left for a negative index, filling any gaps with zeros.