Converting Standard Form (AI SL)
Before you can calculate with standard form, you need to write numbers in it correctly - and read them back out again as ordinary decimals. This page covers converting large and small numbers to the form \(a\times10^k\) and back, with worked examples and the mistakes examiners flag most often. It's part of the broader Standard Form topic.
34 questions on this sub-topic.
What counts as standard form
Covered under IB syllabus reference SL1.1: numbers written in the form \(a\times10^k\), where \(1\le a<10\) and \(k\in\mathbb{Z}\). This is a notation convention rather than a formula-booklet result, so there's nothing to look up in the exam - you just need to apply it consistently.
Standard form
\(a\times10^k,\ 1\le a<10,\ k\in\mathbb{Z}\)
Exactly one non-zero digit sits before the decimal point in \(a\). Calculator notation like \(5.2\text{E}30\) is never acceptable in working - write \(5.2\times10^{30}\).
Converting either way
Move the decimal point until one digit remains before it; \(k\) counts the places moved
Positive \(k\) for numbers greater than 1, negative \(k\) for numbers less than 1. Reverse the process to go from standard form back to an ordinary number.
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 \((3\times10^{2})^{3}\), in standard form.
Worked solution
\((3\times10^{2})^{3} = 3^{3}\times(10^{2})^{3}.\) M1
\(3^{3} = 27\) and \((10^{2})^{3} = 10^{2\times3} = 10^{6}.\) A1
\(27\times10^{6}\) has \(27\ge10\), so shift: \(27 = 2.7\times10^{1}\), giving \(2.7\times10^{1}\times10^{6} = 2.7\times10^{7}.\) A1
A calculator shows \(4.7\text{E}{-}5\) and \(1.2\text{E}9\).
(a)(i) Write \(4.7\text{E}{-}5\) in standard form.
(a)(ii) Write \(1.2\text{E}9\) in standard form.
(b) Write \(1.2\text{E}9\) as an ordinary number.
Worked solution
(a)(i) \(4.7\text{E}{-}5 = 4.7\times10^{-5}.\) A1
(a)(ii) \(1.2\text{E}9 = 1.2\times10^{9}.\) A1
(b) \(1.2\times10^{9}\): move the digits \(9\) places right \(\Rightarrow 1\,200\,000\,000.\) A1
Common mistakes
- Leaving the coefficient outside \(1\le a<10\). Writing \(38.4\times10^{4}\) instead of \(3.84\times10^{5}\) is a very common slip - always check \(a\) is between 1 and 10 before you finish.
- Miscounting the number of places moved. For \(0.000\,56\), it's easy to lose track and shift by 3 or 5 places instead of 4 - count the zeros carefully, including the one before the decimal point.
- Writing calculator notation instead of standard form. A GDC will display \(5.2\text{E}30\); this is not acceptable in written working and must be converted to \(5.2\times10^{30}\).
Ready to practise properly?
34 standard-form conversion questions, marked instantly like the real exam.
Quick answers
What counts as correct 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 in working.
How do you convert a large or small number into standard form?
Move the decimal point until exactly one non-zero digit remains before it, giving the coefficient \(a\). The exponent \(k\) counts how many places the point moved: positive for numbers greater than 1, negative for numbers less than 1.