Standard Form (AI HL)
Standard form packs very large or very small numbers into a single leading digit and a power of 10, which is exactly how a GDC displays them once you switch out of calculator notation. This page covers multiplying, dividing, adding and subtracting numbers in this form, with worked examples and the mismatch that trips students up most. It's part of the broader Standard Form & Approximation topic.
11 questions on this sub-topic.
Standard form notation
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 (e.g. 5.2E30) is not acceptable and must be written as \(5.2\times10^{30}\).
Standard form notation
\(a\times10^k,\ 1\le a<10\)
Not in booklet - notation conventionConverting 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 - definitionNeed bounds notation and the full syllabus wording? See Standard Form & Approximation.
Worked examples
Given \(p = 6.0 \times 10^{8}\) and \(q = 1.5 \times 10^{-3}\), evaluate, giving each answer in standard form:
(a) \(pq\)
(b) \(\dfrac{p}{q}\)
Worked solution
(a) \(pq=(6.0\times10^{8})\times(1.5\times10^{-3})=(6.0\times1.5)\times10^{8+(-3)}.\) M1
\(pq=9.0\times10^{5}.\) A1
(b) \(\dfrac{p}{q}=\dfrac{6.0\times10^{8}}{1.5\times10^{-3}}=(6.0\div1.5)\times10^{8-(-3)}.\) M1
\(\dfrac{p}{q}=4.0\times10^{11}.\) A1
Evaluate \((4.2 \times 10^{5}) + (3.0 \times 10^{4})\), giving the answer in standard form.
Worked solution
\(3.0\times10^{4}=0.30\times10^{5}\), so \((4.2+0.30)\times10^{5}.\) M1
\(=4.5\times10^{5}.\) A1
Light travels at \(3.00 \times 10^{8}\) m s\(^{-1}\). The distance from the Sun to Earth is \(1.50 \times 10^{11}\) m.
Find the time, in seconds and then in minutes, for light to travel this distance.
Worked solution
\(t=\dfrac{1.50\times10^{11}}{3.00\times10^{8}}.\) M1
\(t=0.50\times10^{3}=500\) s. A1
\(t=\dfrac{500}{60}.\) M1
\(t\approx 8.33\) min. A1
A city has population \(2.4 \times 10^{6}\) people, each using on average \(150\) litres of water per day.
(a) Estimate the total daily water use, in standard form.
(b) State the answer in cubic metres (\(1000\) L \(= 1\) m\(^3\)).
Worked solution
(a) \(2.4\times10^{6}\times150=2.4\times10^{6}\times1.5\times10^{2}.\) M1
\(=3.6\times10^{8}\) L. A1
(b) \(3.6\times10^{8}\div10^{3}.\) M1
\(=3.6\times10^{5}\) m³. A1
Common mistakes
- 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.
- Leaving the leading number outside \(1\le a<10\). An answer like \(42\times10^{4}\) or \(0.9\times10^{6}\) is not in standard form - always renormalise so exactly one non-zero digit sits before the decimal point.
- Writing calculator notation as the final answer. A GDC often displays \(5.2\text{E}30\) or \(5.2^{30}\) on screen, but this must be transcribed as \(5.2\times10^{30}\) - the raw calculator display loses marks.
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10 standard-form questions, marked instantly like the real exam.
Quick answers
What is 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 not acceptable and must be rewritten as \(5.2\times10^{30}\).
Can you add two numbers in standard form directly?
Only if the powers of 10 match. If they don't, rewrite one number so both share the same power of 10, then add the leading numbers and keep that power.