Scientific notation and estimation (IB Math AI SL)

Key ideas

• Move the decimal point until exactly one non-zero digit is in front of it; k counts the places moved (positive for large numbers, negative for small ones).
• 0.00045 = 4.5 × 10⁻⁴ and 3 200 000 = 3.2 × 10⁶. A number such as 32 × 10⁵ is not in standard form.
• Multiplying: multiply the a-values and add the powers, then adjust if the new a is 10 or more.
• Your GDC may show 9.468E15 for 9.468 × 10¹⁵. Write it in the a × 10ᵏ form in your answer.
• Percentage error = |approximate value − exact value| ÷ |exact value| × 100%. Divide by the exact value, not the approximation.

Worked example

Light travels at 3.00 × 10⁸ m s⁻¹. A year is about 3.156 × 10⁷ seconds. Find the distance light travels in a year in standard form, then find the percentage error if this distance is rounded to 1 × 10¹⁶ m.

1. Distance = speed × time = (3.00 × 10⁸) × (3.156 × 10⁷).
2. Multiply the numbers: 3.00 × 3.156 = 9.468. Add the powers: 10⁸ × 10⁷ = 10¹⁵.
3. Distance = 9.468 × 10¹⁵ m ≈ 9.47 × 10¹⁵ m (3 s.f.).
4. Percentage error = |1 × 10¹⁶ − 9.468 × 10¹⁵| ÷ (9.468 × 10¹⁵) × 100% = (5.32 × 10¹⁴) ÷ (9.468 × 10¹⁵) × 100% ≈ 5.62%.

Answer: 9.47 × 10¹⁵ m; percentage error ≈ 5.62%.

Exam-style SL AI questions on this topic

• Approximations and Error questions — 45 questions with mark schemes, plus free worked examples
• All SL AI Number and Algebra questions by topic

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Standard form and rounding on your calculator

Related in Number and Algebra

• Arithmetic sequences and series
• Geometric sequences and compound interest
• Financial maths — loans, mortgages and CG50

Revise Scientific notation and estimation for the exam

• SL AI revision notes — every unit summarised, with the key results to learn
• SL AI past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page SL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Are units of measurement discovered in nature or decided by agreement?

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S23 · Problem S87 · Problem S22 · Problem S86. All 22 →