| Content | Guidance, clarification and syllabus links |
|---|---|
| Operations with numbers in the form \( a \times 10^k \) where \( 1 \leq a < 10 \) and \( k \) is an integer. | Calculator or computer notation is not acceptable. For example, 5.2E30 is not acceptable and should be written as \( 5.2 \times 10^{30} \). |
📌 Introduction
Standard form (\(a \times 10^k\)) is essential for efficiently representing very large or small numbers in mathematics and science.
It allows for clear interpretation, maintains precision, and simplifies calculations. Commonly used in Astronomy, Physics, Chemistry, Biology and global finance, this notation avoids lengthy and error-prone digit strings.
📌 Properties & Examples
- Numbers must be written as \(a \times 10^k\) with \(1 \leq a < 10\), \(k \in \mathbb{Z}\).
- \(5 \times 10^{3.2}\): Incorrect, \(k\) must be integer.
- \(0.89 \times 10^{-4}\): Incorrect, \(a\) must be ≥ 1. Correct: \(8.9 \times 10^{-5}\).
- \(2.3 \times 10^8\): Correct, equals 230,000,000.
- Calculator notation such as 5.2E30 is not acceptable; use \(5.2 \times 10^{30}\).
Avoid calculator notation like E30.
📌 Mind Map
📌 Applications in Science and IB Math
- Astronomical distances (e.g. \(1.5 \times 10^{11}\) m: Earth–Sun).
- Microscopic measurements (e.g. \(3 \times 10^{-10}\) m: atom diameter).
- Chemistry (Avogadro’s number: \(6.02 \times 10^{23}\)).
- Physics (order of magnitude calculations, uncertainty).
- Biology (cell sizes and concentrations).
- Finance (global GDPs, populations).
Explicitly state your rounding, and always convert to correct form for final answers.
📌 Worked Examples (IB Style)
Q1. Write 0.0000475 in standard form.
Solution:
\(0.0000475 = 4.75 \times 10^{-5}\)
Here, \(a = 4.75\) is between 1 and 10, \(k = -5\) is an integer.
Final answer: \(4.75 \times 10^{-5}\)
Q2. Express 3,600,000,000 in standard form.
Solution:
\(3,600,000,000 = 3.6 \times 10^{9}\)
Here, \(a = 3.6\) is between 1 and 10, \(k = 9\) is an integer.
Final answer: \(3.6 \times 10^{9}\)
Q3. Simplify \((2.5 \times 10^{4}) \times (3 \times 10^{6})\) and write the answer in standard form.
Solution:
\((2.5 \times 10^{4}) \times (3 \times 10^{6}) = (2.5 \times 3) \times (10^{4} \times 10^{6}) = 7.5 \times 10^{10}\)
Here, \(a = 7.5\) is between 1 and 10, \(k = 10\) is an integer.
Final answer: \(7.5 \times 10^{10}\)
Always check the question requirements and final marking scheme for correct form—partial answers may lose marks if not properly rounded or written.