Standard Form (Scientific Notation)

Statement

Standard form (also called scientific notation) is a way of writing very large or very small numbers in a compact form. A number is in standard form if it is written as:

\[ N \times 10^n \quad \text{with } 1 \leq N < 10, \; n \in \mathbb{Z} \]

Here, \(N\) is called the coefficient and \(n\) is an integer power of ten.

Why it’s true

• Every whole number or decimal can be expressed as a product of a number between 1 and 10 and a power of ten.
• Large numbers use positive indices (e.g. \(3.6 \times 10^8\)), while small numbers use negative indices (e.g. \(5.4 \times 10^{-6}\)).
• This system makes it easier to handle calculations with extremely large or small values, common in science and engineering.

Recipe (how to use it)

1. Move the decimal point so that the number becomes \(N\) with \(1 \leq N < 10\).
2. Count how many places the decimal point was moved.
3. If the decimal point moved left, the power of ten is positive. If it moved right, the power is negative.
4. Write the number as \(N \times 10^n\).

Spotting it

You need to use standard form whenever the question involves “very large” or “very small” numbers, or the exam explicitly says “Give your answer in standard form”.

Common pairings

• Calculator outputs in scientific mode.
• Operations in standard form (multiply, divide, add, subtract).
• Real-world contexts like population size or cell diameters.

Mini examples

1. Given: Write 45,000 in standard form. Answer: \(4.5 \times 10^4\).
2. Given: Write 0.0063 in standard form. Answer: \(6.3 \times 10^{-3}\).

Pitfalls

• Coefficient not between 1 and 10 (e.g. writing \(45 \times 10^3\) instead of \(4.5 \times 10^4\)).
• Forgetting the negative sign in the index when the number is less than 1.
• Not simplifying decimals correctly.

Exam strategy

• Always check the coefficient is in the range \(1 \leq N < 10\).
• Remember: left shift = positive power, right shift = negative power.
• Practise converting quickly — it’s often just one mark but easy to slip on.

Summary

Standard form is essential for handling extreme numbers. The format is \(N \times 10^n\), with \(N\) between 1 and 10 and \(n\) an integer. Use it to simplify calculations, present answers neatly, and handle scientific data efficiently.