GCSE Maths Practice: standard-form

Recognising Correct Standard Form

It’s important not only to convert numbers into standard form, but also to recognise whether a number is written correctly. A valid number in standard form must always follow one rule: the first number (called the coefficient) must be between 1 and 10. For example, 9.876 × 10⁻⁴ is valid, but 98.76 × 10⁻⁵ and 0.9876 × 10⁻³ are not.

Why This Rule Matters

This convention keeps all numbers in a consistent, comparable format. Without this rule, a single number could have many different expressions, making calculations and comparisons more confusing. The standardised format means everyone writes and reads values the same way, whether in science, engineering, or GCSE Maths exams.

Understanding the Coefficient

The coefficient (the number before the × sign) should be at least 1 but less than 10. If it’s too small or too large, the power of ten must be adjusted. This adjustment keeps the representation consistent. For instance, 45 × 10³ is not correct because 45 is greater than 10; we rewrite it as 4.5 × 10⁴. Similarly, 0.45 × 10² is adjusted to 4.5 × 10¹.

Worked Example 1

Which of these is correctly written in standard form?

• A) 7.4 × 10⁵
• B) 74 × 10⁴
• C) 0.74 × 10⁶

Answer: A) 7.4 × 10⁵, because 7.4 is between 1 and 10.

Worked Example 2

Rewrite 0.067 × 10³ in correct standard form.

• Move decimal one place right → 6.7
• Decrease the power by 1 → 6.7 × 10²
• Correct form: 6.7 × 10²

Worked Example 3

Rewrite 45 × 10⁻⁶ correctly.

• Move decimal one place left → 4.5
• Increase the power by 1 → 4.5 × 10⁻⁵

Common Mistakes

• Writing a coefficient outside the range 1–10.
• Forgetting to adjust the exponent when moving the decimal point.
• Thinking 0.45 × 10⁴ and 4.5 × 10³ are different numbers—they are actually equal.

Real-Life Application

Scientists use strict formatting in data to ensure accuracy. For instance, when measuring wavelengths, particle sizes, or voltages, everyone must express results consistently. A small formatting mistake could make a result appear 10× larger or smaller than it actually is!

FAQs

• Can I have more than one correct version? Mathematically yes, but only one is accepted as standard — where the first number is between 1 and 10.
• Why is this called ‘normalisation’? Because it ensures all numbers are presented in a normal, standardised way.
• How do I check my answer? Multiply the number back out to confirm it equals the same value, then check that the coefficient is between 1 and 10.

Study Tip

Before finalising an answer, quickly scan the coefficient. If it’s not between 1 and 10, move the decimal and adjust the power of ten. This habit will help you avoid easy marks lost in your GCSE Maths exam.