Microscopy: Students should be able to express answers in standard form if appropriate.

The idea

Standard form writes a number as A × 10ⁿ, where A is at least 1 and less than 10 and n is an integer. It is useful in microscopy because real cell sizes and magnifications can be very small or very large. For example, 0.00005 m becomes 5 × 10⁻⁵ m, and 250000 becomes 2.5 × 10⁵. To convert, move the decimal point until one non-zero digit remains before it; the number of places moved gives the power of ten, positive for large numbers and negative for small ones. When calculating, keep values in standard form, apply the index laws, and convert the final answer back if a question asks for ordinary numbers. Always check that the coefficient lies between 1 and 10.

What earns the marks

• Write a number in the form A × 10ⁿ where A is at least 1 and less than 10 and n is an integer.
• Convert between ordinary numbers and standard form by counting decimal place movements to determine the sign and size of the index.
• Use standard form appropriately for very small measurements such as cell dimensions and very large values such as magnification.
• Apply index laws when multiplying or dividing numbers in standard form, for example adding indices when multiplying powers of ten.
• Check that the coefficient is at least 1 and less than 10, adjusting the index if it is not.
• Interpret a standard form answer in context, converting to a sensible unit such as micrometres where needed.

Common mistakes and how to put them right

• Writing a coefficient outside the range 1 to 10, such as 25 × 10³.

  Put it right: Correct this to 2.5 × 10⁴ by moving the decimal point and adjusting the index.

• Using the wrong sign for the index when converting small numbers.

  Put it right by checking that numbers less than 1 have a negative index.

• Forgetting to convert units before expressing an answer in standard form.

  Put it right by converting first, then writing the result in standard form.

Exam tips

• Decide whether the original number is large or small before choosing the sign of the index.
• Count decimal place movements carefully and write the index immediately to avoid losing track.
• Check the coefficient is at least 1 and less than 10, and re-read the question to see whether standard form or an ordinary number is required.