Introduction to Standard Form

In mathematics, we often have to deal with numbers that are incredibly large, like the distance between planets, or incredibly small, like the width of a single atom. Writing all those zeros can be exhausting and lead to mistakes! Standard form (also known as scientific notation) is a clever shorthand that allows us to write any number in a clear, consistent way using powers of \(10\).

By the end of this guide, you will be able to convert between "ordinary" numbers and standard form, and perform calculations with them, both with and without a calculator.

What Does Standard Form Look Like?

Every number in standard form must follow this exact structure:

\(A \times 10^n\)

There are two very important rules you must remember:

  1. \(A\) must be between \(1\) and \(10\). Specifically, \(1 \le A < 10\). This means the first number can be \(1\), \(1.5\), or \(9.99\), but it cannot be \(0.5\) or \(10\).
  2. \(n\) must be an integer. This is the "power" or "index". It tells us how many places the decimal point has moved. It is positive for large numbers and negative for small numbers.

Quick Review: Is \(12 \times 10^3\) in standard form? No, because \(12\) is bigger than \(10\). Is \(0.5 \times 10^{-2}\) in standard form? No, because \(0.5\) is less than \(1\).

Working with Large Numbers

When we write a large number in standard form, the power of \(10\) tells us how many places we moved the decimal point to the left to get a number between \(1\) and \(10\).

Step-by-Step: Converting \(54,000,000\) to Standard Form

  1. Identify where the decimal point is now (at the very end: \(54000000.0\)).
  2. Move the decimal point to the left until you have a number between \(1\) and \(10\). Here, we move it \(7\) places to get \(5.4\).
  3. Count the jumps! We moved \(7\) places, so the power is \(7\).
  4. Write it out: \(5.4 \times 10^7\).

Working with Small Numbers

For numbers smaller than \(1\), we move the decimal point to the right. Because the number is small, the power of \(10\) will be negative.

Step-by-Step: Converting \(0.0000072\) to Standard Form

  1. Move the decimal point to the right until you have a number between \(1\) and \(10\). Here, we move it \(6\) places to get \(7.2\).
  2. Count the jumps! We moved \(6\) places. Because it's a tiny number, the power is \(-6\).
  3. Write it out: \(7.2 \times 10^{-6}\).

Memory Tip: If the ordinary number starts with "0.", the power in standard form must be negative!

Calculating without a Calculator (Paper 1)

When you are asked to multiply or divide numbers in standard form on a non-calculator paper, you can treat the numbers and the powers of \(10\) separately. This links directly to the rules of indices found in the Powers, roots and indices chapter.

Multiplication

Example: Work out \((2 \times 10^3) \times (4 \times 10^5)\)

  1. Multiply the numbers: \(2 \times 4 = 8\).
  2. Multiply the powers (add the indices): \(10^3 \times 10^5 = 10^{3+5} = 10^8\).
  3. Combine: \(8 \times 10^8\).

Division

Example: Work out \((6 \times 10^7) \div (3 \times 10^2)\)

  1. Divide the numbers: \(6 \div 3 = 2\).
  2. Divide the powers (subtract the indices): \(10^7 \div 10^2 = 10^{7-2} = 10^5\).
  3. Combine: \(2 \times 10^5\).

Important: Always check your final answer. If you get something like \(20 \times 10^4\), you must convert it back into proper standard form (\(2 \times 10^5\)).

Addition and Subtraction

This is slightly trickier. You cannot just add the powers. Instead, you have two choices:

  1. Convert to ordinary numbers: Change them into "normal" numbers, add or subtract them, and then convert the result back to standard form. (Best for small powers).
  2. Make the powers match: Adjust one number so it has the same power of \(10\) as the other, then add the front numbers.

Example: \((3 \times 10^4) + (5 \times 10^3)\)
Convert to ordinary numbers: \(30,000 + 5,000 = 35,000\).
Convert back: \(3.5 \times 10^4\).

Using a Calculator (Papers 2 & 3)

Most modern calculators have a specific button for standard form, usually labelled \(\times 10^x\) or EXP.

Interpreting the display: Sometimes, a calculator won't show the "\(\times 10\)" part clearly. It might show \(4.5^{06}\) or \(4.5 \text{ E } 06\). You must write this down correctly as \(4.5 \times 10^6\) in your exam. Never write "E" in your final answer!

Key Takeaways
  • Standard form is always \(A \times 10^n\) where \(1 \le A < 10\).
  • Positive \(n\) means a large number; Negative \(n\) means a small number (less than 1).
  • When multiplying, add the powers; when dividing, subtract the powers.
  • Always double-check that your final answer's first number is between 1 and 10.

Common Mistake to Avoid: Many students think \(10 \times 10^5\) is in standard form. It isn't! It should be \(1 \times 10^6\). Don't let the number \(10\) sneak in as the first part of your answer.