Standard Form (Edexcel GCSE Maths): Revision Notes
Standard form
What is standard form?
Standard form (also called standard index form) is a helpful way to write extremely large or extremely small numbers in a more manageable format. Instead of writing out all those zeros, we can express numbers much more efficiently.
For example:
- 56,000,000,000 becomes
- 0.000000345 becomes
While standard form is particularly useful for very big or very small numbers, any number can be written in this format, and understanding how to work with it is essential for GCSE maths.
The format of standard form
A number written in standard form must always follow this exact pattern:
Where:
- A is the front number (coefficient)
- 10 is the base
- n is the power (exponent)
The front number must always be between 1 and 10 (mathematically written as ), and the power of 10 tells us how many places the decimal point moves.
Three essential rules to remember
Understanding these three rules will help you master standard form:
-
The front number must always be between 1 and 10 This is the most important rule - if your front number is 15.2 or 0.34, you haven't written it in proper standard form.
-
The power of 10 shows how far the decimal point moves A positive power means the decimal point moves right, a negative power means it moves left.
-
Use positive powers for big numbers, negative powers for small numbers Numbers greater than 1 need positive powers, numbers less than 1 need negative powers. This is much clearer than trying to remember which direction the decimal point moves.
Converting ordinary numbers to standard form
Let's work through some examples to see how this works in practice. The key function is to move the decimal point to create a number between 1 and 10, then count how many places it moved.
Worked Example: Converting 35,600
- First, move the decimal point to create a number between 1 and 10: 3.56
- Count how many places the decimal point moved: 4 places to the left
- Since 35,600 is a big number, we use a positive power:
Worked Example: Converting 0.000623
- Move the decimal point to get a number between 1 and 10: 6.23
- Count the places moved: 5 places to the right
- Since 0.000623 is a small number, we use a negative power:
Worked Example: Converting 146.3 million
This type of question often catches students out. Take it step by step:
- First write out the full number: 146,300,000
- Move the decimal point: 1.463
- Count the places: 8 places to the left
- Final answer:
Common mistakes to avoid:
- is wrong because 146.3 isn't between 1 and 10
- is wrong because 1463 isn't between 1 and 10
Converting from standard form to ordinary numbers
When you need to convert from standard form back to an ordinary number, the process is straightforward. The main cause of errors here is forgetting which direction to move the decimal point.
The key is to look at the power of 10:
- If the power is negative, you're dealing with a small number (less than 1)
- The power tells you exactly how many places to move the decimal point
- For negative powers, move the decimal point to the left
Worked Example: Converting
- The negative power (-3) means move the decimal point 3 places to the left
- 4.95 becomes 0.00495
Calculations with standard form
Multiplication and division
These operations are more straightforward than you might think. The essential approach is to group the front numbers together and the powers of 10 together.
Worked Example: Multiplication
- Group the front numbers together and the powers of 10 together
- Multiply the front numbers normally:
- Add the powers of 10 (using index laws):
- Result:
Worked Example: Division
- Convert any ordinary numbers to standard form first
- Divide the front numbers normally:
- Subtract the powers of 10 (using index laws):
- Result:
Addition and subtraction
These operations require an extra step because you need to work with the same powers of 10.
Steps for addition and subtraction:
- Make sure both numbers have the same power of 10 (you may need to rewrite one of them)
- Add or subtract the front numbers
- Convert your answer back to standard form if needed
Worked Example: Addition
Calculate
- Rewrite as
- Add the front numbers:
- Convert to standard form:
Using calculators with standard form
Most scientific calculators have special buttons for standard form. Research has found that many students lose marks simply because they don't know how to use these functions properly.
Calculator tips:
- Look for EXP or ×10ˣ buttons
- To enter , press: 2.67 EXP 15
- Your calculator might display answers like as 7.986 E15
Important tip: Always remember to include the "" part when writing down your final answer, even if your calculator displays it differently.
Key takeaways
Key Points to Remember:
- Standard form follows the pattern where A is between 1 and 10
- Big numbers use positive powers, small numbers use negative powers
- The power tells you how many places the decimal point moves
- For calculations, group front numbers and powers separately
- Always check your final answer is in proper standard form
- Practice using your calculator's standard form functions