Standard Form (Scientific Notation) – The Language of Extremes
Welcome to the fascinating world of Standard Form! This is a core topic in the Number section (C1.8/E1.8) and is absolutely essential for dealing with the enormous and the minuscule numbers that populate science and mathematics.
If you’ve ever wondered how scientists write the mass of the sun (a gigantic number) or the size of an atom (a super tiny decimal) without filling pages with zeros, Standard Form is the answer. It’s a clean, efficient way to handle numbers easily and accurately. Don't worry if this seems tricky at first—it’s just a clever use of indices!
1. Defining Standard Form: The Golden Rule
The Structure
A number is written in Standard Form when it is expressed like this:
$$A \times 10^n$$
The Two Crucial Rules
-
A (The first part): This must be a number greater than or equal to 1, but strictly less than 10.
$$1 \le A < 10$$ Example: A could be 3.5, 9.99, or 1. But A cannot be 0.4 or 12. - n (The Index/Exponent): This must be an integer (a whole number, positive, negative, or zero). This tells us how many places the decimal point has moved.
Quick Review: Key Characteristics
Standard Form always looks like: (A single digit) . (some decimals) \(\times 10^{\text{power}}\)
2. Converting Numbers INTO Standard Form (A \(\times 10^n\))
Converting involves finding the correct value for $A$ and counting the decimal shifts to find $n$.
Case 1: Converting Very Large Numbers (Positive \(n\))
When you start with a number greater than 10, the exponent ($n$) will be positive.
Step-by-Step Example: Convert 4,500,000 into Standard Form.
-
Find A: Move the decimal point (which is currently after the last zero) until the number is between 1 and 10.
$$4,500,000.$$ Move left: $4.500000$
So, $A = 4.5$. -
Find n: Count how many places you moved the decimal point.
We moved it 6 places to the left.
Since the original number was BIG, the exponent is positive. So, $n = 6$. -
Write in Standard Form:
$$4.5 \times 10^6$$
Memory Aid: If the number is BIG, the power is POSITIVE. You move the decimal point LEFT.
Case 2: Converting Very Small Numbers (Negative \(n\))
When you start with a decimal number between 0 and 1, the exponent ($n$) will be negative.
Step-by-Step Example: Convert 0.000078 into Standard Form.
-
Find A: Move the decimal point until the number is between 1 and 10.
$$0.00007.8$$ Move right: $7.8$
So, $A = 7.8$. -
Find n: Count how many places you moved the decimal point.
We moved it 5 places to the right.
Since the original number was TINY, the exponent is negative. So, $n = -5$. -
Write in Standard Form:
$$7.8 \times 10^{-5}$$
Memory Aid: If the number is TINY (starts 0.0...), the power is NEGATIVE. You move the decimal point RIGHT.
⚠ Common Mistake Alert!
The number $A$ must be $1 \le A < 10$. Writing $45 \times 10^5$ or $0.78 \times 10^{-4}$ is mathematically correct but not in Standard Form, and you will lose marks! Always make sure there is only one non-zero digit before the decimal point in $A$.
3. Converting Numbers FROM Standard Form
To convert from Standard Form back to an ordinary number, you simply reverse the process based on the sign of $n$.
Rule: The Sign of \(n\) Tells You Direction
- If $n$ is Positive (e.g., $10^5$): The number gets bigger. Move the decimal point $n$ places to the right.
- If $n$ is Negative (e.g., $10^{-3}$): The number gets smaller. Move the decimal point $|n|$ places to the left.
Example 1 (Positive \(n\)): Convert \(6.02 \times 10^3\)
The exponent is 3 (positive), so move the decimal 3 places right.
$$6.02 \rightarrow 6020.$$
Example 2 (Negative \(n\)): Convert \(1.5 \times 10^{-4}\)
The exponent is -4 (negative), so move the decimal 4 places left, adding zeros as placeholders.
$$1.5 \rightarrow 0.00015$$
Did you know? Standard Form is sometimes called "scientific notation" because it is the notation of choice for physicists and astronomers. For instance, the speed of light is roughly \(3 \times 10^8\) metres per second!
4. Calculations Involving Standard Form
You must be able to perform all four basic operations (+, –, \(\times\), \(\div\)) with numbers in Standard Form.
4.1. Multiplication and Division (The Easy Ones)
When multiplying or dividing, you treat the $A$ values and the $10^n$ powers separately, using the rules of indices.
A. Multiplication: Multiply the $A$s and Add the $n$s.
$$ (A \times 10^m) \times (B \times 10^n) = (A \times B) \times 10^{(m+n)} $$
Example: Work out \((3 \times 10^4) \times (2 \times 10^5)\)
- Multiply the $A$ values: $3 \times 2 = 6$.
- Add the exponents: $4 + 5 = 9$.
- Result: \(6 \times 10^9\). (This is already in Standard Form, as $6$ is between 1 and 10).
B. Division: Divide the $A$s and Subtract the $n$s.
$$ (A \times 10^m) \div (B \times 10^n) = (A \div B) \times 10^{(m-n)} $$
Example: Work out \((8 \times 10^{-3}) \div (4 \times 10^2)\)
- Divide the $A$ values: $8 \div 4 = 2$.
- Subtract the exponents: $-3 - 2 = -5$.
- Result: \(2 \times 10^{-5}\).
Key Takeaway for \(\times\) and \(\div\)
Always check your final answer to ensure $A$ satisfies $1 \le A < 10$. If the new $A$ value is, say, 15, you must adjust the exponent $n$ accordingly (e.g., $15 \times 10^7$ becomes $1.5 \times 10^8$).
4.2. Addition and Subtraction (The Tricky Ones)
You cannot add or subtract numbers in Standard Form directly unless they have the same power of 10. Think of it like adding different currencies—you must convert them to the same currency first!
Step-by-Step Method:
- Match the Powers: Choose the larger power and convert the smaller number so its power matches.
- Combine A values: Add or subtract the $A$ values.
- Revert to Standard Form: If needed, adjust the result so $A$ is between 1 and 10.
Example: Work out \((3.5 \times 10^5) + (4.1 \times 10^4)\)
-
Match the Powers: We want both numbers to have the power $10^5$.
The second number has $10^4$. To increase the power by 1 (from 4 to 5), we must move the decimal point in $4.1$ one place to the left (making the $A$ value smaller).
$$4.1 \times 10^4 = 0.41 \times 10^5$$ -
Combine A values: Now we have:
$$(3.5 \times 10^5) + (0.41 \times 10^5)$$
Combine the $A$ values: $3.5 + 0.41 = 3.91$. -
Final Result:
$$3.91 \times 10^5$$ (This is already in Standard Form, as $3.91$ is between 1 and 10).
Analogy: \(3.5 \times 10^5\) is like 350,000, and \(4.1 \times 10^4\) is like 41,000. If you try to add 3.5 and 4.1, you get 7.6, which is wrong. You must align the place values first!
Calculator Use (Specific to Core Paper 3 and Extended Paper 4)
While you may need to perform calculations manually or on non-calculator papers (Core Paper 1, Extended Paper 2), in the calculator exams, you can use the built-in Standard Form function (often labelled EXP or EE).
- To enter \(6.5 \times 10^{-7}\), you type: 6.5 [EXP] (-) 7.
- The calculator will often display the answer in Standard Form (e.g., 3.14 E 8, which means \(3.14 \times 10^8\)).
- Crucially: Even when using a calculator, you must ensure your final answer is written correctly in the format $A \times 10^n$ where $1 \le A < 10$. If the calculator gives you $12.3 \times 10^5$, you must write the final answer as $1.23 \times 10^6$.
Standard Form Summary Checklist
- Definition: \(A \times 10^n\).
- Rule for A: $1 \le A < 10$.
- Rule for n: $n$ is an integer.
- Big numbers: Positive exponent, move decimal left.
- Tiny numbers: Negative exponent, move decimal right.
- Addition/Subtraction: Powers of 10 must be matched first.