🎬 Move the decimal, count the places
Standard form keeps one non-zero digit before the decimal point.
🎯 By the end of this lesson...
You should be able to:
- Explain why standard form is useful.
- Recognise numbers written correctly in standard form.
- Write large ordinary numbers in standard form.
- Write small ordinary numbers in standard form.
- Convert standard-form numbers into ordinary numbers.
- Compare numbers written in standard form.
- Multiply numbers written in standard form.
- Divide numbers written in standard form.
- Add and subtract numbers written in standard form.
- Use a calculator correctly with standard form.
- Solve practical problems involving very large and very small values.
🤔 Think About This...
Numbers with many zeros
Scientists may work with numbers such as:
Writing long strings of zeros repeatedly is inconvenient and makes errors more likely.
Is there a shorter way to write these numbers without changing their values?
1️⃣ What Is Standard Form?
Standard form is a compact way of writing very large or very small numbers.
Large number
Small number
2️⃣ The Standard-Form Rule
A number in standard form is written as:
where:
- a is greater than or equal to 1 but less than 10;
- n is an integer, which means it may be positive, negative or zero.
3️⃣ Is It Written in Standard Form?
✅ Correct
❌ Incorrect
The numbers in the second column have not been written in standard form because their first numbers do not satisfy:
🧠 StudyNest Method: Recognising Standard Form
Step 2: Look at the first number.
Step 3: Confirm that it is at least 1 but smaller than 10.
Step 4: Confirm that the power of 10 is a whole-number exponent, which may be positive, negative or zero.
4️⃣ Checking Examples
Example 1: Which numbers are in standard form?
💡 Did You Know?
Standard form is used extensively in science, medicine, astronomy, engineering and technology.
Earth to the Sun
Mass of an electron
Speed of light
Standard form helps scientists compare and calculate with numbers that would otherwise contain many zeros.
5️⃣ Writing Large Numbers in Standard Form
To write a large number in standard form, move the decimal point until only one non-zero digit remains before it.
Example 2: Write 6 300 000 in standard form
The decimal point moved 6 places.
Because the original number is large, the exponent is positive.
Example 3: Write 450 000 in standard form
The decimal point moved 5 places.
🧠 StudyNest Method: Large Numbers
Step 2: Count how many places the decimal point moved.
Step 3: Use that number as the exponent.
Step 4: Use a positive exponent because the original number was greater than or equal to 10.
Step 5: Check that the first number is at least 1 but less than 10.
6️⃣ Writing Small Numbers in Standard Form
For a small decimal, move the decimal point to the right until the first number is at least 1 but less than 10.
Example 4: Write 0.00082 in standard form
The decimal point moved 4 places to the right.
Because the original number is between 0 and 1, the exponent is negative.
Example 5: Write 0.0037 in standard form
The decimal point moved 3 places to the right.
🧠 StudyNest Method: Small Numbers
Step 2: Place the decimal point after that digit.
Step 3: Count how many places the decimal point moved.
Step 4: Use a negative exponent because the original number was between 0 and 1.
Step 5: Check that the first number is at least 1 but less than 10.
🔑 An Easy Way to Remember
Large ordinary number
→ Positive exponentSmall positive decimal
→ Negative exponent7️⃣ Converting Standard Form to an Ordinary Number
The exponent tells us how far and in which direction to move the decimal point.
Example 6: Write 2.6 × 10⁵ as an ordinary number
The exponent is positive, so move the decimal point 5 places to the right.
Example 7: Write 8.3 × 10⁻⁴ as an ordinary number
The exponent is negative, so move the decimal point 4 places to the left.
🧠 StudyNest Method: Returning to Ordinary Form
Step 2: For a positive exponent, move the decimal point to the right.
Step 3: For a negative exponent, move the decimal point to the left.
Step 4: Move it by the number of places shown by the exponent.
Step 5: Insert zeros where necessary.
8️⃣ Comparing Numbers in Standard Form
When comparing positive standard-form numbers, compare their powers first.
Example 8: Which number is larger?
Since 10⁶ is greater than 10⁵:
Example 9: Compare numbers with the same power
The powers are equal, so compare 4.7 and 6.1.
9️⃣ Multiplying Numbers in Standard Form
Multiply the first numbers and add the powers of ten.
Example 10
Multiply the ordinary numbers:
Add the exponents:
🔟 Correcting the Final Answer
After multiplying, the first number may not be between 1 and 10. Rewrite it if necessary.
Example 11
This is not yet in standard form because 20 is greater than 10.
🧠 StudyNest Method: Multiplication
Step 2: Add the exponents.
Step 3: Combine the two results.
Step 4: Check whether the first number is between 1 and 10.
Step 5: Adjust the power if the first number needs to be rewritten.
1️⃣1️⃣ Dividing Numbers in Standard Form
Divide the first numbers and subtract the powers.
Example 12
Divide the first numbers:
Subtract the exponents:
Example 13
The first number must be at least 1.
🧠 StudyNest Method: Division
Step 2: Subtract the exponent in the denominator from the exponent in the numerator.
Step 3: Combine the results.
Step 4: Rewrite the answer so the first number is at least 1 but less than 10.
1️⃣2️⃣ Adding and Subtracting in Standard Form
Numbers can be added or subtracted directly only when their powers of ten are the same.
Example 14: Equal powers
Keep the common power and add the first numbers.
Example 15: Different powers
Rewrite the second number using 10⁶.
1️⃣3️⃣ Using a Calculator
Many scientific calculators have an EXP, EE or ×10ˣ button.
🌍 Real-Life Example: Digital Storage
Example 16
A storage device holds:
Place the decimal point after the first digit.
The decimal point moved 10 places.
🌍 Real-Life Example: A Tiny Organism
Example 17
A microorganism has a length of:
The decimal point moves 6 places to produce 6.2.
🌍 Real-Life Calculation
Example 18
A machine produces 3 × 10⁴ components each day.
How many components will 2 × 10² identical machines produce in one day?
🧠 The StudyNest Standard-Form Strategy
Step 2: For conversion, move the decimal point and count the places.
Step 3: For multiplication, multiply the first numbers and add exponents.
Step 4: For division, divide the first numbers and subtract exponents.
Step 5: For addition or subtraction, first make the powers equal.
Step 6: Rewrite the final answer so that:
1 ≤ a < 10.
Step 7: Check whether the sign and size of the exponent make sense.
🎯 Exam Tips
⚠ Common Mistakes
Leaving the first number outside the range 1 ≤ a < 10.
Using a positive exponent for a positive number smaller than 1.
Counting the digits instead of counting the number of places the decimal point moves.
Moving the decimal point in the wrong direction when returning to ordinary form.
Multiplying the first numbers but forgetting to add the exponents.
Adding exponents during division instead of subtracting them.
Adding numbers that have different powers without first rewriting one of them.
Giving a mathematically equal answer that has not been corrected into standard form.
🎮 Your Turn!
🟢 Foundation Questions
- Write 3 200 000 in standard form.
- Write 560 000 in standard form.
- Write 0.00061 in standard form.
- Write 0.0045 in standard form.
-
State which of the following are already in standard form:
4 × 10³, 12 × 10⁴, 0.7 × 10², 5.8 × 10⁻⁶. - Write 7.2 × 10⁵ as an ordinary number.
- Write 9.1 × 10⁻⁴ as an ordinary number.
- Write 3 × 10⁰ as an ordinary number.
🔵 Developing Questions
- Write 980 000 000 in standard form.
- Write 0.00000043 in standard form.
- Write 6.04 × 10⁷ as an ordinary number.
- Write 2.8 × 10⁻⁶ as an ordinary number.
-
Calculate:
(3 × 10⁴)(5 × 10³) -
Calculate:
(4 × 10⁶)(2 × 10⁻³) -
Calculate:
(8 × 10⁹) ÷ (2 × 10⁴) -
Calculate:
(6 × 10³) ÷ (3 × 10⁷) -
Calculate:
2.4 × 10⁵ + 3.1 × 10⁵ -
Calculate:
7.8 × 10⁶ − 2.3 × 10⁶
🔴 Challenge Questions
-
Calculate and give your answer in standard form:
(6 × 10⁷)(4 × 10⁵) -
Calculate and give your answer in standard form:
(2.5 × 10⁻³)(4 × 10⁸) -
Calculate and give your answer in standard form:
(3 × 10⁴) ÷ (6 × 10⁸) -
Calculate:
5.2 × 10⁶ + 7 × 10⁵ -
Calculate:
8.4 × 10⁻³ − 2.5 × 10⁻⁴ - The speed of light is approximately 300 000 000 m/s. Write this in standard form.
- A bacterium measures 0.000000009 m. Write this in standard form.
- A factory produces 4.5 × 10⁵ items each month. How many items does it produce in 2 × 10¹ months?
- The mass of one particle is 3 × 10⁻⁸ g. Find the mass of 4 × 10⁶ such particles.
- Explain why 15 × 10⁵ is not written in standard form and rewrite it correctly.
✅ Show Answers
- 3.2 × 10⁶
- 5.6 × 10⁵
- 6.1 × 10⁻⁴
- 4.5 × 10⁻³
-
Correct:
4 × 10³
5.8 × 10⁻⁶ - 720 000
- 0.00091
- 3
- 9.8 × 10⁸
- 4.3 × 10⁻⁷
- 60 400 000
- 0.0000028
-
(3 × 5) × 10⁴⁺³
= 15 × 10⁷
= 1.5 × 10⁸ -
(4 × 2) × 10⁶⁻³
= 8 × 10³ -
(8 ÷ 2) × 10⁹⁻⁴
= 4 × 10⁵ -
(6 ÷ 3) × 10³⁻⁷
= 2 × 10⁻⁴ -
(2.4 + 3.1) × 10⁵
= 5.5 × 10⁵ -
(7.8 − 2.3) × 10⁶
= 5.5 × 10⁶ -
24 × 10¹²
= 2.4 × 10¹³ -
10 × 10⁵
= 1 × 10⁶ -
0.5 × 10⁻⁴
= 5 × 10⁻⁵ -
7 × 10⁵ = 0.7 × 10⁶
5.2 × 10⁶ + 0.7 × 10⁶
= 5.9 × 10⁶ -
2.5 × 10⁻⁴ = 0.25 × 10⁻³
8.4 × 10⁻³ − 0.25 × 10⁻³
= 8.15 × 10⁻³ - 3 × 10⁸ m/s
- 9 × 10⁻⁹ m
-
(4.5 × 10⁵)(2 × 10¹)
= 9 × 10⁶
= 9 × 10⁶ items -
(3 × 10⁻⁸)(4 × 10⁶)
= 12 × 10⁻²
= 1.2 × 10⁻¹ g -
15 is not less than 10.
15 × 10⁵
= 1.5 × 10⁶
🧠 Remember This
The first number must satisfy 1 ≤ a < 10.
Large ordinary numbers normally use positive exponents.
Positive decimals smaller than 1 normally use negative exponents.
During multiplication, multiply the first numbers and add exponents.
During division, divide the first numbers and subtract exponents.
For addition or subtraction, make the powers equal first.
Always correct the final answer into proper standard form.
🎯 Before Moving On...
- ✅ Can I explain what standard form is?
- ✅ Can I identify correctly written standard form?
- ✅ Can I convert a large number into standard form?
- ✅ Can I convert a small decimal into standard form?
- ✅ Can I return a standard-form value to ordinary form?
- ✅ Can I compare standard-form numbers?
- ✅ Can I multiply numbers in standard form?
- ✅ Can I divide numbers in standard form?
- ✅ Can I add and subtract numbers in standard form?
- ✅ Can I correct my final answer into standard form?
🎉 Well Done!
You can now represent and calculate with very large and very small numbers.
Move the decimal carefully → Use the correct exponent → Apply the index rule → Correct the final form
📈 Your Progress
Lesson 15 of 19