Expressing Very Small Numbers
Very small numbers like 0.000000045 are difficult to read and write. Just as we use standard form (scientific notation) to express very large numbers, we can express very small numbers using negative exponents.
In standard form, a very small number is written as a × 10⁻ⁿ, where 1 ≤ a < 10 and n is a positive integer. The negative exponent tells us how many places to move the decimal point to the left.
This notation is widely used in science — for example, the size of an atom, the mass of an electron, or the thickness of a human hair. It makes very small numbers easy to read, compare, and calculate with.
What is Expressing Very Small Numbers?
Definition: A very small number is expressed in standard form (scientific notation) as:
a × 10⁻ⁿ
Where:
- a = a number such that 1 ≤ a < 10
- 10⁻ⁿ = 1/10ⁿ (a fraction with denominator 10ⁿ)
- n = a positive integer (the number of places the decimal moves)
Understanding negative exponents:
- 10⁻¹ = 1/10 = 0.1
- 10⁻² = 1/100 = 0.01
- 10⁻³ = 1/1000 = 0.001
- 10⁻⁶ = 1/1000000 = 0.000001
Methods
Converting a small decimal to standard form:
- Move the decimal point to the right until you get a number between 1 and 10.
- Count the number of places you moved the decimal. Call this n.
- Write the number as a × 10⁻ⁿ.
Example: Express 0.00056 in standard form.
- Move decimal 4 places right: 5.6
- n = 4
- 0.00056 = 5.6 × 10⁻⁴
Converting standard form back to decimal:
- Look at the negative exponent −n.
- Move the decimal point n places to the left.
- Add zeros as needed.
Example: Convert 3.2 × 10⁻⁵ to decimal.
- Move decimal 5 places left: 0.000032
Solved Examples
Example 1: Example 1: Basic conversion
Problem: Express 0.0078 in standard form.
Solution:
- Move decimal 3 places right: 7.8
- 0.0078 = 7.8 × 10⁻³
Answer: 0.0078 = 7.8 × 10⁻³.
Example 2: Example 2: Very small number
Problem: Express 0.00000039 in standard form.
Solution:
- Move decimal 7 places right: 3.9
- 0.00000039 = 3.9 × 10⁻⁷
Answer: 0.00000039 = 3.9 × 10⁻⁷.
Example 3: Example 3: Converting back to decimal
Problem: Write 6.02 × 10⁻⁵ in usual decimal form.
Solution:
- Exponent is −5, so move decimal 5 places left.
- 6.02 → 0.0000602
Answer: 6.02 × 10⁻⁵ = 0.0000602.
Example 4: Example 4: Comparing small numbers
Problem: Which is larger: 3.5 × 10⁻⁴ or 8.1 × 10⁻⁶?
Solution:
- Compare exponents: −4 > −6
- So 10⁻⁴ > 10⁻⁶
- Therefore 3.5 × 10⁻⁴ > 8.1 × 10⁻⁶
Answer: 3.5 × 10⁻⁴ is larger.
Example 5: Example 5: Mass of an electron
Problem: The mass of an electron is 0.00000000000000000000000000000091 kg. Express in standard form.
Solution:
- Count decimal places moved: 31 places right gives 9.1
- Mass = 9.1 × 10⁻³¹ kg
Answer: Mass of electron = 9.1 × 10⁻³¹ kg.
Example 6: Example 6: Multiplying small numbers
Problem: Simplify (3 × 10⁻⁴) × (2 × 10⁻³).
Solution:
- Multiply coefficients: 3 × 2 = 6
- Add exponents: (−4) + (−3) = −7
- Result = 6 × 10⁻⁷
Answer: (3 × 10⁻⁴) × (2 × 10⁻³) = 6 × 10⁻⁷.
Example 7: Example 7: Dividing in standard form
Problem: Simplify (8.4 × 10⁻⁵) ÷ (2.1 × 10⁻²).
Solution:
- Divide coefficients: 8.4 ÷ 2.1 = 4
- Subtract exponents: (−5) − (−2) = −5 + 2 = −3
- Result = 4 × 10⁻³
Answer: (8.4 × 10⁻⁵) ÷ (2.1 × 10⁻²) = 4 × 10⁻³.
Example 8: Example 8: Thickness of paper
Problem: A sheet of paper is 0.0016 cm thick. Express in standard form and find the thickness of 500 sheets.
Solution:
- 0.0016 = 1.6 × 10⁻³ cm
- 500 sheets = 500 × 1.6 × 10⁻³
- = 800 × 10⁻³
- = 8 × 10² × 10⁻³
- = 8 × 10⁻¹ = 0.8 cm
Answer: One sheet = 1.6 × 10⁻³ cm; 500 sheets = 0.8 cm.
Example 9: Example 9: Converting fraction to standard form
Problem: Express 7/10000000 in standard form.
Solution:
- 7/10000000 = 7/10⁷ = 7 × 10⁻⁷
Answer: 7/10000000 = 7 × 10⁻⁷.
Example 10: Example 10: Ordering small numbers
Problem: Arrange in ascending order: 5 × 10⁻³, 2 × 10⁻⁵, 9 × 10⁻⁴.
Solution:
- 2 × 10⁻⁵ = 0.00002 (smallest exponent → smallest number)
- 9 × 10⁻⁴ = 0.0009
- 5 × 10⁻³ = 0.005 (largest exponent → largest number)
Ascending order: 2 × 10⁻⁵ < 9 × 10⁻⁴ < 5 × 10⁻³
Answer: 2 × 10⁻⁵, 9 × 10⁻⁴, 5 × 10⁻³.
Real-World Applications
Real-world uses of standard form for small numbers:
- Physics: Expressing sizes of atoms, wavelengths of light, and masses of subatomic particles.
- Chemistry: Molar masses of molecules and concentrations of solutions.
- Biology: Size of cells, bacteria, and viruses (e.g., COVID virus ≈ 1.2 × 10⁻⁷ m).
- Engineering: Tolerances in precision manufacturing measured in micrometres.
- Medicine: Dosages of medicines and concentrations of drugs in blood.
Key Points to Remember
- Very small numbers are written as a × 10⁻ⁿ where 1 ≤ a < 10.
- Move the decimal point right to convert a small decimal to standard form; count the moves as n.
- The negative exponent tells you the number is less than 1.
- To convert back: move the decimal n places to the left.
- When multiplying: multiply coefficients and add exponents.
- When dividing: divide coefficients and subtract exponents.
- To compare small numbers, first compare the exponents of 10.
- The less negative exponent gives the larger number.
Practice Problems
- Express 0.000045 in standard form.
- Write 2.7 × 10⁻⁶ in decimal form.
- Which is larger: 4 × 10⁻³ or 9 × 10⁻⁵?
- Simplify: (5 × 10⁻³) × (4 × 10⁻⁵).
- Simplify: (9.6 × 10⁻⁴) ÷ (3.2 × 10⁻²).
- The diameter of a red blood cell is 0.000007 m. Express in standard form.
- Arrange in descending order: 3 × 10⁻², 7 × 10⁻⁵, 1 × 10⁻³.
- Express 1/50000 in standard form.
Frequently Asked Questions
Q1. How do you express a very small number in standard form?
Move the decimal point right until you get a number between 1 and 10. Count the moves as n. Write as a × 10⁻ⁿ.
Q2. What does a negative exponent mean?
A negative exponent means the number is a fraction. 10⁻ⁿ = 1/10ⁿ. For example, 10⁻³ = 1/1000 = 0.001.
Q3. How do you compare very small numbers in standard form?
Compare exponents first. The number with the less negative (larger) exponent is bigger. If exponents are equal, compare the coefficients.
Q4. What is the standard form of 0.001?
0.001 = 1 × 10⁻³.
Q5. Can standard form be used for numbers greater than 1?
Yes. For numbers ≥ 1, the exponent is positive or zero. For example, 4500 = 4.5 × 10³.
Q6. How do you multiply numbers in standard form?
Multiply the coefficients and add the exponents. (a × 10ᵐ) × (b × 10ⁿ) = (ab) × 10ᵐ⁺ⁿ.
Q7. What is the size of a hydrogen atom in standard form?
The diameter of a hydrogen atom is approximately 1.2 × 10⁻¹⁰ metres.
Q8. Is 0.5 × 10⁻³ in correct standard form?
No. The coefficient must be between 1 and 10. Correct form: 5 × 10⁻⁴.
