Scientific Notation
What is Scientific Notation?
Scientific notation is a standard way of writing very large or very small numbers in a more concise form. It's written as:
Where:
- a is a number between 1 and 10 (1 ≤ a < 10)
- n is an integer (positive or negative)
Rules for Writing Numbers in Scientific Notation
- Move the decimal point to follow the first non-zero digit
- Count the number of places the decimal point was moved
- If you moved the decimal point to the left, the exponent is positive
- If you moved the decimal point to the right, the exponent is negative
Examples
Converting Large Numbers to Scientific Notation
Example 1:
Convert 5,280,000 to scientific notation
Step 1: Move the decimal point to follow the first non-zero digit
5.280000
Step 2: Count the number of places moved (6 places to the left)
Step 3: The exponent is positive: 5.28 × 106
Example 2:
Convert 7,430,000,000 to scientific notation
Step 1: Move the decimal point to follow the first non-zero digit
7.430000000
Step 2: Count the number of places moved (9 places to the left)
Step 3: The exponent is positive: 7.43 × 109
Converting Small Numbers to Scientific Notation
Example 3:
Convert 0.000082 to scientific notation
Step 1: Move the decimal point to follow the first non-zero digit
8.2 × 10-5
Step 2: Count the number of places moved (5 places to the right)
Step 3: The exponent is negative: 8.2 × 10-5
Example 4:
Convert 0.00000000457 to scientific notation
Step 1: Move the decimal point to follow the first non-zero digit
4.57 × 10-9
Step 2: Count the number of places moved (9 places to the right)
Step 3: The exponent is negative: 4.57 × 10-9
Operations with Scientific Notation
Multiplication
To multiply numbers in scientific notation, multiply the coefficients and add the exponents.
(a × 10n) × (b × 10m) = (a × b) × 10(n+m)
Example:
Multiply (2.5 × 104) × (3.0 × 10-2)
Step 1: Multiply the coefficients: 2.5 × 3.0 = 7.5
Step 2: Add the exponents: 4 + (-2) = 2
Step 3: Result: 7.5 × 102 = 750
Division
To divide numbers in scientific notation, divide the coefficients and subtract the exponents.
(a × 10n) ÷ (b × 10m) = (a ÷ b) × 10(n-m)
Example:
Divide (8.4 × 105) ÷ (2.1 × 102)
Step 1: Divide the coefficients: 8.4 ÷ 2.1 = 4.0
Step 2: Subtract the exponents: 5 - 2 = 3
Step 3: Result: 4.0 × 103 = 4,000
Addition and Subtraction
To add or subtract numbers in scientific notation, convert them to the same power of 10, then add or subtract the coefficients.
Example of Addition:
Add (5.2 × 103) + (3.7 × 102)
Step 1: Convert to the same power: (5.2 × 103) + (0.37 × 103)
Step 2: Add the coefficients: 5.2 + 0.37 = 5.57
Step 3: Result: 5.57 × 103 = 5,570
Example of Subtraction:
Subtract (9.3 × 104) - (6.8 × 103)
Step 1: Convert to the same power: (9.3 × 104) - (0.68 × 104)
Step 2: Subtract the coefficients: 9.3 - 0.68 = 8.62
Step 3: Result: 8.62 × 104 = 86,200
Powers of Scientific Notation
To raise a number in scientific notation to a power, raise the coefficient to that power and multiply the exponent by the power.
(a × 10n)p = ap × 10(n×p)
Example:
Calculate (2.0 × 103)2
Step 1: Calculate the coefficient: 2.02 = 4.0
Step 2: Calculate the exponent: 3 × 2 = 6
Step 3: Result: 4.0 × 106 = 4,000,000
Real-world Applications
Astronomy
The distance from Earth to the Sun is approximately 1.496 × 108 kilometers.
Physics
The mass of an electron is approximately 9.11 × 10-31 kilograms.
Chemistry
Avogadro's number is 6.022 × 1023 particles per mole.
Special Cases and Tips
Numbers Between 1 and 10
For numbers between 1 and 10, the scientific notation has an exponent of 0.
Example: 5.67 = 5.67 × 100
Working with Zeros
Zero in scientific notation is usually written as 0 × 100 or simply 0.
Example: 0 = 0 × 100
Test Your Knowledge: Scientific Notation Quiz
Key Takeaways
- Scientific notation expresses numbers in the form a × 10n where 1 ≤ a < 10
- Positive exponents represent large numbers (moved decimal left)
- Negative exponents represent small numbers (moved decimal right)
- Multiplication: multiply coefficients, add exponents
- Division: divide coefficients, subtract exponents
- Addition/Subtraction: convert to same power, then add/subtract coefficients
- Powers: raise coefficient to power, multiply exponent by power