Significant Figures with Scientific Notation in Addition, Subtraction Multiplication and Division

Introduction to Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is especially useful in chemistry and physics, where precision is essential. The general form of scientific notation is:

N×10ⁿ

Where:

• N is a number between 1 and 10.
• n is an integer (positive or negative) that represents the exponent.

For example:

• 1.34×10⁵ = 134,000
• 8.6×10⁻⁴ = 0.00086

Counting Significant Figures in Scientific Notation

The number of significant figures is determined by N, not by the exponent.

• 1.34×10⁵ has 3 significant figures.
• 8.6×10⁻⁴ has 2 significant figures.

To express a number with a certain number of significant figures, zeros may be added after the decimal point:

• 1.34×10⁵ with 6 significant figures = 1.34000×10⁵

Rounding Off to Significant Figures

If a number has more digits than required, it is rounded appropriately.

• Example: Express 46897 in two significant figures.
  • The first two digits are 46.
  • Since the next digit is 8 (greater than 5), we round up.
  • Final result: 4.7×10⁴.

Operations in Scientific Notation

Addition and Subtraction

1. If the exponents are the same, simply add or subtract the coefficients.
   • Example: (2.65×10³)+(6.4×10³)
   • (2.65+6.4)×10³=9.05×10³
   • Rounded to one decimal place: 9.1 × 10³
2. If the exponents are different, adjust one number to match the other.
   • Example: (9.578×10³)−(5.326×10²)
   • Convert to decimal: 9578 – 532.6 = 9045.4
   • Convert back to scientific notation: 9.045 × 10³

Multiplication and Division

1. Multiplication: Multiply the coefficients and add the exponents.
   • Example: (3.4×10⁻⁶)×(2.5×10⁴)
   • (3.4×2.5)×10⁻⁶⁺⁴ = 8.5×10⁻²
2. Division: Divide the coefficients and subtract the exponents.
   • Example: (6.0×10⁵)÷(2.0×10²)
   • (6.0÷2.0)×10⁵⁻² = 3.0×10³

Conceptual Questions

1. Why do we use scientific notation?
   • To simplify the representation of very large or small numbers.
   • To avoid misplacing zeros.
2. What determines the number of significant figures in scientific notation?
   • The coefficient N (not the exponent).

Do You Know?

• The speed of light in vacuum is approximately 3.00 × 10⁸ m/s.
• Avogadro’s number is 6.022 × 10²³, representing the number of atoms in a mole.
• The charge of an electron is 1.602 × 10⁻¹⁹ C.

Worksheet

Q1: Convert the following into scientific notation:

a) 0.000678

b) 123,400

c) 5,690,000

Q2: Perform the following calculations and express the result in scientific notation:

a) (2.5×10⁴)×(3.6×10⁻²)

b) (5.7×10⁶)÷(1.9×10²)

Test Paper (Total: 10 Marks)

Q1: Convert into scientific notation: (2 Marks)

a) 0.00034

b) 789,000

Q2: Express with three significant figures: (2 Marks)

a) 12.347

b) 0.008659

Q3: Solve: (3 Marks)

(4.5×10³)+(6.3×10³)

Q4: Perform the following multiplication: (3 Marks)

(3.2×10⁻⁴)×(1.1×10²)

Quick Revision Points

• Scientific notation is N×10ⁿ, where N is between 1 and 10.
• Significant figures are counted in the coefficient N, not in the exponent.
• In addition/subtraction, make exponents the same before performing operations.
• In multiplication, add exponents; in division, subtract them.
• Always round answers to match the smallest number of significant figures.

Test Your Knowledge (Quiz)

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