Scientific notation Knowledge-Based Study

Study notes

1. What is Scientific Notation?

   In physics, we frequently encounter quantities that are unimaginably large or extremely small. For instance, the mass of the Sun is approximately 1,989,000,000,000,000,000,000,000,000,000 kg, while the mass of an electron is roughly 0.0000000000000000000000000000009109 kg. Writing out all of these zeros is not only tedious, but it also increases the likelihood of calculation errors.

   To overcome this, scientists use scientific notation (also called standard index form), which expresses any physical quantity in the standard form:

   $M \times 10^n$

   • Coefficient (Mantissa), $M$: A real number such that $1 \le |M| < 10$. It has exactly one non-zero digit to the left of the decimal point.
   • Base: Always 10.
   • Exponent (Power), $n$: An integer (positive, negative, or zero) indicating the number of places the decimal point was shifted.

   Handling Large Quantities

   When a measured value is greater than or equal to 10, the exponent $n$ is a positive integer.

   Step-by-Step Procedure:

   1. Locate the original decimal point (at the far right if no decimal point is shown).
   2. Shift the decimal point to the left until exactly one non-zero digit remains to its left. This gives the coefficient $M$.
   3. Count the total number of places moved. That count is the positive exponent $n$.

   Physics Examples:

   • Speed of light in a vacuum:
     $300\,000\,000\text{ m/s} = 3.0 \times 10^8\text{ m/s}$
     (The decimal moved 8 places to the left.)
   • Mean distance from Earth to the Sun:
     $149\,600\,000\,000\text{ m} = 1.496 \times 10^{11}\text{ m}$
     (The decimal moved 11 places to the left.)

   Handling Submicroscopic Quantities

   When a measured value lies between 0 and 1, the exponent $n$ is a negative integer.

   Step-by-Step Procedure:

   1. Locate the decimal point.
   2. Shift the decimal point to the right until it is directly after the first non-zero digit. This gives the coefficient $M$.
   3. Count the number of places moved. The exponent $n$ is the negative of this count.

   Physics Examples:

   • Elementary electric charge:
     $0.0000000000000000001602\text{ C} = 1.602 \times 10^{-19}\text{ C}$
     (The decimal moved 19 places to the right.)
   • Typical atomic diameter:
     $0.0000000001\text{ m} = 1 \times 10^{-10}\text{ m}$
     (The decimal moved 10 places to the right.)

   Performing Calculations

   Physics problems regularly require mathematical operations involving scientific notation. Follow these core rules:

   1. Multiplication and Division

   • Multiplication: Multiply the coefficients and add the exponents.
     $(A \times 10^a) \times (B \times 10^b) = (A \times B) \times 10^{a+b}$
   • Division: Divide the coefficients and subtract the denominator exponent from the numerator exponent.
     $\frac{A \times 10^a}{B \times 10^b} = \left(\frac{A}{B}\right) \times 10^{a-b}$
   • Note: Always readjust the final coefficient so that $1 \le |M| < 10$.

   2. Addition and Subtraction

   • Before adding or subtracting, both terms must be rewritten to share the same exponent of 10.
   • Once the exponents match, add or subtract the coefficients and keep the common power of 10:
     $(2.5 \times 10^4\text{ m}) + (3.1 \times 10^3\text{ m}) = (2.5 \times 10^4\text{ m}) + (0.31 \times 10^4\text{ m}) = 2.81 \times 10^4\text{ m}$