Scientific notation Knowledge-Based Study

Study notes

1. What is Scientific Notation?

   Scientific notation is a standardized way of writing very large or very small numbers compactly using powers of ten. Scientists, astronomers, and engineers regularly use this format to handle numbers such as the distance between celestial bodies or the mass of an atom without writing long strings of zeros.

   The Standard Form

   A number written in scientific notation has the form:

   $a \times 10^{n}$

   where:

   • Coefficient ($a$): A real number such that $1 \le |a| < 10$. This means the coefficient must have exactly one non-zero digit to the left of the decimal point.
   • Base: Always 10.
   • Exponent ($n$): An integer (positive, negative, or zero).

   For example, the quantity 45,000 is written as $4.5 \times 10^4$, whereas 0.0072 is written as $7.2 \times 10^{-3}$.

   Converting Standard Numbers to Scientific Notation

   To convert a standard decimal number into scientific notation, determine the value of the coefficient $a$ and the exponent $n$ using the following steps:

   1. Place the decimal point: Move the decimal point so that there is exactly one non-zero digit to its left. This produces the coefficient $a$.
   2. Count the shifts: Count how many places you moved the decimal point. This count is the absolute value of the exponent $|n|$.
   3. Determine the sign of the exponent:
      • If the original number has an absolute value greater than or equal to 10, the exponent $n$ is positive (the decimal moved to the left).
      • If the original number has an absolute value between 0 and 1, the exponent $n$ is negative (the decimal moved to the right).

   Examples

   • Example 1 (Large Number): Convert 684,000,000.
     Move the decimal point 8 places to the left to obtain 6.84.
     Result: $6.84 \times 10^8$.
   • Example 2 (Small Number): Convert 0.0000305.
     Move the decimal point 5 places to the right to obtain 3.05.
     Result: $3.05 \times 10^{-5}$.

   Converting to Standard Decimal Form

   To expand a number from scientific notation back into standard form, observe the exponent of 10 to determine the direction and number of places to move the decimal point.

   • Positive Exponent ($n > 0$): Move the decimal point $n$ places to the right. Fill any empty place values with zeros. This yields a number greater than or equal to 10 (or less than or equal to -10).
   • Negative Exponent ($n < 0$): Move the decimal point $|n|$ places to the left. Fill any empty place values with zeros. This yields a number between -1 and 1.
   • Zero Exponent ($n = 0$): Because $10^0 = 1$, the decimal point does not move.

   Examples

   • Example 1: Expand $5.12 \times 10^5$.
     Shift the decimal point 5 places to the right: 512,000.
   • Example 2: Expand $8.9 \times 10^{-4}$.
     Shift the decimal point 4 places to the left: 0.00089.

   Multiplying and Dividing in Scientific Notation

   Computations in scientific notation apply the basic laws of exponents.

   Multiplication

   To multiply two numbers in scientific notation, multiply their coefficients and add their exponents:

   $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$

   Example: $(3.0 \times 10^4) \times (2.0 \times 10^3) = (3.0 \times 2.0) \times 10^{4+3} = 6.0 \times 10^7$.

   If the resulting coefficient is 10 or greater, adjust it so that $1 \le |a| < 10$ and increase the exponent accordingly.

   Division

   To divide two numbers in scientific notation, divide the first coefficient by the second and subtract the exponent of the denominator from that of the numerator:

   $\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

   Example: $\frac{8.4 \times 10^6}{2.1 \times 10^2} = \left(\frac{8.4}{2.1}\right) \times 10^{6-2} = 4.0 \times 10^4$.