3.9 Scientific Notation

Scientific notation simplifies working with very large or small numbers. It's a compact way to express values using powers of 10, making calculations easier and more intuitive. This system is crucial for handling extreme scales in science and math.

From cosmic distances to atomic sizes, scientific notation helps us grasp and compare vastly different magnitudes. It's a powerful tool for scientists, engineers, and anyone dealing with numbers that stretch beyond our everyday experience.

Scientific Notation

Conversion of scientific notation

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• Expresses very large or very small numbers compactly using a number between 1 and 10 multiplied by a power of 10 in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer ($1.5 \times 10^6$ for 1,500,000)
• Convert from standard form to scientific notation by moving the decimal point to obtain a number between 1 and 10 and counting the number of places moved
  • Positive exponent if moved to the left (123,000 = $1.23 \times 10^5$)
  • Negative exponent if moved to the right (0.0042 = $4.2 \times 10^{-3}$)
• Convert from scientific notation to standard form by moving the decimal point to the right for positive exponents or to the left for negative exponents, filling in zeros as needed ($4.56 \times 10^{-3}$ = 0.00456)
• Normalization ensures consistency by adjusting the coefficient to be between 1 and 10

Calculations with scientific notation

• Addition and subtraction require numbers to have the same exponent (power of 10)
  • Adjust one number's exponent to match the other, then add or subtract the coefficients ($(1.5 \times 10^3) + (4.2 \times 10^2) = (1.5 \times 10^3) + (0.42 \times 10^3) = 1.92 \times 10^3$)
• Multiplication involves multiplying the coefficients and adding the exponents ($(2.0 \times 10^4) \times (3.0 \times 10^{-2}) = 6.0 \times 10^2$)
• Division involves dividing the coefficients and subtracting the exponents ($(8.0 \times 10^5) \div (2.0 \times 10^{-3}) = 4.0 \times 10^8$)

Real-world applications of scientific notation

• Expresses distances in the universe
  • Light-years represent the distance light travels in one year (approx. $9.46 \times 10^{15}$ meters)
  • Parsecs equal approx. 3.26 light-years or $3.09 \times 10^{16}$ meters
• Measures sizes of atoms and subatomic particles
  • Angstroms ($1 \times 10^{-10}$ meters) used to measure atomic radii
  • Electron radius approximately $2.82 \times 10^{-15}$ meters
• Represents time scales
  • Age of the universe estimated around $13.8 \times 10^9$ years old
  • Half-life of radioactive elements like carbon-14 is $5,730$ years or $5.73 \times 10^3$ years

Understanding magnitude and precision

• Base 10 is used in scientific notation to represent powers of ten
• Magnitude refers to the size or scale of a number in scientific notation
• Order of magnitude describes approximate size differences between values
• Precision in scientific notation is determined by the number of significant figures in the coefficient