Introduction to Scientific Notation

What is Scientific Notation?

• A method to represent very large or very small numbers.
• Useful for simplifying numbers and calculations.

Converting Large Numbers to Scientific Notation

1. Example: 45,000

   • Move decimal between the first two digits: 4 and 5.
   • Move decimal 4 places to the left.
   • Result: $4.5 \times 10^4$

2. Example Problems:

   • 3,750: Move 3 spaces to the left: $3.75 \times 10^3$
   • 580,000: Move 5 spaces to the left: $5.8 \times 10^5$
   • 72,000,000: Move 7 spaces to the left: $7.2 \times 10^7$
   • 9.3 billion: Move 9 spaces to the left: $9.3 \times 10^9$

Converting Small Numbers to Scientific Notation

1. Example: 0.0023

   • Move decimal 3 places to the right.
   • Result: $2.3 \times 10^{-3}$

2. Example Problems:

   • 0.00076: Move 4 spaces to the right: $7.6 \times 10^{-4}$
   • 0.049: Move 2 spaces to the right: $4.9 \times 10^{-2}$
   • 0.00000541: Move 6 spaces to the right: $5.41 \times 10^{-6}$
   • 0.000000000835: Move 10 spaces to the right: $8.35 \times 10^{-10}$

Converting from Scientific Notation to Standard Notation

1. Example: $2.4 \times 10^2$

   • Positive exponent: move decimal 2 spaces to the right.
   • Result: 240

2. Example Problems:

   • $3.56 \times 10^3$: Move 3 spaces to the right: 3,560
   • $4.27 \times 10^5$: Move 5 spaces to the right: 427,000
   • $3.96 \times 10^7$: Move 7 spaces to the right: 39,600,000

Converting Negative Exponents to Standard Notation

1. Example: $3.7 \times 10^{-3}$

   • Negative exponent: move decimal 3 spaces to the left.
   • Result: 0.0037

2. Example Problems:

   • $4.16 \times 10^{-5}$: Move 5 spaces to the left: 0.0000416

Mixed Review

• Small number: $7.35 \times 10^{-3}$
  • Move 3 spaces to the right: 0.00735
• Large number: $3.64 \times 10^5$
  • Move 5 spaces to the left: 364,000
• Small number: $1.5 \times 10^{-2}$
  • Move 2 spaces to the left: 0.015
• Large number: $2.8 \times 10^3$
  • Move 3 spaces to the left: 2,800

Practice Problems

• $1.8 \times 10^{-3}$: Move 3 spaces to the left: 0.0018
• $4.1 \times 10^2$: Move 2 spaces to the right: 410
• $1.2 \times 10^{-5}$: Move 5 spaces to the left: 0.000012
• $2.7 \times 10^4$: Move 4 spaces to the right: 27,000

Key Takeaways

• Positive exponents in scientific notation indicate large numbers.
• Negative exponents indicate small numbers.
• Moving the decimal point to the right increases the number's value; moving it to the left decreases it.