Introduction to Scientific Notation

Converting from Standard Format

Converting a number from its "standard format" (the way we usually write numbers) to scientific notation depends on whether the number is very large or very small.

Large Numbers (Positive Exponents)

When dealing with large numbers, you move the decimal point to the left until only one non-zero digit remains to its left. The number of spaces you move the decimal becomes a positive exponent.

Example: Convert 52,000 to scientific notation.

1. Start with $52000.$
2. Move the decimal 4 places to the left: $5.2000$
3. Since we moved 4 places, the exponent is $4$.
4. Result: $5.2 \times 10^{4}$.

Small Numbers (Negative Exponents)

When dealing with numbers much smaller than 1, you move the decimal point to the right until it is placed after the first non-zero digit. The number of spaces moved becomes a negative exponent.

Example: Convert 0.00378 to scientific notation.

1. Start with $0.00378$
2. Move the decimal 3 places to the right: $3.78$
3. Since we moved 3 places to the right, the exponent is $-3$.
4. Result: $3.78 \times 10^{-3}$.

Visualizing small scales: The width of a human hair can be expressed as $8 \times 10^{-6} \text{ m}$.

Understanding Powers of Ten

To master these conversions, you must understand how powers of ten relate to standard numbers. The exponent acts as a shorthand for how many times 10 is multiplied or divided.

Comparison of Scales

Scientific notation allows us to compare vastly different scales side-by-side without getting lost in a sea of zeros.

• Diameter of the Earth: $12,800,000 \text{ m} \rightarrow 1.28 \times 10^{7} \text{ m}$
• Mass of a human: $68 \text{ kg} \rightarrow 6.8 \times 10^{1} \text{ kg}$
• Length of a virus: $0.00003 \text{ cm} \rightarrow 3 \times 10^{-5} \text{ cm}$

Practice Problems

Test your understanding by attempting these conversions.

Part 1: Standard to Scientific

Convert the following to scientific notation:

1. $0.00000008 \text{ m}$
2. $72,000 \text{ L}$

Solutions:

1. Move the decimal 8 places to the right: $8 \times 10^{-8} \text{ m}$.
2. Move the decimal 4 places to the left: $7.2 \times 10^{4} \text{ L}$.

Part 2: Scientific to Standard

Convert the following to standard format:

1. $2.0 \times 10^{-2} \text{ s}$
2. $1.8 \times 10^{5} \text{ g}$

Solutions:

1. Move the decimal 2 places to the left: $0.020 \text{ s}$.
2. Move the decimal 5 places to the right: $180,000 \text{ g}$.