Scientific Notation and Unit Conversions

Scientific Notation

• Scientific notation is useful for handling very large or very small numbers.

• Steps to convert a number to scientific notation:

  1. Move the decimal place behind the first non-zero number.

  2. Count the number of spaces moved. This number becomes the power of 10.

  3. If the original number is greater than 10, the exponent is positive. If it's less than 10, the exponent is negative.

  4. Round the value to the correct number of significant figures.

• Example: Convert 202 to scientific notation.

  • First non-zero number is 2.

  • Move the decimal two places: 2.02

  • Write as $2.02 \times 10^2$$$2.02 \times 10^2$$

  • Since 202 is greater than 10, the exponent is positive.

• If two significant figures are desired, round to $2.0 \times 10^2$$$2.0 \times 10^2$$

• Only the numbers before the "$\times 10$$$\times 10$$ to the power of something" part count for significant figures.

Why Scientific Notation is Important

• Scientific notation makes it easier to work with extremely large or small numbers, such as the number of atoms in the human body.

• Example:

  • Instead of writing 602,000,000,000,000,000,000,000, write $6.02 \times 10^{23}$$$6.02 \times 10^{23}$$

• Calculations become easier.

  • Instead of $(5 \times 10^{34}) / (6.02 \times 10^{23})$$$(5 \times 10^{34}) / (6.02 \times 10^{23})$$, write $5/6.02 \times 10^{34-23} = 0.83 \times 10^{11}$$$5/6.02 \times 10^{34-23} = 0.83 \times 10^{11}$$

• When using a calculator, there's often a button for scientific notation.

Examples of Converting to Scientific Notation

• Convert 12.33 to scientific notation.

  • Move the decimal one place: 1.233

  • Write as $1.233 \times 10^1$$$1.233 \times 10^1$$ (positive because 12.33 > 10)

• Convert 9,234,442.3 to scientific notation.

  • Move the decimal six places: 9.2344423

  • Write as $9.2344423 \times 10^6$$$9.2344423 \times 10^6$$ (positive because 9,234,442.3 > 10)

• Convert 0.0023 to scientific notation.

  • Move the decimal three places: 2.3

  • Write as $2.3 \times 10^{-3}$$$2.3 \times 10^{-3}$$ (negative because 0.0023 < 10)

• Convert 1230 to scientific notation.

  • Decimal is at the end: 1230.

  • Move the decimal three places: 1.230

  • Write as $1.23 \times 10^3$$$1.23 \times 10^3$$ (positive because 1230 > 10)

Practice Problem

• Convert 55,500,000,000 to scientific notation.

  • Move the decimal 10 places: 5.55

  • Write as $5.55 \times 10^{10}$$$5.55 \times 10^{10}$$

Units of Measurement

• Every measurement in science needs a unit.

• Science uses metric units.

• Main units to know:

  • Length: meter (m) - Roughly equal to one yard (approximately 39.25 inches)

  • Mass: kilogram (kg) - Roughly equal to 2.2 pounds.

  • Time: second (s)

  • Temperature: Kelvin (K) - $273 K = 0^\circ C = 32^\circ F$$$273 K = 0^\circ C = 32^\circ F$$

• These are standard units called SI units (International System of Units).

• Other units are built upon these.

Metric Prefixes

• Metric prefixes scale units (similar to inches to feet, feet to yards, etc.).

• Prefixes replace the "$\times 10$$$\times 10$$ to the power of something" part of scientific notation.

• If you have a number with scientific notation and a unit (e.g., $2 \times 10^{-2}$$$2 \times 10^{-2}$$ seconds), you can replace the "$\times 10^{-2}$$$\times 10^{-2}$$ " with the prefix "centi" (c), resulting in "2 centiseconds" or "2 cs".

• Prefixes have pre-ordained meanings/values related to powers of 10.

• Prefixes need to be memorized, including their symbols and scientific notation equivalents.

• Main prefixes for this exam:

  • Kilo (k) - $10^3$$$10^3$$

  • Centi (c) - $10^{-2}$$$10^{-2}$$

  • Milli (m) - $10^{-3}$$$10^{-3}$$

  • Micro ($\mu$) - $10^{-6}$$$10^{-6}$$

• Equation: 1 prefix = $10^n$$$10^n$$

• Example:

  • 1 kilogram = $10^3$$$10^3$$ grams

  • 10 centimeters = $10 \times 10^{-2}$$$10 \times 10^{-2}$$ meters

Volume

• Volume is a combination of three lengths (width, length, and height).

• Originally derived from the concept of displaced water.

• A milliliter (mL) is equal to one centimeter cubed ($cm^3$$$cm^3$$). Both measure volume.

Density

• Density is the relationship between mass and volume.

• Formula: $d = \frac{m}{v}$$$d = \frac{m}{v}$$, where:

  • d = density

  • m = mass

  • v = volume

• Example: 250 mL of a liquid has a mass of 322.0 grams. What is its density?

  • $d = \frac{322.0 \text{ g}}{250.0 \text{ mL}} = 1.288 \text{ g/mL}$$$d = \frac{322.0 \text{ g}}{250.0 \text{ mL}} = 1.288 \text{ g/mL}$$

• Always include units in calculations; units must be included in the final answer as well.

Relationships Between English and Metric Units

• 1 pound ≈ 500 grams (or 0.5 kg)

• 1 inch ≈ 2.5 centimeters

• 1 gallon ≈ 4 liters

Conversion Factors

• Conversion factors are used to change units.

• Conversions involve relationships, equalities, or ratios.

• Two ways to write conversions:

  • Equality: stating equal values with different units e.g., 1 pound = 453 grams

  • Ratio: a balanced fraction with units e.g., 60 miles per hour (60 mph).

• You can write equalities as ratios.

  • Example: 60 seconds = 1 minute can be written as $\frac{60 \text{ seconds}}{1 \text{ minute}}$$$\frac{60 \text{ seconds}}{1 \text{ minute}}$$ or $\frac{1 \text{ minute}}{60 \text{ seconds}}$$$\frac{1 \text{ minute}}{60 \text{ seconds}}$$

• Prefixes can also be represented as ratios.

Dimensional Analysis

• Dimensional analysis uses conversion factors to cancel out one unit to get to the desired unit by multiplying and dividing.

• Steps for dimensional analysis:

  1. Identify the desired unit.

  2. Identify the given unit.

  3. Identify helpful conversions.

• Example: Convert 100 meters to centimeters.

  • Given: 100 meters

  • Desired: Centimeters

  • Conversion: $1 \text{ cm} = 10^{-2} \text{ m}$$$1 \text{ cm} = 10^{-2} \text{ m}$$

• Set up the dimensional analysis:

  • $\frac{100 \text{ m}}{1} \times \frac{1 \text{ cm}}{10^{-2} \text{ m}} = 10000 \text{ cm}$$$\frac{100 \text{ m}}{1} \times \frac{1 \text{ cm}}{10^{-2} \text{ m}} = 10000 \text{ cm}$$

  • Meters cancel out, leaving centimeters.

  • Multiply all numbers on top:
    $\frac{100 \times 1}{1} \text { cm} = 10,000 \text{ cm}$$$\frac{100 \times 1}{1} \text { cm} = 10,000 \text{ cm}$$