Scientific notation and Dimensional Analysis Mr. Conkey Fall Semester Astronomy Scientific notation and Dimensional Analysis Mr. Conkey Fall Semester

Scientific Notation Scientific notation is typically used when dealing with very large quantities or numbers as well as very small ones It is useful because it expresses those amounts without having to enter the whole number into a calculator Example: If I had a sample of 350,000,000 (three hundred-fifty million in standard notation) microscopic meteorites, the number expressed in scientific notation would be 3.5 x 10 ⁸ meteorites

Scientific Notation Rules The Rules – 3.5 is the base number and MUST be between 1 and 10 (this tells you where to move the decimal point to). x 10 is your power of 10 you are multiplying the base number by ⁸ is the exponent (how many powers of 10 you are multiplying by) For large numbers, when going from standard notation to scientific, you would move the decimal to the left which ALWAYS gives a positive exponent 3.5 x 10 ⁸

Scientific notation rules (Cont.) When working with numbers less than one and converting from standard to scientific notation, you would move the decimal to the right which ALWAYS gives you a negative exponent “How do I know what my exponent should be?” This depends on how many times you have to move the decimal place in order to make the base number between 1 and 10

Standard to Scientific notation examples Example: convert 584,200. grams to scientific notation The arrows show how many times the decimal point moves to the left, giving us 5.84200 x 105 Example: convert 0.00546 grams to scientific notation The arrows show how many times the decimal point moves giving us 5.46 x 10 ˉ³ grams

Lets try some… 245,385 0.013 385 0.193 5.9 2.45385 x 105 1.3 x 10ˉ2 3.85 x 102 1.93 x 10ˉ1 5.9 x 100

Going back to Standard notation In order to convert from scientific notation to standard, you would move the decimal place in the OPPOSITE direction than before Example 1: convert 5.462 x 104 to standard notation The 4 tells us we need to move the decimal four times to the right In order to do this, we have to add a zero to the right side = 5.4620 This gives us 54,620.

Going back to Standard Notation (cont.) Example 2: convert 2.793 x 10ˉ2 to standard notation The ˉ2 tells us we need to move the decimal place two times to the left To do this, we add one zero to the left side = 02.793 This gives us 0. 02793

Lets do this!!! 1.893 x 105 1.0181 x 10ˉ3 5.32 x 100 9.8372 x 107 0.15692 x 105 189,300 0.0010181 5.32 98,372,000 This one is tricky! The correct scientific notation for this is 1.5692 x 104 because of the decimal being in the wrong spot (remember the between 1 and 10 rule!); therefore the standard notation is 1,569.2

Shifting gears: Dimensional analysis Dimensional analysis is used to compare one thing to another using a fundamental unit (i.e. grams, liters, meters, etc.) In order to do this, you must first know the conversion factors to use: 1000 base units = 1 kilo- 10 of these = 1 base unit 100 of these = 1 base unit 1000 of these = 1 base unit

Dimensional analysis examples (single and multiple steps) Example: Convert 1500 centimeters (cm) to meters (m) When setting up the table, you begin with the given number and unit in the upper left Next, look up how many cm in 1 m on the conversion ladder and put this number in the lower right Finally, put the 1 m in the top right 1500 cm 1 m Multiply the top 100 cm Divide the Bottom = 15 m

Dimensional analysis examples (single and multiple steps(cont.)) Example: Convert 2630 centigrams (cg) to kilograms (kg) This is similar to the last example except there will be multiple conversions in the table 2630 cg 1 g 1 kg 100 cg 1000 g = 0.0263 kg