Scientific notation and significant digits

1. Scientific notation and significant digits

2. Scientific notation a great way for writing really big numbers. Instead of ... 602,300,000,000,000,000,000,000 you could write: 6.02 x 1023

3. stem exponent 6.02 x 10 23 base every number in scientific notation has three parts: the stem the base the exponent

4. multiplying numbers in scientific notation... • MULTIPLY the bases • ADD the exponents • REDUCE to proper scientific notation 3.24 x 1017 x 1.52 x 1015 = 4.93 x 1032 (already in proper scientific notation) 7.87 x 105 x 8.20 x 1015 = 64.5 x 1020 reduce to proper form = 6.45 x 1021

5. it’s also great for really small numbers, like 0.000 000 000 000 000 000 000 000 000 000 000 034 = 3.40 x 10-35 • multiplying is done same way as previous example: • multiply the stems • add the exponents • reduce to proper form 3.40 x 10-35 x 6.02 x 1023 =20.5 x 10-12 but must reduce to proper form – move decimal one to the left note moving left means adding to the exponent so (-12 + 1) or -11 =2.05 x 10-11

6. using one significant digit... you can quickly multiply very daunting numbers by using a single sig.dig. for example, with NO CALCULATOR, multiply the following: 137,445,000,000,000,000,000,000 x 342,223,111,000,000,000,000,000 = (1 x 1023) x (3 x 1023) = 3 x 1046 sure, you lose some precision by dropping down to one significant digit. But with numbers this big, that precision probably isn’t necessary.

7. some tricky items... anything to the exponent zero equals ONE so “7” in scientific notation would be 7.00 x 100

8. Significant digits: 7.00 x 100 actually MEANS that the true measurment is between 6.995 and 7.005 x 100 that’s a possible “spread” of 0.01 that’s very different from saying 7.00000 x 100 which would MEAN between 6.999995 and 7.0000005 x 100 that’s a possible “spread” of 0.00001 so if you’re working with three significant digits, always round off to keep those three significant digits.

9. SI exponents First, MEMORIZE these: billionth millionth thousandth thousand million billion nano- micro- milli- (base measurement – e.g. meter) kilo- mega- giga-

10. subtract 3 zeros or divide by 1000 or subtract 3 from exponent subtract 3 zeros or divide by 1000 or subtract 3 from exponent subtract 3 zeros or divide by 1000 or subtract 3 from exponent subtract 3 zeros or divide by 1000 or subtract 3 from exponent subtract 3 zeros or divide by 1000 or subtract 3 from exponent subtract 3 zeros or divide by 1000 or subtract 3 from exponent nm nanometers one billionth um micrometers one millionth mm millimeters one thousandth m meters km kilometers one thousand Mm megameters one million Gm gigameters one billion add 3 zeros or mult by 1000 or add 3 to exponent add 3 zeros or mult by 1000 or add 3 to exponent add 3 zeros or mult by 1000 or add 3 to exponent add 3 zeros or mult by 1000 or add 3 to exponent add 3 zeros or mult by 1000 or add 3 to exponent add 3 zeros or mult by 1000 or add 3 to exponent examples: 6.7 m is ___ um? 6,700,000 or 6.7 x 106 um 42 um is ____ Mm? 42 x 10-12 Mm or 4.2 x 10-11 Mm 77 km = ____ mm? 77 x 106 mm = 7.7 x 107 mm rem: numbers in scientific notation always have a stem, a base, and an exponent. The stem always has one digit to the left of the decimal. the number of digits to the right of the decimal in the stem depends on the number of significant digits. in most science applications, 3 significant digits is sufficient – one digit to the left and 2 digits to the right of the decimal point. example: 2.45 x 1012 or 6.02 x 1023 or 4.00 x 10-14