McMahon: Basic Mathematics for Science

Basic Mathematics for Science I. Measurements 1. To ‘measure’ something means that you are comparing some property of the thing or event to a known standard

For example, if you buy a pair of pants that have an inseam of 32”, the only reason that number means something is because the inch is a definite length. Any shoe size ‘8’ will be about the same size, because shoemakers agree on the measurement.

2. There are 2 parts to every measurement. These are: a. Quantity/magnitude, or amount, which is a number that has been measured. b. units, which indicate what property of matter or energy is being measured.

For example, the number 365 indicates that someone counted something, but you cannot tell what has been counted. You cannot compare 365 days to $365 or 365 0 F. The units indicate that these three numbers are completely different concepts. Students also often make the mistake of saying ‘degrees’ for temperature, but this doesn’t really mean anything. Circles, latitude and longitude are measured in degrees. And 22 0 Fahrenheit is a very cold day, while 22 0 Celsius is the temperature of a warm room!

3. The units must be written every time the number is written 3. The units must be written every time the number is written. They must be included in every measurement. ******** A measurement is incorrect if it does not include the units.

Check: Fill in each blank: “She took the job because it pays 10 _______________”. What part of measurement was missing? _______________ “He lost almost 10 ________ by running 5 miles/day.” “ The school year is about _____ days long” What part of measurement was missing? ____________ “One kilogram equals about 2.2 pounds. Which one is mass? _________ “Xena hates it when she tried to buy pants online and the size 8 fits more like a size 0, because the different stores don’t have the same _s__________ so the _m___________ are meaningless.”

II. Scientific notation and significant digits A. Scientific Notation 1. Scientific notation is the scientists’ shorthand for showing very large and very small numbers using exponents (powers of ten). For example, the size of an electron is 10 −18 m, which is written as: 0.000000000000000001 m. Not an easy number!!! On the other hand, the distance to the end of our galaxy is 10 18 km. Written out, this is 10, 000, 000, 000, 000, 000, 000 km.

2. Any number can be written in scientific notation by taking the numeral, placing it in standard form, and multiplying it by a power of ten: *** Standard form means the first numeral is always in the units place and all other numerals are written to the right of the decimal: Example: 630 becomes 6.3 x 10 2 0.0063 becomes 6.3 x 10 −3

3. Using exponents: a. The number indicates how many times you move the decimal point. example: 10 3 and 10 −3 both mean the decimal point is moved 3 times.

b. The ‘sign’ indicates which direction the decimal will be moved b. The ‘sign’ indicates which direction the decimal will be moved. A “-“ (negative) sign means that you have a very small value and the decimal will be moved to the left. If there is no sign, it means it is a positive exponent and the decimal will be moved to the right. From the example above, the first number becomes 1000 and the second is 0.001 c. 100 equals 1

Practice: write 0.083 in standard form: ______________ How many times did you move the decimal? ______ which direction? ________ what is the exponent? _________ Write 234, 800 in scientific notation ________________ And write 4.007 x 10-4 as a numeral __________________________

B. Significant Digits:

1. Significant digits indicate how precise a measurement is. a. This property is indicated by stating what decimal place the instrument (and measurement) are correct to. Example: $482.31 indicates that money was counted to the nearest ____________. But $480 indicates it was only counted to the nearest ______________________ and $500 indicates it was estimated to the nearest ___________________ dollars. All of these are accurate, but not with the same precision (decimal place).

b. All instruments are accurate to a certain decimal placed based on a standard. You must know how precise your measurements are going to be before you begin using an instrument. You must record the data to the correct decimal place for the instrument you are using.

c. When you do any mathematical operations (addition, division, etc c. When you do any mathematical operations (addition, division, etc.) on data, the answer can only be correct to the accuracy of the least precise measurement. (for addition and subtraction, the decimal point determines the least precise. For multiplication and division, the number of significant digits determines the least precise.) example 1: add the dollar amounts from the example above. ________________________ Is this answer accurate? ______ But should it be rounded? ____________ to what? _____________ Now multiply those 3 values. _______________ Is this correct? ______ How many significant digits should the answer have? ___________

And about zeroes……. Most students look at zeroes as if they are all place holders. This is not correct. Zeroes are only place holders when they are used to count spaces from the decimal point in scientific notation.

Example 306000 equals 3.06 x 105 . But 306000.00 equals 3.0600000 x 105   And 0.00467 equals 4.67 x 10-3 But 0.00467000 equals 4.67000 x 10-3