Numbers and measurement Chapter 2 p

Units of Measurement Measurements must have both a number and a unit!!!! Example: 26.7 m Like units have a standard value.

SI system of measurement Used by all scientists worldwide Why? Revised version of metric system Based on powers of ten

SI base units A defined unit based on an object or event

SI Base Units UnitUnit symbolQuantityQuantity symbol metermlength, distance l or d kilogramkgmass m secondstime t KelvinKtemperature T molemolamount of substance n ampereAelectric current I candelacdluminous intensity I

Derived units All other measurements have derived units made by combining base units Example: volume - m 3 (m·m·m)

Appropriate units It is necessary to use the most appropriate unit when measuring. Example: centimeters to measure the length of a pencil rather than kilometers Sometimes this means that we must convert units to use the measurement in a calculation

SI Prefixes Prefixes are used to show the relationships between units of the same quantity Example: 100 cm = 1 m (length) Prefixes are added or removed as needed See handout of prefixes

Practice Give the meaning and symbol of each prefix listed below: – Deka – Micro – Nano – Kilo – Centi – Tera

Scientific notation Makes using very large or very small numbers much easier Look at the correct form of a number written in scientific notation: 1.88 x 10 4 kg

General formula The general formula for a number in scientific notation is M x 10 n where – M = number between 1 and 10 1 to 9.999… – n = an integer

Step 1: find the decimal, may be unwritten at end of number Step 2: determine where the decimal should be moved so that there is one non-zero digit in front of the decimal Step 3: move the decimal and count places Step 4: put the number into the form M x 10 n

Examples Change the following numbers into scientific notation: – m – m – m – m – m – Numbers >1 or = 1 have + exponents, and numbers <1 have - exponents

Regular notation The exponent tells you how many places to move the decimal and in which direction Positive- move right Negative- move left 3.45 x 10 5 m 2.14 x km

Using scientific notation Calculator makes this easy! You must enter the numbers and functions correctly

Calculators and scientific notation Find the exponent key on your calculator (EXP, EE, x10 - may be a second function). This is the key that allows you to enter a number in scientific notation. Do NOT use the ^ key!

Entering numbers in scientific notation 1.88 x 10 4 m – Enter 1.88 (as usual) – Hit the exponent key (may be a 2 nd function) Will indicate that you can now enter the exponent value ( , 1.88E, x10 00 ) DO NOT punch in x 10 !!! – Enter 4 – Ready to enter a math function

Calculating with measurements All measurements, regardless of scientific notation Addition and subtraction- units must be the same Multiplication and division- you must multiply or divide units too – cm x cm = cm 2 – kg ÷ mL = kg/mL

(2.65 x kg) ÷ (4.92 x L) – Enter 2.65 – Hit exponent key – Enter -5 – Hit ÷ – Enter 4.92 – Hit exponent key – Enter -2 – Hit =

(4.77 x 10 9 m) x (3.02 x 10 6 s) – Enter 4.77 – Hit exp key – Enter 9 – Hit x – Enter 3.02 – Hit exp key – Enter 6 – Hit =

Converting between units Dimensional analysis or factor- label method Based on the relationship between UNITS!!! Which unit is larger? Is there a prefix? Or two? Prefix value?

Equalities measurements that represent the same quantity Written as numerical statements Example: 1 yard = 3 feet 1 m = 100 cm

Determining equalities See SI conversion cheat sheet

Conversion factors Are used to convert between units Equalities written as fractions Example: 1 m = 100 cm becomes the factors: 1 m and 100 cm 100 cm 1 m To make conversion factors, you have to know how units are equal

Check prior knowledge How many seconds are in 45 minutes? – 2700 s How many yards is 750 inches? – 20.8 yd

Using conversion factors What is the original unit given? What unit am I converting to? How are these two units equal? What are my conversion factors? Which conversion factor do I use? – Remember: going from large to small- multiply by 10 n going from small to large- divide by 10 n

Example Convert mg into g Original unit: mg New unit: g Equal?: 1000 mg = 1 g Conversion factors: 1000 mg and 1 g 1 g 1000 mg

Factor to use? Look at the units! You need the original unit to cancel out, so you need that unit in the numerator (given) and the denominator (conversion factor) mg x 1 g = g 1000 mg

Two prefixes? Convert km to mm (opposite sides of base unit – Kilo means 1000, milli means 1/1000- that gives me 6 zeros, so my decimal must move 6 places – Large to small, decimal moves right Convert 2550 µg to cg (same side of base unit) – Micro has 6 zeros, centi has 2 zeros – 6-2 = 4, my decimal moves four places – Small to large, decimal moves left

Other ways to convert Stair step method works with prefixes from kilo to milli

Density Ratio of an object’s mass to its volume D = m/V Unit- g/mL, kg/L, g/cm 3, etc. Example problems p. 29 Do # 73 and 74 p. 50

Accuracy How close a measurement is to an accepted value

Precision How close a series of measurements are to one another

Limits in measurements Human error Limitations of measuring instrument All measurements have an estimation

Percent error Percent error = experimental – accepted x 100% accepted value See p. 38

Significant digits For a measurement includes all known digits plus one estimated digit

Rounding with Sigdigs Round as usual

Calculations with sigdigs Adding and subtraction – Count decimal places Multiplying and dividing – Count total number of sigdigs in measurements