Quantity and Unit

Quantity & Unit

Quantity

Base

Measuring Instrumen

t

MeasurementDerived

Vector

Scalar

Unit

MKS SI

CGS

Dimension

2

Measurement

Scientific Notation

Uncertainty (Δx)

Precise & Accurate

Random

Systemic

Significant Figure

Exact & Measured

Adding & Subtracting

Multiplying & Dividing

Scientific Notation

• Scientific notation is a convenient way to write a very small or a very large number.

• Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power.

• 215 is written in scientific notation as: 215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x

102

Chapter Two 4

• Alternative writing method• Using standard form• N × 10n where 1 N < 10 and n is an integer

This galaxy is about 2.5 × 106 light years from the Earth.

The diameter of this atom is about 1 × 10−10 m.

Scientific Notation

Commonly used scientific notation (Prefix)Factor Decimal Representation Prefix Symbol1018 1,000,000,000,000,000,000 exa E1015 1,000,000,000,000,000 peta P1012 1,000,000,000,000 tera T109 1,000,000,000 giga G106 1,000,000 mega M103 1,000 kilo k102 100 hecto h101 10 deka da100 1    

10-1 0.1 deci d10-2 0.01 centi c10-3 0.001 milli m10-6 0.000 001 micro m10-9 0.000 000 001 nano n10-12 0.000 000 000 001 pico p10-15 0.000 000 000 000 001 femto f10-18 0.000 000 000 000 000 001 atto a

Chapter Two 7

Two examples of converting standard notation to scientific notation are shown below.

Chapter Two 8

Two examples of converting scientific notation back to standard notation are shown below.

Convert To Scientific Notation• 0.00034 =• 0.00145 =• 0.0000985 =• 0.016856 =• 0.0003967 =• 0.0000002 =• 0.00040 =• 0.00600 =

3.4 x 10 41.45 x 10 3 9.85 x 10 5 1.6856 x 10 23.967 x 10 4 2 x 10 74.0 x 10 4 6.00 x 10 3

Convert To Scientific Notation• 3400 =• 36,000,000 =• 367,800,000,000 =• 58 =• 65789 =• 1,000,000,000 =• 2,000 =

3.4 x 103

3.6 x 107

3.678 x 1011

5.8 x 101

6.5789 x 104

1 x 109

2 x 103

Convert to Numerical Values• 7.4 x 103 =• 5.6 x 105 =• 6.674 x 1010 =• 5.1 x 104 =• 6.5559 x 101 =• 3.64186 x 104 =• 1 x 103 =

7,400560,00066,740,000,00051,00065.55936,418.61,000

Convert to Numerical Values• 7.4 x 103 =• 5.6 x 105 =• 6.674 x 108 =• 5.1 x 104 =• 6.5559 x 101 =• 3.641 x 104 =• 1 x 103 =

0.00740.0000560.000000066740.000510.655590.00036410.001

05/02/2023

Example…

Chapter Two 13

Convert the following quantities to their SI unit.a. 510 nm

510 nm = (510 x ) m = 51 x m = 5.1 x m = 5.10 x m = 5.100 x m

b. 2,3 mmc. 78 MHz

Scientific notation

Chapter Two 14

• Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.

• The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.

• Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4.

• Scientific notation can make doing arithmetic easier. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A.

Science, Measurement, Uncertainty and Error 15

Scientific Data

The precision and accuracy are limited by the

instrumentation and data gathering

techniques.

Scientists always want the most precise and accurate experimental data.

Science, Measurement, Uncertainty and Error 16

Bad news…• No matter how good

you are… there will always be errors.

• This error in measurement is called as uncertainties

• The uncertainties are the natural behavior in a measurements

Science, Measurement, Uncertainty and Error 17

Dealing with Errors• Identify the errors and their

magnitude.

• Try to reduce the magnitude of the error.

HOW? • Better instruments• Better experimental design• Collect a lot of data

Science, Measurement, Uncertainty and Error 18

Precision and Accuracy in Measurements

• Precision How reproducible are

measurements?

• Accuracy How close are the measurements to the

true value.

Science, Measurement, Uncertainty and Error 19

Dartboard analogy• Imagine a person throwing darts, trying to

hit the bulls-eye. 

Not accurateNot precise

AccurateNot precise

Not accuratePrecise

AccuratePrecise

Type of uncertainties• Random Uncertainties

a. Different Peopleb. Measurement Equipmentc. Different Measurement Result

• Systemic Uncertainties, it is caused by the measurement equipment has never been calibrated

Science, Measurement, Uncertainty and Error 20

Significant Figures• There are 2 different types of numbers

• Exact• Measured

• Exact numbers are infinitely important• Measured number = they are measured with a

measuring device (name all 4) so these numbers have ERROR.

21

Exact NumbersAn exact number is obtained when you count

objects or use a defined relationship.

22

Counting objects are always exact

2 soccer balls4 pizzasExact relationships, predefined values, not

measured1 foot = 12 inches1 meter = 100 cmFor instance is 1 foot = 12.000000000001

inches? No 1 ft is EXACTLY 12 inches.

2.4 Measurement and Significant Figures• Every experimental

measurement has a degree of uncertainty.

• The volume, V, at right is certain in the 10’s place, 10mL<V<20mL

• The 1’s digit is also certain, 17mL<V<18mL

• A best guess is needed for the tenths place.

Chapter Two 23

What is the Length?

• We can see the markings between 1.6-1.7cm• We can’t see the markings between the .6-.7• We must guess between .6 & .7• We record 1.67 cm as our measurement• The last digit an 7 was our guess...stop there

24

1 2 3 4 cm

Learning Check

What is the length of the wooden stick?

1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

Measured Numbers• Do you see why Measured Numbers have error…

you have to make that Guess!• All but one of the significant figures are known

with certainty. The last significant figure is only the best possible estimate.

• To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

26

Chapter Two 27

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

Note the 4 rulesWhen reading a measured value, all nonzero digits

should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not.

• RULE 1. Zeros in the middle of a number are like any other digit; they are always significant.

94.072 g has five significant figures.• RULE 2. Zeros at the beginning of a number are

not significant; they act only to locate the decimal point.

0.0834 cm has three significant figures0.029 07 mL has four.

Chapter Two 28

• RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant.

138.200 m has six significant figures.If the value were known to only four significant figures, we would write 138.2 m.

• RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.

1000 g has 1 significant figure1000 g has two sigure figureChapter Two 29

Practice Rule #1 Zeros

45.87360.000239 0.00023900 48000. 48000 3.982106 1.00040

6355246

• All digits count• Leading 0’s don’t• Trailing 0’s do• 0’s count in decimal form• 0’s don’t count w/o decimal

• All digits count• 0’s between digits count as well as trailing in decimal form

Answers to question A)

1. 2.832. 36.773. 14.04. 0.00335. 0.026. 0.24107. 2.350 x 10

– 2

8. 1.000099. 310. 0.0056040

3 4 3 21446

infinite5

Rounding Off Numbers• Often when doing arithmetic on a pocket

calculator, the answer is displayed with more significant figures than are really justified.

• How do you decide how many digits to keep?• Simple rules exist to tell you how.

Chapter Two 32

• Once you decide how many digits to retain, the rules for rounding off numbers are straightforward:

• RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.

• RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater.

• If a calculation has several steps, it is best to round off at the end.

Chapter Two 33

Practice Rule #2 RoundingMake the following into a 3 Sig Fig number

1.5587

.0037421

1367

128,522

1.6683 106

1.56

.00374

1370

129,000

1.67 106

Your Final number must be of the same value as the number you started with,129,000 and not 129

Examples of RoundingFor example you want a 4 Sig Fig number

4965.03 780,582 1999.5

0 is dropped, it is <5

8 is dropped, it is >5; Note you must include the 0’s5 is dropped it is = 5; note you need a 4 Sig Fig

4965

780,600

2000.

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

Chapter Two 36

•RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.

Chapter Two 37

Multiplication and division

32.27 1.54 = 49.69583.68 .07925 = 46.43533121.750 .0342000 = 0.059853.2650106 4.858 = 1.586137 107 6.0221023 1.66110-24 = 1.000000

49.746.4.059851.586 107 1.000

Addition/Subtraction

25.5 32.72 320 +34.270 ‑ 0.0049 + 12.5 59.770 32.7151

332.5 59.8 32.72 330

__ ___ __

Addition and Subtraction

.56 + .153 = .71382000 + 5.32 = 82005.3210.0 - 9.8742 = .1258010 – 9.8742 = .12580

.7182000.10

Look for the last important digit

8.52 + 4.1586 18.73 + 153.2 =

Mixed Order of Operation

8.52 + 4.1586 18.73 + 153.2 =

(8.52 + 4.1586) (18.73 + 153.2) =

239.6

2180.

= 8.52 + 77.89 + 153.2 = 239.61 =

= 12.68 171.9 = 2179.692 =