Periodic Table of the Elements Chemistry Reference Sheet California Standards Test

Sodium 22.99

Na 11

Element symbol

* Element name

Hydrogen 1.01

H 1

Lithium 6.94

Li 3

Sodium 22.99

Na 11

39.10

19

K 58.69

Ni 28

Rubidium 85.47

Rb 37

140.91

Pr 59

(261)

Rf 104

95.94

Mo 42

(258)

Md 101

72.61

Ge 32

1 1A

2 2A

1

2

3

4

7 7B

11 1B

12 2B

13 3A

16 6A

8

5

6

7

9 8B

10

14 4A

15 5A

17 7A

18 8A

3 3B

4 4B

5 5B

6 6B

Copper 63.55

Cu 29

Cobalt 58.93

Co 27

Helium 4.00�

He 2

Boron 10.81

B 5

Carbon 12.01�

C 6

Nitrogen 14.01

N 7

Oxygen 16.00

O 8

19.00�

F 9

Neon 20.18�

Ne 10

26.98

Al 13

Silicon 28.09�

Si 14

30.97

P 15

Sulfur 32.07

S 16

35.45

Cl 17

Argon 39.95

Ar 18

Calcium 40.08

Ca 20

Scandium 44.96

Sc 21

Titanium 47.87

Ti 22

Chromium 52.00

Cr 24

Iron 55.85

Fe 26

Zinc 65.39

Zn 30

Gallium 69.72

Ga 31

Arsenic 74.92

As 33

Selenium 78.96

Se 34

Bromine 79.90

Br 35

83.80

Kr 36

Strontium 87.62

Sr 38

88.91

Y 39

Zirconium 91.22

Zr 40

Niobium 92.91

Nb 41

(98)

Tc 43

Ruthenium 101.07

Ru 44

Rhodium 102.91

Rh 45

106.42

46

107.87

Ag 47

Cadmium 112.41

Cd 48

Indium 114.82

In 49

Tin 118.71

Sn 50

121.76

Sb 51

127.60

52

Iodine 126.90

I 53

Xenon 131.29

Xe 54

Cesium 132.91

Cs 55

137.33

Ba 56

138.91

La 57

Hafnium 178.49

Hf 72

180.95

73

183.84

W 74

Rhenium 186.21

Re 75

Osmium 190.23

Os 76

192.22

Ir 77

195.08

Pt 78

Gold 19 7

79

200.59

Hg 80

Thallium 204.38

Tl 81

Lead 207.2

Pb 82

208.98

Bi 83

(209)

84

Astatine (210)

At 85

Pd

Radon (222)

Rn 86

(223)

Fr 87

Radium (226)

Ra 88

Actinium (227)

Ac 89

Dubnium (262)

Db 105

Seaborgium (266)

Sg 106

(264)

Bh 107

Hassium (269)

Hs 108

(268)

Mt 109

140.12

Ce 58

Neodymium 144.24

Nd 60

Promethium (145)

Pm 61

150.36

Sm 62

Europium 151.96

Eu 63

Gadolinium 157.25

Gd 64

158.93

Tb 65

Dysprosium 162.50

Dy 66

Holmium 164.93

Ho 67

Erbium 167.26

Er 68

Thulium 168.93

Tm 69

Ytterbium 173.04

Yb 70

Lutetium 174.97

Lu 71

232.04

Th 90

Protactinium 231.04

91

238.03

U 92

Neptunium (237)

Np 93

Plutonium (244)

Pu 94

(243)

Am 95

(247)

Cm 96

(247)

Bk 97

(251)

Cf 98

Einsteinium (252)

Es 99

(257)

Fm 100

Nobelium (259)

No 102

(262)

Lr 103

Magnesium 24.31

Mg 12

9.01

Be 4

50.94

V 23

Manganese 54.94

Mn 25

*

Atomic number

Average atomic mass

Potassium Nickel

Praseodymium

Rutherfordium

Molybdenum

Mendelevium

Germanium

Key

Fluorine

Aluminum Phosphorus Chlorine

Krypton

Yttrium Technetium Palladium Silver Antimony Tellurium Te

Barium Lanthanum Tantalum Ta

Tungsten Iridium Platinum 6.9

Au Mercury Bismuth Polonium

Po

Francium Bohrium Meitnerium

Cerium Samarium Terbium

Thorium Pa

Uranium Americium Curium Berkelium Californium Fermium Lawrencium

Beryllium

Vanadium

If this number is in parentheses, then it refers to the atomic mass of the most stable isotope.

Copyright © 2003 California Department of Education

Fe3+

Fe2+iron (III)

iron (II)

26atomicnumber

ionchargeionname

symbol (IUPAC)

KEY

58 59 60 61 62 63 64 65 66 67 68 69 70 71

90 91 92 93 94 95 96 97 98 99 100 101 102 103

Ce3+

ceriumPr3+

praseodymiumNd3+

neodymiumPm3+

promethium

Sm3+samarium(III)

samarium(II)

Eu3+

Eu2+europium (III)

europium (II)

Th4+

thorium

Pa5+

Pa4+protactinium(V)

protactinium(IV)

U6+

U4+uranium (VI)

uranium (IV)

Gd3+

gadoliniumTb3+

terbiumDy3+

dysprosiumHo3+

holmiumEr3+

erbiumTm3+

thulium

Yb3+

Yb3+ytterbium(III)

ytterbium(II)Sm2+

Lu3+

lutetium

Np5+ Pu4+

Pu6+plutonium(IV)

Am3+

Am4+americium(IV)

americium(III)neptunium

Bk3+

Bk4+berkelium(IV)

berkelium(III)Cm3+

curiumCf3+

californiumEs3+

einsteiniumFm3+

fermium

Md2+

Md3+mendelevium (II)

mendelevium (III)

Lr3+

lawrencium

No2+

No3+nobelium(II)

nobelium(III)plutonium(VI)

PERIODIC TABLE OF IONSacetatearsenatearsenitebenzoateboratebromatecarbonatechloratechloridechloritechromatecyanatecyanidedichromate

CH3COO–

AsO43–

AsO33–

C6H5COO–

BO33–

BrO3 –

CO32–

ClO3 –

Cl–

ClO2 –

CrO42–

CNO–

CN–

Cr2O72–

oxalateperchlorateperiodatepermanganateperoxidephosphatepyrophosphatesulfatesulfitethiocyanatethiosulfate

ammoniumhydronium

C2O42–

ClO4 –

IO4 –

MnO4 –

O22–

PO43–

P2O74–

SO42–

SO32–

SCN–

S2O32–

NH4+

H3O+

POSITIVE POLYATOMIC IONS

TABLE OF POLYATOMIC IONS

H2PO4 –

HCO3 –

HC2O4 –

HSO4 –

HS–

HSO3 –

OH–

ClO–

IO3 –

HPO42–

NO3 –

NO2 –

SiO44–

hydrogen carbonatehydrogen oxalatehydrogen sulfatehydrogen sulfidehydrogen sulfitehydroxidehypochloriteiodate

nitratenitriteorthosilicate

monohydrogen phosphate

dihydrogen phosphate

1 2

3 4 5 6 7 8 9 10

13 14 15 16 17 1811 12

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36Ti4+titanium (IV)

Ti3+titanium (III)

V3+vanadium(III)

V5+vanadium (V)

Cr3+

Cr2+chromium (III)

chromium (II)

Mn2+

Mn4+manganese(II)

manganese(IV)

Fe3+

Fe2+iron (III)

iron (II)

Co2+

Co3+cobalt (II)

cobalt (III)

Ni2+

Ni3+nickel (II)

nickel (III)

Cu2+

Cu+copper (II)

copper (I)

Ga3+

galliumSc3+

scandium

Y3+

yttrium

La3+

lanthanum

Ac3+

actinium

Zr4+

zirconium

Hf4+

hafnium

Nb5+

Nb3+niobium (V)

niobium(III)

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

Mo6+molybdenum

Rh3+

rhodium

Ru3+

Ru4+ruthenium(III)

ruthenium(IV)

Pd2+

Pd4+paladium(II)

paladium(IV)

Ag+

silverCd2+

cadmium

Pt4+

Pt2+platinum(IV)

platinum(II)

55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86Au3+

Au+gold (III)

gold (I)

Hg2+

Hg+mercury (II)

mercury (I)

Tl+

Tl3+thallium (I)

thallium(III)

Pb2+

Pb4+lead (II)

lead (IV)

Bi3+

Bi5+bismuth(III)

bismuth(V)

Sn4+

Sn2+tin (IV)

tin (II)

Sb3+

Sb5+antimony(III)

antimony(V)

Tc7+

Ta5+

tantalumW6+

tungstenRe7+

rheniumOs4+

osmiumIr4+

iridium

87 88 89

H+hydrogen

Li+lithium

Be2+beryllium

Na+sodium

Mg2+magnesium

K+

potassiumCa2+

Rb+

rubidiumSr2+

strontium

Cs+cesium

Ba2+

calcium

barium

Fr+francium

Ra2+radium

Bboron

Ccarbon nitride

N3-oxideO2-

fluorideF-

neonNe

Al3+aluminum

Sisilicon phosphide

P3-sulfiide

S2-chloride

Cl-argonAr

heliumHe

Zn2+

zinc

In3+

indium

Ge4+

germaniumAs3-

arsenide selenideSe2-

bromideBr-

kryptonKr

tellurideTe2-

Po2+polonium(II)

polonium(IV)Po4+

iodideI-

xenonXe

astatideAt-

radonRn

hydrideH-

1

1

2

3 4 5 6 7 8 9 10 11 12

13 14 15 16

17 18

technetium

List of Common Multivalent Ions

The following elements form multivalent ions, and therefore require a

Roman numeral charge when writing the name of the compound. Rare and

synthetic elements are not included in this list.

Element Sym. Possible Charges

Element Sym. Possible Charges

Titanium Ti +2, +3, +4 Tin Sn +2, +4 Vanadium V +2, +3, +4, +5 Rhenium Re +4, +6, +7 Chromium Cr +2, +3, +6 Osmium Os +3, +4 Manganese Mn +2, +3, +4, +7 Iridium Ir +3, +4

Iron Fe +2, +3 Platinum Pt +2, +4 Cobalt Co +2, +3 Gold Au +1, +3 Nickel Ni +2, +3 Mercury Hg +1, +2 Copper Cu +1, +2 Thallium Tl +1, +3

Niobium Nb +2, +5 Lead Pb +2, +4 Molybdenum Mo +3, +6 Bismuth Bi +3, +5

Palladium Pd +2, +4 Polonium Po +2, +4

Steps to Determine Charge from the Chemical Formula

1. Find total negative charge on all anions.

2. Divide value by number of cations to give charge on one multivalent

cation.

Lab Safety

Safety is essential in a science lab, due to the serious risks that can occur from mishandled chemicals,

glassware and other material. It is vital that all students are aware of the lab’s safety equipment, in

particular where it is stored and how to use it. Additionally, students need to act in a conscientious and

cautious manner when conducting an experiment or working in the lab.

Safety Equipment

Eyewash Station

An eyewash is used to flush out one or both eyes if they have been contacted with a chemical or

abrasive substance. In serious cases, flushing should occur immediately for at least five minutes (or

more, depending on the nature of the substance) using a water faucet.

First Aid Kit

1. The first aid kit contains a number of items for dealing with minor cuts, burns and scrapes. Inform

Ms. Hayduk of any injuries incurred during a lab, regardless of how severe they are.

Fire Extinguisher

Fire extinguishers should be used for putting out small to medium sized fires that are uncontrolled.

Before using a fire extinguisher, it is important to ensure it is appropriate for the type of fire.

Class Use Symbol

A Ordinary combustible materials (e.g. paper,

wood, cardboard and most plastics) Green triangle

B Flammable or combustible liquids (e.g.

gasoline, kerosene, grease and oil) Red square

C Electrical equipment Blue circle

D Combustible metals (e.g. magnesium,

potassium and sodium) Yellow decagon

K Kitchen fires (e.g. cooking oil, trans-fats, fats) Black hexagon

An ABC fire extinguisher, or an all-purpose fire extinguisher, is the most common, because it will put

out the most common types of fire.

Spill Kit

2. A spill kit contains a number of absorbent substances that can be used to neutralize and safety clean

up chemical spills in the lab. A serious chemical spill includes substances that cannot be handled

safely (highly corrosive) or an unknown mixture of chemicals. If a serious chemical spill occurs, find

Mrs. Hayduk immediately; do not attempt to clean up the spill. Ensure that other students remain

clear of the area.

Personal Protective Equipment

3. Personal protective equipment refers to all clothing, helmets and eyewear designed to protect the

wearer from injury. Prior to any lab, Ms. Hayduk will inform you which equipment is required to be

worn during the activity. Students must wear all of this equipment for the duration of the lab, even

if they have completed the activity.

Specific Emergency Procedures

Fire

If the fire is small, use the appropriate fire extinguisher on the fire or extinguish it with a lid or

blanket.

To use a fire extinguisher, follow the acronym PASS: pull out the pin, aim the nozzle at the base of

the fire, squeeze the trigger and sweep the spray along the base of the fire until it is extinguished.

If the fire is large and cannot be controlled, leave the room immediately and pull the fire alarm. All

students should file out of the building in a calm manner to the designated meeting spot outside.

The last student out of the room should shut the door to impede spread of the fire.

If a fire alarm goes off during an experiment, students should shut off all gas and heat sources

before exiting the lab.

Spill

If the spill is a harmless substance (e.g. water, vinegar), clean it up immediately with rags or paper

towel.

If the spill is a more hazardous substance, inform Ms. Hayduk immediately. She will clean the spill

with the spill kit. If possible, block the spill from spreading and remove any books, bags or personal

items from the area.

If more than one chemical spills in the same area, tell Ms. Hayduk and evacuate the room

immediately. She will assess the danger and take steps to decontaminate the area

Injury

For minor injuries, use the first aid kit. Make sure to clean any cuts or scrapes before applying a

bandage.

For serious injuries, inform Ms. Hayduk immediately and use a cell phone or the office phone to call

911. Make sure to tell EMS if the injury was caused by contact with a chemical (or broken glass

contaminated with chemical).

Laboratory Safety Procedures

This list of safety procedures is general, and does not cover all aspects of safety in the lab. It is important

that students use common sense and caution when working in the science lab, and ask for help when

instructions or procedures are unclear.

1. Behave in a calm, professional, responsible manner at all times.

2. No food in the lab at any time.

Beverages are allowed provided they are in re-sealable containers.

Never eat any materials being used for experiments.

3. Use the appropriate personal protective equipment for the activity you (or others) are performing.

Do not remove your PPE until you are instructed to do so by the teacher.

4. Keep yourself, your equipment and your workstation clean before, during and after the lab.

Handle equipment with care.

Wash glassware thoroughly with soap and water.

After handling chemicals, wash hands thoroughly with soap and water.

Keep aisles and table tops clear of bags and books.

5. Dispose of materials properly.

Do not dump any chemicals down the drain unless instructed to by the teacher.

Do not put any solid material in the drains.

Sharp materials (e.g. dissection pins, broken glass) should be disposed of in the proper

waste container – never in the garbage can.

6. Do not touch any chemicals or equipment you have not been instructed to handle.

Do not smell or taste chemicals.

Do not try any unauthorized experiments.

Do not enter the science storage room.

7. Never leave your lab station unattended.

8. Dress appropriately.

Tie back long hair.

Avoid wearing loose or dangling clothes or jewelry around chemicals or open flames.

Wear closed-toed shoes.

9. Report any accident or incident immediately.

Students who do not follow these safety procedures will be asked to leave the lab, and will be given an

alternate written assignment to complete.

WHMIS

WHMIS stands for Workplace Hazardous Material Information System. It is a program designed to protect

workers (e.g. students and teachers) who are handling chemicals on a regular basis. There are three key

elements to WHMIS:

1. Labels,

2. Material safety data sheets; and,

3. Education and training.

As a student, your primary concern will be with labels and material safety data sheets (MSDS).

Labels

The purpose of a WHMIS label is to identify the product as controlled and alert the user to the hazards and

safe handling procedures of the product. The label provides a summary of the important information about

the substance, including:

1. Name of chemical and common name

2. Hazard symbols

3. Risks associate with the substance and precautions to take when using it

4. First aid measures

5. Supplier information

6. Reference to MSDS, stating that more information is available

Chemicals that are going to be stored in beakers, flasks or test tubes should be labelled with a name, date

and some identifier of what is contained in it. If you are doing work in the lab and are planning to store

chemicals, you must ensure this is done.

Hazard Symbols

Class Controlled Product Dangers

A

Compressed Gas

It is a gas kept under pressure

Heat may cause the container to explode

Drop or impact may cause the container to explode

Compressed gases can be dangerous because the contents are

under pressure, but also because the gas may be poisonous

B

Flammable and Combustible Material

Material may burn at a low temperature, and may catch on fire

from sparks, flame or friction

May burst into flame in air or release a flammable gas if it comes

into contact with water

C

Oxidizing Material

Substance is a fire or explosion risk near flammable or

combustible material

May not burn itself, but could release oxygen or other gases that

might make other materials burn more easily

D

Poisonous and Infectious Material

Material is a potentially fatal poisonous substance that could kill

or cause permanent damage if inhaled, swallowed or absorbed

through skin

Division 1: Material Causing Immediate and Serious Toxic Effects

Immediately dangerous to life and health

Division 2: Materials Causing Other Toxic Effects

Material is poisonous but not immediately dangerous

May cause death or permanent damage as a result of repeated

exposure over time

May cause irritation

Division 3: Biohazardous Infectious Material

An organism and the toxins they produce that may cause disease

E

Corrosive Material Acidic or caustic materials that can eat through skin or corrode

metals

F

Dangerously Reactive Material

Substance could undergo dangerous reactions when subject to

heat, pressure, shock or contact with other materials

International Hazard Symbols

Not all products and substances are controlled by WHMIS, so they may not have WHMIS labels or

symbols. These are other symbols you may see on other household products. The border of the

symbol represents the level of danger, and the symbol inside represents the specific hazard.

“Danger”, shown with an octagon, is the biggest threat. “Caution”, given by an upside down triangle,

is a smaller threat but should still be considered dangerous!

Danger Warning Caution

Poison

Flammable

Explosive

Corrosive

Material Safety Data Sheets

Material safety data sheets (MSDS) are used to give more detailed information about the product

than the information on the WHMIS label. The information includes:

A description of the chemical

Hazardous ingredients

Physical and chemical data

Fire and explosive hazard (how easily it will catch on fire or explode)

Reactivity data

Toxicological properties (how toxic it is to humans and other animals)

Preventative measures to take when handling and storing it

First aid measures to be taken if exposed

When it was made and who to contact for more information

MSDS are available online, as well as in a binder in the school. They are all current within three

years for all of the chemicals available in the school. It is expected that all students will review the

MSDS for important chemicals prior to every lab so they know the safety hazards and the personal

protective equipment they should wear.

Experimental Error and Measurements

Accuracy and Precision

Experimental error is the difference between a measurement and the true value, or between two

measured values. This is measured by accuracy and precision.

Accuracy tells how close a measured value is to the theoretical, accepted or “true” value. If this

value is unknown, then the accuracy of a measurement cannot be determined.

Precision measures how close two or more measurements are to each other. Precision can also be

called “repeatability” or “reproducibility”. A measurement that is highly precise, or highly

reproducible, will give values that are very close.

Experimental Error

Human errors are avoidable mistakes that come from mistakes, blunders or miscalculations. These

are not considered sources of experimental error, since they can be eliminated by careful technique

and by repeating the procedure correctly. Examples include spilling chemicals, using the wrong

calculation or missing a step in the procedure.

Systematic errors, or bias, are a type of experimental error that affect accuracy of a measurement.

They are generally caused by flaws in equipment or in reading measurements, and are generally

difficult to detect. These errors are “one-sided”, meaning they will usually give results that are close

to each other, but are not close to the true value. Systematic errors can be caused by poor

calibration of measuring instruments, poorly maintained or poorly constructed instruments and

faulty reading of measurements by the user (e.g. reading volume at an angle).

Random errors affect the precision of a measurement. These errors are “two-sided” because they

can fluctuate above or below the true value in repeated trials. Random errors can be reduced by

repeating the procedure and taking average values or by using better quality instruments. Common

sources of random errors include estimating a measurement that is between graduations on an

instrument or recording a value that fluctuates during the reading.

Sample Sources of Error

Limited accuracy of measuring tools (how many decimal places the measurement can reasonably have)

Accidental spills or residue from pouring from one container to another

Contaminants on equipment or in chemicals Lots of measurements that increase uncertainty in calculations

Measuring tools not calibrated properly Not reading volumes and temperatures directly at eye-level

Impurities in chemicals Reaction does not go to completion

Not enough trials or data Volumes are too small to read easily

Calculating Experimental Error

Accuracy of Equipment

All measuring equipment has precision depending on the smallest unit of measurement on the

instrument. Unless otherwise stated, the precision of a measurement can stated be up to 1/2 of the

smallest unit, and the precision of that measurement is ±0.5 of the unit. For example, a ruler that

has graduations of 1 mm can be used to measure a line that is 12.5 mm long, and that length is

precise to ±0.5 mm. This is stated as 12.5 ±0.5 mm, which means the true value of the length is

between 12 and 13 mm.

Digital equipment and chemistry glassware generally has a specified accuracy that is written on the

instrument. With glassware, some equipment is deliberately very precise (e.g. volumetric flasks),

while others are not really intended to be used for measurements (e.g. beakers).

Percent Error

Percent error is the difference in accuracy between a measured, or experimental, value and the

true, or accepted, value. It is calculated as follows:

% 𝐸𝑟𝑟𝑜𝑟 = |𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 − 𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑|

𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑× 100

The vertical lines on the top of the equation indicates absolute value, which means that negative

signs are ignored. If you get a negative answer, record the value without the negative sign.

Percent error can only be used if the true value is known, or can be calculated.

Percent Difference

Percent difference is used in place of percent error when the true value is unknown. It is used to

find the precision of repeated measurements (experimental values, E). The equation used is:

% 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = |𝐸1 − 𝐸2|

(𝐸1 + 𝐸2

2 )× 100

A Short Guide to Significant Figures

What is a “significant figure”?

The number of significant figures in a result is simply the number of figures that are known with somedegree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said tohave 4 significant figures.

Rules for deciding the number of significant figures in a measured quantity:

(1) All nonzero digits are significant:

1.234 g has 4 significant figures,1.2 g has 2 significant figures.

(2) Zeroes between nonzero digits are significant:

1002 kg has 4 significant figures,3.07 mL has 3 significant figures.

(3) Zeroes to the left of the first nonzero digits are not significant; such zeroes merely indicate theposition of the decimal point:

0.001o C has only 1 significant figure,0.012 g has 2 significant figures.

(4) Zeroes to the right of a decimal point in a number are significant:

0.023 mL has 2 significant figures,0.200 g has 3 significant figures.

(5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are notnecessarily significant:

190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. Thepotential ambiguity in the last rule can be avoided by the use of standard exponential, or ”scientific,”notation. For example, depending on whether 3, 4, or 5 significant figures is correct, we could write50,6000 calories as:

5.06× 104 calories (3 significant figures)5.060× 104 calories (4 significant figures), or5.0600× 104 calories (5 significant figures).

What is a ”exact number”?

Some numbers are exact because they are known with complete certainty.

Most exact numbers are integers: exactly 12 inches are in a foot, there might be exactly 23 studentsin a class. Exact numbers are often found as conversion factors or as counts of objects.

Exact numbers can be considered to have an infinite number of significant figures. Thus, number ofapparent significant figures in any exact number can be ignored as a limiting factor in determining thenumber of significant figures in the result of a calculation.

Rules for mathematical operations

In carrying out calculations, the general rule is that the accuracy of a calculated result is limited bythe least accurate measurement involved in the calculation.

1

(1) In addition and subtraction, the result is rounded off to the last common digit occurring furthest tothe right in all components. For example, 100 (assume 3 significant figures) + 23.643 (5 significantfigures) = 123.643, which should be rounded to 124 (3 significant figures).

(2) In multiplication and division, the result should be rounded off so as to have the same number ofsignificant figures as in the component with the least number of significant figures. For example, 3.0(2 significant figures ) 12.60 (4 significant figures) = 37.8000 which should be rounded off to 38 (2significant figures).

Rules for rounding off numbers

(1) If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example,

12.6 is rounded to 13.

(2) If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example,

12.4 is rounded to 12.

(3) If the digit to be dropped is 5, and if any digit following it is not zero, the last remaining digit isincreased by one. For example,

12.51 is rounded to 13.

(4) If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit is increased byone if it is odd, but left as it is if even. For example,

11.5 is rounded to 12,12.5 is rounded to 12.

This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is alwaysrounded to the even digit. The rationale is to avoid bias in rounding: half of the time we round up,half the time we round down.

General guidelines for using calculators

When using a calculator, if you work the entirety of a long calculation without writing down anyintermediate results, you may not be able to tell if a error is made and, even if you realize that one hasoccurred, you may not be able to tell where the error is.

In a long calculation involving mixed operations, carry as many digits as possible through the entireset of calculations and then round the final result appropriately. For example,

(5.00 / 1.235) + 3.000 + (6.35 / 4.0)=4.04858... + 3.000 + 1.5875=8.630829...

The first division should result in 3 significant figures; the last division should result in 2 significantfigures; the three numbers added together should result in a number that is rounded off to the lastcommon significant digit occurring furthest to the right (which in this case means the final result shouldbe rounded with 1 digit after the decimal). The correct rounded final result should be 8.6. This finalresult has been limited by the accuracy in the last division.

Warning: carrying all digits through to the final result before rounding is critical for many math-ematical operations in statistics. Rounding intermediate results when calculating sums of squares canseriously compromise the accuracy of the result.

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Significant Digits Flowchart

Unit Conversions and Scientific Notation

Metric Unit Conversions

To convert between metric units in the same “category”, use the staircase method:

Tera Giga Mega Kilo Hecto Deca BASE Deci Centi Milli Micro Nano Pico

T G M k h da d c m µ n p

1012 109 106 1000 100 10 1 0.1 0.01 0.001 10-6 10-9 10-12

To move to a smaller unit, multiply by the number of steps it takes to get to the right prefix.

For example, 5 km = (5×10000) cm = 50000 cm, since it takes five steps to get from kilo to centi.

To move to a larger unit, divide by the number of steps it takes to get to the right prefix. For example, 30 mg = (30÷1000) g = 0.03 g, since it takes three steps to get from milli to “base”.

Scientific Notation

Scientific notation is used for very large or very small numbers, which are common in science! It is written

as a product (multiplication) between a number between 1 and 10 and a power of 10.

3.45 × 105 is a big number. It has a positive exponent.

2.31 × 10-4 is a small number. It has a negative exponent.

Many scientific calculators have a button that allows the user to enter numbers in scientific notation. It may

be “EXP”, “×10x”, “EE” or “y1/x”. It is important that students learn to use this function on their own

calculator, to make calculations with scientific notation easier.

To convert to scientific notation:

1. Create a number between 1 and 10 by moving the decimal (for whole numbers, it is after the last

digit) to the left. There should be only one non-zero number before the decimal.

2. Count the number of spaces the decimal moves to determine the exponent on the 10.

a. If the decimal moves left, the exponent is positive.

b. If the decimal moves right, the exponent is negative.

Example: 3 346 000 000 = 3.346 × 109

The decimal moved from the right of the last zero nine places to the left, which gives the exponent

of 9. This number has four significant digits, but more could be added by putting zeros or

removed by rounding (e.g. 3.346 to 3.35).

To convert to standard form:

1. Multiply the two terms (the decimal between 1 and 10 and the power of 10) together; to ensure you

are multiplying or dividing properly. Ask, “Is it reasonable?” after doing a calculation. If you get an

answer of 1000 km in 1 m, and you know kilometres are much bigger than metres, you should

recognize that the answer does not make sense and there was an error in the calculation.

Comparing Values of Scientific Notation

Numbers with higher exponents on the 10 are greater:

10 > 3

4 > -1

-2 > -5

For numbers with the same exponent, numbers with a larger decimal value are greater:

6.43 × 105 > 2.17 × 105

3 × 10-2 > 1 × 10-2

Factor-Label Method

The factor-label method, also called dimensional analysis, is a problem solving method used for unit

conversions and stoichiometric calculations. It is based on the idea of “cancelling” units to get the desired

result. To get from one unit to another, conversion factors are used to change the units to the correct ones.

Example: How many inches are in 32 km?

The unit conversions here are:

1.6 km = 1 mile 1 mile = 5280 ft 1 ft = 12 inches

A grid is used to show the work – horizontal lines indicate “divide” (like in a fraction) and vertical lines

indicate “multiply” (multiplication signs can also be used).

32 km 1 mile 5280 ft 12 inches 1.6 km 1 mile 1 ft

Units that are on both the top and the bottom can be crossed out, like this:

32 km 1 mile 5280 ft 12 inches 1.6 km 1 mile 1 ft

This leaves the answer in inches. To get the value, multiply the numbers on the top together, and divide by

the numbers on the bottom:

= (32 × 1 × 5280 × 12) ÷ (1.6 × 1 × 1)

= 2027520 ÷ 1.6

= 1 267 200 inches

Remember to ALWAYS include units in the final answer!!

Common Unit Conversions

Length 1 in = 2.54 cm 3.28 ft = 1 m

1 mile = 1.61 km

1 mile = 5280 ft 1 ft = 12 in

Mass 1 oz = 28.3 g 1 kg = 2.20 lb

1 lb = 16 oz

Volume 1 fl oz = 29.6 mL 1 cup = 237 mL

1 tbsp = 3 tsp 1 cup = 16 tbsp = 8 fl oz

1 US pint = 2 cups 1 US quart = 4 cups

1 US gallon = 16 cups

Temperature °𝐶 = (°𝐹 − 32) ×5

9 °𝐹 =

9

5× °𝐶 + 32