Problem Solving – a Math Problem Solving – a Math ReviewReview

Unit 1Unit 1

Significant Figures, Scientific Significant Figures, Scientific Notation & Dimensional AnalysisNotation & Dimensional Analysis

Significant FiguresSignificant Figures

In science, we describe a value as having a In science, we describe a value as having a certain number of significant figures or certain number of significant figures or digits.digits.– Includes all the #’s that are certain and 1 Includes all the #’s that are certain and 1

uncertain digit (the LAST one).uncertain digit (the LAST one).

– There are rules that dictate which #’s are There are rules that dictate which #’s are considered significant!considered significant!

Rules for Significant FiguresRules for Significant Figures

Any non-zero # is considered significantAny non-zero # is considered significant Zeroes!Zeroes!

– Any Any zeroes between 2 numbers zeroes between 2 numbers is significantis significant Ex. 205 has 3 sig. figs. Ex. 205 has 3 sig. figs. Ex. 4060033 has 7 sig. figs.Ex. 4060033 has 7 sig. figs. Ex. 10.007 has 5 sig. figs.Ex. 10.007 has 5 sig. figs.

– Any Any zeroes before a number zeroes before a number are NOT are NOT significantsignificant Ex. 0.054 has 2 sig. figs. Ex. 0.054 has 2 sig. figs. Ex. 0.000 005 has 1 sig. fig.Ex. 0.000 005 has 1 sig. fig.

Rules for Significant FiguresRules for Significant Figures

Zeroes! ContinuedZeroes! Continued– Any zeroes after numbers may or may not be Any zeroes after numbers may or may not be

significant.significant. If there is If there is a decimal point in the number, then YES, they are

significant!– Ex. 12.000 has 5 sig. figs.Ex. 12.000 has 5 sig. figs.

– Ex. 0.1200 has 4 sig. figs.Ex. 0.1200 has 4 sig. figs.

– Ex. 530.0000 has 7 sig. figs.Ex. 530.0000 has 7 sig. figs.

If there is If there is no decimal point in the number, then NO, they aren’t no decimal point in the number, then NO, they aren’t significantsignificant!!

– Ex. 120 has 2 sig. figs.Ex. 120 has 2 sig. figs.

– Ex. 430 000 000 000 has 2 sig. figs.Ex. 430 000 000 000 has 2 sig. figs.

Adding/ Subtracting and Significant Adding/ Subtracting and Significant FiguresFigures

The ruleThe rule– When adding or subtracting When adding or subtracting

Look at the Significant Figures AFTER the decimal Look at the Significant Figures AFTER the decimal point. Which one has the least amount? That’s how point. Which one has the least amount? That’s how many significant figures your answer can havemany significant figures your answer can have

ExamplesExamples

17.34 + 4.900 + 23.1 = 45.34 17.34 + 4.900 + 23.1 = 45.34

(1 sig. fig after decimal) (1 sig. fig after decimal) = 45.3= 45.3

9.80 – 4.782 = 5.3189.80 – 4.782 = 5.318

(2 sig. figs. After decimal) (2 sig. figs. After decimal) = 5.32= 5.32

Multiplying/ Dividing and Significant Multiplying/ Dividing and Significant FiguresFigures

The ruleThe rule– When multiplying or dividing, check out how When multiplying or dividing, check out how

many significant figures (all of them) each many significant figures (all of them) each number has. Which one has the least amount? number has. Which one has the least amount? That’s how many significant figures your answer That’s how many significant figures your answer can have.can have.

ExamplesExamples

3.9 3.9 × 6.05 × 420 = 9909.9× 6.05 × 420 = 9909.9

(2 sig. figs total) (2 sig. figs total) = 9900= 9900

= 9.9 × 10= 9.9 × 1033

14.2 ÷ 5 = 2.8214.2 ÷ 5 = 2.82

(1 sig. fig total) (1 sig. fig total) = 3= 3

Scientific NotationScientific Notation

Do you know this number?Do you know this number?

– 300 000 000 m/s300 000 000 m/s– It’s the speed of light.It’s the speed of light.

Do you know this number?Do you know this number?

– 0.000 000 000 752kg0.000 000 000 752kg– It’s the mass of a dust particle.It’s the mass of a dust particle.

Scientific NotationScientific Notation

Instead of counting zeroes and getting confused, Instead of counting zeroes and getting confused, we use scientific notation to write really big or we use scientific notation to write really big or small numbers.small numbers.– 3.00 3.00 × 10× 108 8 m/sm/s– 7.53 × 107.53 × 10-10 -10 kgkg

– The 1The 1stst number is the COEFFICIENT- it is always a number is the COEFFICIENT- it is always a number between 1 and 10.number between 1 and 10.

– The 2The 2ndnd number is the BASE- it is the number 10 raised number is the BASE- it is the number 10 raised to a power, the power being the number of decimal to a power, the power being the number of decimal places moved.places moved.

Using a calculator with scientific Using a calculator with scientific notationnotation

A number written in scientific notation is A number written in scientific notation is NOT a math problem, it is a number in its NOT a math problem, it is a number in its own right. We put it into the calculator in a own right. We put it into the calculator in a specific way!specific way!

IF you have a scientific calculator, find the IF you have a scientific calculator, find the button that says EE or EXP.button that says EE or EXP.

Scientific CalculatorsScientific Calculators

Scientific CalculatorsScientific Calculators

The EE or EXP button fills in for the The EE or EXP button fills in for the × 10 × 10 part of part of the number written in scientific notation.the number written in scientific notation.

Let’s say you are adding these two numbersLet’s say you are adding these two numbers3.21 × 103.21 × 1077 + 6.99 × 10 + 6.99 × 1066 = =This is how you would enter it into your calculatorThis is how you would enter it into your calculator3.21 EE 7 + 6.99 EE 6 =3.21 EE 7 + 6.99 EE 6 =And you would get your answer.And you would get your answer.3.91 × 103.91 × 1077

Scientists generally work in metric units. Scientists generally work in metric units. Common prefixes used are the following:Common prefixes used are the following:

Dimensional AnalysisDimensional Analysis

is a problem-solving method that uses the is a problem-solving method that uses the fact that any number or expression can be fact that any number or expression can be multiplied by one without changing its value. multiplied by one without changing its value. It is a useful technique. The only danger is It is a useful technique. The only danger is that you may end up thinking that chemistry that you may end up thinking that chemistry is simply a math problem - which it definitely is simply a math problem - which it definitely is not. is not.

Dimensional AnalysisDimensional Analysis

Unit factors may be made from any two Unit factors may be made from any two terms that describe the same or equivalent terms that describe the same or equivalent "amounts" of what we are interested in. For "amounts" of what we are interested in. For example, we know thatexample, we know that

1 inch = 2.54 centimeters1 inch = 2.54 centimeters

We can make two unit factors from this We can make two unit factors from this information: information:

Now, we can solve some problems. Set up each Now, we can solve some problems. Set up each problem by writing down what you need to find problem by writing down what you need to find with a question mark. Then set it equal to the with a question mark. Then set it equal to the information that you are given. The problem is information that you are given. The problem is solved by multiplying the given data and its units solved by multiplying the given data and its units by the appropriate unit factors so that only the by the appropriate unit factors so that only the desired units are present at the end. desired units are present at the end.

(1) How many centimeters are in 6.00 (1) How many centimeters are in 6.00 inches? inches?

(2) Express 24.0 cm in inches. (2) Express 24.0 cm in inches.

You can also string many unit factors You can also string many unit factors together. together.

(3) How many seconds are in 2.0 years?(3) How many seconds are in 2.0 years?

Density- What is it?Density- What is it?

Density is the ratio of mass to volume of a Density is the ratio of mass to volume of a substance. substance. – It can be used to identify a substance.It can be used to identify a substance.

– Ex. Water has a density of 1.00 g/mLEx. Water has a density of 1.00 g/mL– Ex. Gold has a density of 19.30 g/mLEx. Gold has a density of 19.30 g/mL– Ex. Pumice has a density of 0.65 g/mLEx. Pumice has a density of 0.65 g/mL

Density & TemperatureDensity & Temperature

Density = mass/ volumeDensity = mass/ volume

d = m/Vd = m/V

Temperature = measure of the average kinetic Temperature = measure of the average kinetic energy a substance hasenergy a substance has– 3 scales3 scales

Fahrenheit (Fahrenheit (°F)°F) Celsius (Celsius (°C)°C) Kelvin (K)Kelvin (K)

Temperature Scale ConversionsTemperature Scale Conversions

From From °C to °F°C to °F TT°F°F = 1.8(T = 1.8(T°C°C) + 32°) + 32°

From °F to °CFrom °F to °C TT°C°C = .56(T = .56(T°F°F - 32°) - 32°)

From °C to KFrom °C to K T = T + 273T = T + 273

From K to °CFrom K to °C TT°C°C = T = TKK - 273 - 273

There are 3 There are 3 temperature scales:temperature scales:

– Fahrenheit (°F)Fahrenheit (°F)– Celsius (°C)Celsius (°C)– Kelvin (K)Kelvin (K)