Prerequisite

Science Skills

Dr. Tenery

• A measurement is a number with a unit attached.

• It is not possible to make exact measurements, thus

all measurements have uncertainty.

• We will generally use metric system units. These

include:

– The meter, m, for length measurements

– The gram, g, for mass measurements

– The liter, L, for volume measurements

PSS.1 Measurements

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• Let’s measure the length of a candy cane.

• Ruler A has 1 cm divisions, so we can estimate the

length to ± 0.1 cm. The length is 4.2 ± 0.1 cm.

• Ruler B has 0.1 cm divisions, so we can estimate

the length to ± 0.05 cm. The length is 4.25 ± 0.05

cm.

Length Measurements

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• Ruler A: 4.2 ± 0.1 cm; Ruler B: 4.25 ± 0.05 cm.

• Ruler A has more uncertainty than Ruler B.

• Ruler B gives a more precise measurement.

What does this mean?

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• The mass of an object

is a measure of the

amount of matter it

possesses.

• Mass is measured

with a balance and is

not affected by gravity.

• Mass and weight are

not interchangeable.

Mass Measurements Balances

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• Mass and weight are not the same.

– Weight is the force exerted by gravity on an

object.

Mass vs. Weight

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Volume Measurements

• Volume is the amount of space occupied by a solid,

a liquid, or a gas.

• There are several instruments for measuring

volume, including:

- Graduated cylinder

- Syringe

- Buret

- Pipet

- Volumetric flask

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PSS.2 Significant Digits

The term significant digit (significant figures) refers to

the number of digits reported for the value of measured

or calculate quantity, indicating precision of the value.

Rules:

1. All digits are significant.

i.e. 9.12 cm 3 s.f.

2. Leading zeros do not count as significant.

- zeros at the beginning of the number

i.e. 0.912 cm, 0.00912 cm, 0.0000912 cm 3 s.f. 8

Rules con’t:

2. Terminal zeros

- zeros at the end of the number

- terminal zeros at the right or left of the decimal point

are significant.

- terminal zeros w/o the decimal point are not significant

i.e. 9.00 cm, 9.10 cm, and 90.0 cm 3 s.f.

i.e. 900 cm 1 s.f. vs. 900. cm 3 s.f.

3. Captive zeros

- zeros b/t numbers and they are always significant.

i.e. 1.008 cm 4 s.f.

1.012 cm 4 s.f 9

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EXAMPLE EXERCISE PSS.2 Significant Digits

State the number of significant digits in the following

measurements:

(a) 12,345 cm (b) 0.123 g

(c) 0.5 mL (d) 102.0 s

State the number of significant digits in the following

measurements:

(a) 2005 cm (b) 25.000 g

(c) 25.0 mL (d) 0.25 s

Practice Exercise

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EXAMPLE EXERCISE PSS.3 Significant Digits

State the number of significant digits in the following

measurements:

(a) 0.025 cm (b)0.2050 g

(c) 25.0 mL (d)2500 s

State the number of significant digits in the

following measurements:

(a) 0.050 cm (b)0.0250 g

(c) 50.00 mL (d)1000 s

Practice Exercise

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Rounding If the digit

following the last

digit to be

retained is:

Then the last digit

should:

Examples (round

to 3 s.f.)

Greater than 5 be increased by 1 42.68 g

Less than 5 stay the same 17.32 m

Exact numbers Numbers without any uncertainty.

i.e. 9 coins in a bottle

i.e. 1 in = 2.54 cm

Significant figures rules do not apply to exact numbers.

PSS.3 Rounding Off Nonsignificant Digit

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EXAMPLE EXERCISE PSS.4 Rounding Off

Round off the following numbers to three significant

digits:

(a) 22.250 (b)0.34548

(c) 0.072038 (d)12,267

Round off the following numbers to three

significant digits:

(a) 12.514748 (b)0.6015261

(c) 192.49032 (d)14652.832

Practice Exercise

i.e. 5.44 m – 2.6103 m = 2.8297 m 2.83 m

When you are adding and subtracting numbers in a

calculation:

Report the answer with the least number of decimal places

PSS.4 Adding and Subtracting Measurements

2 dec 4 dec 2 dec

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EXAMPLE EXERCISE PSS.5 Addition/Subtraction and Rounding Off

Add or subtract the following measurements and round off your answer:

(a) 106.7 g + 0.25 g + 0.195 g (b) 35.45 mL – 30.5 mL

Add or subtract the following measurements and round off your

answer:

(a) 8.6 cm + 50.05 cm (b) 34.1 s – 0.55 s

Practice Exercise

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i.e. 2.4 g/mL x 15.82 mL = 37.968 g 38 g

When you are multiplying and dividing numbers in a

calculation:

Report the answer with the least number of s.f.

PSS.5 Multiplying and Dividing Measurements

2 s.f. 2 s.f. 4 s.f.

EXAMPLE EXERCISE PSS.6 Multiplication/Division and Rounding Off

Multiply or divide the following measurements and round off your answer:

(a) 50.5 cm 12 cm (b) 103.37 g/20.5 mL

Multiply or divide the following measurements and round off your

answer:

(a) (359 cm) (0.20 cm) (b) 73.950 g/25.5 mL

Practice Exercise

• Exponents are used to indicate that a number has been multiplied by itself.

• Exponents are written using a superscript; thus, (2)(2)(2) = 23.

• The number 3 is an exponent and indicates that the number 2 is multiplied by itself 3 times. It is read “2 to the third power” or “2 cubed”.

• (2)(2)(2) = 23 = 8

PSS.6 Exponential Numbers

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EXAMPLE EXERCISE PSS.7 Converting to Powers of 10

Express each of the following ordinary numbers as

a power of 10:

(a) 100,000 (b)0.000 000 01

Express each of the following ordinary numbers

as a power of 10:

(a) 10,000,000 (b)0.000 000 000 001

Practice Exercise

PSS.7 Scientific Notation

Numbers associated with scientific measurements are often

too large or very small. Do we have time to write all the zeros?

Let’s look at this number 1000,000,000,000,000,000,000

How do we write scientific notation?

M x 10 exponent

Number between

1 and less than 10 Negative exp means the number is less than 1

Positive exp means the number is greater than 1

1000,000,000,000,000,000,000 = 1.0 x 1021

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EXAMPLE EXERCISE PSS.8

Converting to Ordinary Numbers

Express each of the following powers of 10 as an

ordinary number:

(a) 1 104 (b)1 10–9s

Express each of the following powers of 10 as an

ordinary number:

(a) 1 1010 (b)1 10–5

Practice Exercise

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EXAMPLE EXERCISE PSS.9 Scientific Notation Express each of the following values in scientific notation:

(a) There are 26,800,000,000,000,000,000,000 helium atoms

in a one liter balloon filled with helium gas.

(b) The mass of one helium atom is 0.000 000 000 000 000

000 000 006 65 g.

Express each of the following values in ordinary numbers:

(a) The mass of one mercury atom is 3.33 10–22 g.

(b) The number of atoms in 1 mL of liquid mercury is 4.08

1022.

Practice Exercise

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Homework Assignment

Key Terms

Exercises

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 37, 39, 41

PSS Self-Test