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Using Numbers in Science

Metric System and Instruments Used for Measurement

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Parts of a Measurement

1. The value (numerical portion)

2. The unit (describes what units)

3. The name of substance being measured

EX:

1 teaspoon salt

2 liters of pop

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Scientific Measurements

• Made in metric units• Referred to as the International System or

SI units• Based on the decimal system• Related by powers of 10• Each type has a base unit• Use prefixes to refer to a unit larger or

smaller than the base by some power of 10

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kilo hecto deca Base Unit

deci centi milli micro

k h da gram (g) d c mm μ

1000 100 10 liter (L) .1 .01 .001 0.000001

103 102 101 meter (m) 10-1 10-2 10-3 10-4 10-5 10-6

Metric Unit Conversions

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Kilometer

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Meter

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Centimeter

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Review of Metric System

• Nano = 1/100,000,000 = 0.000000001 • Micro = 1/1,000,000 = 0.000001 • XXX• XXX• Milli = 1/1,000 = 0.001• Centi = 1/100 = 0.01• Deci = 1/10 = 0.1• BASE UNIT = 1• Deca = 10 = • Hecto = 100• Kilo = 1000

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Comparison Ideas

equals

equals

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SI (International System) unitsLength meter m

Mass kilogram kg

Temperature Kelvin K

Time second S

Volume Liter L

Centigrade or Celsius = ºC This is the unit we will use in lab for temperatures!

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Uncertainty in Measurement

• No matter how careful you are

• No matter how carefully you read the measuring instrument

• No measurement is perfectly accurate

• The quality of our measurements are stated in terms of accuracy and precision

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Measured numbers

-are an estimated amount

-measured to a certain number of significant figures

-a numerical value with attached units that expresses a physical quantity such as length, mass, volume, time or temperature.

Uncertainty in Measurement

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•Error: is the difference between the true value and our estimate, or measurement, of the value.

•Accuracy: is the degree of agreement between the true value and the measured value.

•Precision: is a measure of the agreement of replicate measurements

•Uncertainty: is the degree of doubt in a single measurement.

Uncertainty in Measurements

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Accuracy

= of a measurement is how close that measurement is to the true or “exact” value

EX: Standard weight = 5.00g

4.98g more accurate than 5.12 g

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Accuracy

• Also subject to the reliability of the measuring instrument

• The smaller the increments of units on the instrument, the more accurate

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Length Measurements

• Ruler A has more uncertainty and gives less precise measurements.

• Ruler B has less uncertainty and gives more precise measurements.

Measuring the length of a metal rod

Metric Rulers for Measuring Length.

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Precision

• Precision = making reproducible or repetitive measurements of the same quantity

• How fine the divisions are

• There will always be some uncertainty because of the limits in the accuracy of your instruments

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It is also possible to have an accurate

measurement without being

precise.

Imprecise andinaccurate

Precise andaccurate

Precise butinaccurate

Precision versus Accuracy

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“Accuracy is telling the truth…..

Precision is telling the same story over and over again.”

Yiding Wang

yiang@mtu.edu

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• Strive for measurements that are accurate and precise

• Measurements you perform will be used in subsequent calculations

• In scientific measurements all the digits known w/certainty, plus the one estimated digit, are known as significant figures or significant digits.

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Significant figures: is defined to be all digits in a number representing data or results that are known with certainty plus the first uncertain digit.

0 1 2 3 4 5 6 7 8 9 10 cm

5.48 cm

0 1 2 3 4 5 6 7 8 9 10 cm

5.4 cm

Significant Figures

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Significant figures or Significant digits

• ANY numbers generated by means of a measurement (length, volume, time, etc) should be expressed in the correct number of significant figures.

• This reflects how close the measured values are to the true values.

Significant Figures (digits)= reliable figures obtained by

measurement= all digits known with certainty plus

one estimated digit

Taking the measurement

• Is always some uncertainty

• Because of the limits of the instrument you are using

EXAMPLE: mm ruler

Is the length of the line between 4 and 5 cm? Yes, definitely.Is the length between 4.0 and 4.5 cm? Yes, it looks that way. But is the length 4.3 cm? Is it 4.4 cm?

• It is important to be honest when reporting a measurement, so that it does not appear to be more accurate than the equipment used to make the measurement allows.

• We can achieve this by controlling the number of digits, or significant figures, used to report the measurement.

As we improve the sensitivity of the equipment used to make a measurement,

the number of significant figures increases. Postage Scale

3 g

1 g 1 significant figure

Two-pan balance

2.53 g

0.01 g 3 significant figures

Analytical balance

2.531 g

0.001g 4 significant figures

Which numbers are Significant?

How to count them!

55.00 mm

0.003g

9000 L

5,551,213

Non-Zero integers

• Always count as significant figures

1235 has 4 significant digits

Zeros – there are 3 types

Leading zeros (place holders)

The first significant figure in a measurement is the first digit other than zero counting from left to right

0.0045g

(4 is the 1st sig. fig.)

“0.00” are place holders.

The zeros are not significant

Captive zeros

Zeros within a number at always significant – 30.0809 g

All digits are significant

Trailing zeros – at the end of numbers but to the right of the decimal point

2.00 g - has 3 sig. digits (what this means is that the measuring instrument can measure exactly to two decimal places.

100 m has 1 sig. digit

Zeros are significant if a number contains decimals

Exact Numbers

Are numbers that are not obtained by measuring

Referred to as counting numbers

EX : 12 apples, 100 people

Exact Numbers

Also arise by definition

1” = 2.54 cm or 12 in. = 1 foot

Are referred to as conversion factors that allow for the expression of a value using two different units

Rules for sig figs.:•Count the number of digits in a measurement from left to right:

•Start with the first nonzero digit•Do not count place-holder zeros.

•The rules for significant digits apply only to measurements and not to exact numbers

Sig figs is short for significant figures.

Significant Figures

Determining Significant Figures

State the number of significant figures in the following measurements:

2005 cm

25,000 g

25.0 ml

0.25 s

0.00250 mol

4

2

3

2

3

0.050 cm

0.0280 g

50.00 ml

1000 s

1000. mol

2

3

4

1

4

Rounding Numbers

• To express answer in correctly

• Only use the first number to the right of the last significant digit

Rounding

• Always carry the extra digits through to the final result

• Then round

EX:

Answer is 1.331 rounds to 1.3

OR

1.356 rounds to 1.4

Rounding off sig figs (significant figures):

Rule 1: If the first non-sig fig is less than 5, drop all non-sig fig.Rule 2: If the first sig fig is 5, or greater that 5, increase the last sig fig by 1 and drop all non-sig figs.

Round off each of the following to 3 significant figures:

12.514748 12.5 0.6015261 0.602

192.49032 192 14652.832 14,700

Significant Figures

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Measurements With a Ruler orMeter Stick – Look at it FIRST! –

Where is “0”

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Protractor for Measuring Angles

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Measuring Angles Units are degrees (º)

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Practice Time

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Using a Vernier Caliperhttp://phoenix.phys.clemson.edu/labs/cupol/vernier/

•Used to accurately determine the fraction part of the least count division.

•Length of an object, the outer diameter (OD) of a round or cylindrical object, the inner diameter (ID) of a pipe, and the depth of a hole.

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Parts of a Caliper

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• The caliper consists of a main scale engraved on a fixed ruler and an auxiliary caliper scale engraved on a movable jaw

• The movable auxiliary scale is free to slide along the length of the fixed ruler.

• The main scale is calibrated in centimeters with the smallest division in millimeters.

• The auxiliary scale has 10 divisions that cover the same distance as 9 divisions on the main scale. Therefore, the length of the auxiliary scale is 9.0 mm.

Auxillary (Venier) Scale

Main scale

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• When the caliper is closed and properly zeroed the first mark (zero) on the main scale is aligned with the first mark on the auxiliary scale.

• The last mark on the auxiliary scale will then coincide with the 9 mm-mark on the main scale.

• This is read 0.00 cm.

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• Once the caliper is positioned to make a reading, make a note of where the first mark on the auxiliary scale falls on the main scale.

• We see that the object's length is between 1.2 cm and 1.3 cm because the first auxiliary mark is between these two values on the main scale.

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• The last digit (tenths of a millimeter) is found by noting which line on the auxiliary scale coincides with a mark on the main scale.

• The last digit is 3 because the third auxiliary mark lines up with a mark on the main scale. T

• The length of the object is 1.23 cm.

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Practice Time!

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