mathematics the language of science
DESCRIPTION
MATHEMATICS THE LANGUAGE OF SCIENCE. SIGNIFICANT FIGURES. Defined as all of the digits that can be read directly from the instrument used in making the measurement plus one uncertain digit that is obtained by estimating the fraction of the smallest division of the instrument’s scale. - PowerPoint PPT PresentationTRANSCRIPT
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MATHEMATICS THE LANGUAGE OF SCIENCE
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SIGNIFICANT FIGURES
• Defined as all of the digits that can be read directly from the instrument used in making the measurement plus one uncertain digit that is obtained by estimating the fraction of the smallest division of the instrument’s scale
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Significant figures
• Rules: – NON-ZERO digits
• 1, 2, 3, 4, 5, 6, 7. & 9 are always significant.
– DIGIT ZERO• Zero may or may not be significant, depending on
whether they mark the decimal point or indicate a measured value.
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Significant figures
• The DIGIT zeros– Leading zeros:
• Zeros located at the beginning of a number are NEVER significant. They merely locate the decimal point.
– Ex. 0.0254 – 3 significant numbers (2, 5, 4)
– Confines zeros• Zeros located between non-zero digits are
ALWAYS significant. – Ex. 104. 6 m – 4 significant numbers (1, 0, 4, 6)
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Significant figures
• The digit zero– Trailing zeros:
• Zeros located at the end of a number are significant only if the number has an explicitly shown decimal.
– Ex. 2705.00 – 6 significant numbers ( 2,7,0, 5,0,0)
• In whole numbers without a decimal point that end in one or more zeros, the zeros may not be significant
– Ex. 5000 – 1 significant number (5)
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Significant figures
• The digit zeros– Trailing zeros
• Numbers expressed in scientific notation with zero, follows the rule in decimal number.
– Ex. 5.0 X 105 – 2 significant numbers (5, 0)– 2.30 X 10 -8 – 3 significant numbers (2, 3, 0)
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HOW MANY SIGNIFICANT DIGITS ARE THERE?
1.25.25
2.200.5
3.0.0025
4.0.0250
5.300
6.1.48
7.800 x 10-4
8.0.4904
9.980476
10. 6739.30 x 10-5
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Rounding off numbers
1. If the next digit after the last significant figure is 5 or greater, round up: Increase the last significant figure by 1.
ex. 2.136 become 2.14 rounded to 3 significant figures
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Rounding off numbers
2. If the next digit after the last significant figure is less than 5, round down: do not change the last significant figure.
ex. 2.132 become 2.13 rounded to 3 significant figures
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Round off the following numbers to the nearest 10th
1.2469.4508
2.1.805
3.4. 3849
4.487.554
5.89320.444
6. 13.873
7. 3245.8739
8. 45.135
9. 499.502
10. 4.0009
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ACCURACY & PRECISION
• Reasons why the measurement or physical quantity is always subject to some degrees of uncertainty:
1.The limitations inherent in the construction of the measuring instrument.
2.The conditions under which the measurement is made.
3.The different ways in which the person uses or read the measuring instrument.
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ACCURACY
• Refers to the closeness of a measurement to the accepted value for a specific physical quantity. It is expressed as either an absolute error or a relative error.
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ACCURACY
I. Absolute error (Ea) is the actual difference between the measured value and the accepted value.
Ea = I O – A I
Ea = absolute error
O = observed or measured value
A = accepted value
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ACCURACY
II. Relative error (Er) is often called percentage error
Ea
Er = ------- X 100%
A
where: Er = relative error
Ea = absolute error
A = accepted value
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PRECISION
• Is the agreement among several measurements that have been made in the same way. It tells how reproducible the measurements are and is expressed in terms of deviation.
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PRECISION
I. Absolute deviation (Da) is the difference between a single measured value and the average of several measurements made in the same way.
Da = absolute deviation
O = observed value
M = mean average of several measurements
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PRECISION
II. Relative deviation (Dr) is the percentage average deviation of a set of measurements
Da (average)
Dr = ------------------ X 100%
M Where: Dr = relative deviation
Da (average)= the average absolute deviation of a set of measurements
M = mean or average of several readings
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EXERCISE
• Accepted value = 13.20 g
Trial Mass Absolute error (Ea)
Absolute Deviation
(Da)
1 13.26 0.06 0.13
2 13.18 0.02 0.05
3 12.95 0.25 0.18
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COMPUTATION
• ABSOLUTE ERROR = l O – A l– TRIAL 1: l 13.26 – 13.20 l = 0.06– TRIAL 2: l 13.18 – 13.20l = 0.02– TRIAL 3: l 12.95 – 13.20 l = 0.25
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Computation
• ABSOLUTE DEVIATION (Da) = l O – M l– M = (13.26 + 13.18 + 12.95)÷3 = 13.13– Trial 1: l 13.26 – 13.13 l = 0.13– Trial 2: l 13.18 – 13.13 l = 0.05– Trial 3: l 12.95 – 13.13 l = 0.18
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EXERCISE
Based on the given on slide number 19, COMPUTE THE FOLLOWING:
1.Er of trial 1 and 3
Er of trial 1 = (0.06 ÷ 13.20) X 100% = 0.45%
Er of trial 3 = (0.25 ÷ 13.20) X 100% = 1.89%
1.Average of Mass: Answer = 13.13
2.Dr of the data: [Da (average) ÷ M] X 100%
{[( 0.13 + 0.05 + 0.18) ÷3] ÷ 13.13} X 100%
= 0.91%
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FUNDAMENTAL AND DERIVED QUANTITIES/UNITS
1. FUNDAMENTAL QUANTITIES / UNITS– The simplest quantities and units that are
convenient to use as the basis for explaining or defining other quantities and units
– 7 fundamental quantities and units (metric)
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7 FUNDAMENTAL QUANTITIES / UNITS (METRIC)
QUANTITY UNIT SYMBOL
Length Meter m
Mass Kilogram kg
Time Second s
Temperature Kelvin K
Electric current Ampere A
Amount of substance Mole mol
Luminous intensity Candela cd
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Derived quantities / units
• Are quantities and units defined in terms of the fundamental quantities and units are said to be derived quantities.
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QUANTITY UNIT SYMBOLVolume 1Cubic meter or
1000 litersm3, L
Density Kilogram per cubic meter
kg/m3
Speed or velocity Meter per second m/s
Concentration Moles per cubic meter
mol/m3
Force Newton N (kg m/s2)
Energy Joule J (kg m2/s2)
Power Watt W (J/s2)
Quantity of electricity
Coulomb Coul (A.s)
Electric potential Volt V (W/A)
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SYSTEM OF UNITS
• current: International System of Units (SI)– To standardize and simplify measurements
and promote advances in science and technology
• Metric system – 2 system of untis:– MKS (meter, kilogram & second)– CGS (centimeter, gram & second)
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PREFIX SYMBOL MEANING
Nano n 10-9 (billionth)
Micro u 10-6 (millionth)
Milli m 10-3 (thousandth)
Centi c 10-2 (hundredth)
Deci d 10-1 (tenth)
Deka dk 101 (ten)
Hecto h 102 (hundred)
Kilo k 103 (thousand)
Mega M 106 (million)
Giga G 109 (billion)
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CONVERSION OF UNITS
1. Conversion factors• Characteristics of conversion factors:
• Ratios that specify how units are related to each other
• Derived from equations that relate units 1 minute = 60 seconds
• Come in pairs, one member of one pair being the reciprocal of the other 1 min/60 sec & 60 sec/1 min
Given quantity X conversion factor = desired quantity
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CONVERSION OF UNITS
• 5 inches convert to centimeter
• 1 inch = 2.54 cm
• 1 inch or 2.54 cm
2.54 cm 1 inch
5 inches X 2.54 cm = 12.70 cm
1 inch
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CONVERSION OF UNITS
• Example
15 cm convert to kilometer
15 cm X 1 m X 1 km = 1.5 x 10-4 km
10-2 cm 103 m
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Some useful equivalent units
LENGTH EQUIVALENT
1 inch 2.54 cm
1 foot 30.48 cm
1 yard 0.9188 m
1 mile 1.609 km
1 meter 39.37 in
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Some useful equivalent units
VOLUME EQUIVALENT
1 quart 0.946 l
1 gallon 3.785 l
MASS EQUIVALENT
1 oz 28.35 g
1 lb 453.6 g
1 kg 2.2 lb
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EXERCISE
a. Convert 1.25 km to cm = 125,000
b. How many liters are there in exactly 25 m3 ANS: 25000 LITERS (conversion factor is 1000 liters = 1 cubic meter)
c. Convert 2 yards to mm
d. Express 5 ft and 3 inch in cm
e. Convert 130 lbs to kg
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CONVERSION OF UNITS
2. FOR TEMPERATURE
• The SI unit of temperature is the Kelvin (K). However, thermometer is never marked with the kelvin scale.
• To convert Celsius to Fahrenheit
Tf = 1.8 Tc + 32°F
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CONVERSION OF UNITS
• To convert from Fahrenheit to Celsius
Tc = Tf – 32 / 1.8• To convert to Kelvin
TK = Tc + 273.15 K
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EXERCISE
a. Normal body temperature is 37° C, convert it to °F
b. Nitrogen boils at -196 °C. What is this temperature in Kelvin scale?