“You can be a mathematician without a lot of science…

However, you can’t be a scientist

without math…”

T.Webb HHS

The “Grammar and Etiquette” of Scientific

Math

Part 1 - Terminology in Basic Data Part 1 - Terminology in Basic Data AnalysisAnalysis

Quantitative analysisQuantitative analysis is expressing data in is expressing data in numerical form. The data is measured within a numerical form. The data is measured within a given degree of accuracy and precision. This is an given degree of accuracy and precision. This is an objective form of data. (Quantity – “how much” objective form of data. (Quantity – “how much” answered with a number)answered with a number)

Qualitative analysisQualitative analysis involves descriptive terms involves descriptive terms that rely on the senses, such as sight, touch, that rely on the senses, such as sight, touch, sound, smell and taste. It is a subjective form of sound, smell and taste. It is a subjective form of data that may be biased by personal experiences. data that may be biased by personal experiences. (Quality – a value answered by a description)(Quality – a value answered by a description)

Applications and terms used in Applications and terms used in quantitative analysisquantitative analysis::

Relative ErrorRelative Error: the magnitude, degree or size : the magnitude, degree or size of an error; the deviation from the true or of an error; the deviation from the true or accepted value as compared to the derived accepted value as compared to the derived value (your answer). The derived value is value (your answer). The derived value is subtracted from the accepted/true value, and subtracted from the accepted/true value, and then divided by the accepted/true value; then divided by the accepted/true value;

multiplied by 100 to get the percent error.multiplied by 100 to get the percent error.

Ideally, one should strive for a high Ideally, one should strive for a high degree of BOTH accuracy and precision!degree of BOTH accuracy and precision!

PrecisionPrecision: the reproducibility of results (data and : the reproducibility of results (data and measurements). A caliper may have the precision measurements). A caliper may have the precision of measuring to 0.01 cm every time. Automated of measuring to 0.01 cm every time. Automated assembly lines have a high degree of precision –assembly lines have a high degree of precision –they can do the same thing every time.they can do the same thing every time.

AccuracyAccuracy: the correctness of a measurement to : the correctness of a measurement to the desired result. The closer the result is to the the desired result. The closer the result is to the true, standard or accepted valuetrue, standard or accepted value, the higher its , the higher its degree of accuracy.degree of accuracy.

Average or meanAverage or mean: obtained by adding : obtained by adding together all results and dividing by the sum of together all results and dividing by the sum of the number of results.the number of results.

ToleranceTolerance: the amount of accepted variation in : the amount of accepted variation in the precision and accuracy in reference to a the precision and accuracy in reference to a measuring instrument – the instrument’s measuring instrument – the instrument’s limitations. For example, the average bathroom limitations. For example, the average bathroom scale will be accurate up to 350 lb, and precise scale will be accurate up to 350 lb, and precise to 0.10 lb if used properly. It would not be to 0.10 lb if used properly. It would not be suitable if you weighed more than 350 lb, or suitable if you weighed more than 350 lb, or you wanted to get the mass of a toothpick.you wanted to get the mass of a toothpick.

Part 2 - Digital Integrity 101Part 2 - Digital Integrity 101……

aka aka Significant FiguresSignificant Figures All digits (as in numbers, not just your fingers and All digits (as in numbers, not just your fingers and

toes) count as significant when obtained from a toes) count as significant when obtained from a properly taken measurement. The last reasonably properly taken measurement. The last reasonably measured digit is “uncertain”, since we do not measured digit is “uncertain”, since we do not know the next number. The value of 56.9 cm has know the next number. The value of 56.9 cm has 3 sf, with the “9” being the uncertain digit. 3 sf, with the “9” being the uncertain digit.

For another example, we would not record a For another example, we would not record a measurement of 34.9384633 cm taken by a measurement of 34.9384633 cm taken by a common ruler – as it is unreasonable and beyond common ruler – as it is unreasonable and beyond the ruler’s limitation of measure. The last the ruler’s limitation of measure. The last reasonably measured digit would be the “3”… reasonably measured digit would be the “3”…

How to count significant digits…How to count significant digits…

Count all reasonably measured digits from 1 – 9 Count all reasonably measured digits from 1 – 9 as significant.as significant.

Exact numbersExact numbers are not uncertain, and have an are not uncertain, and have an infinite number of sig figs. These are defined infinite number of sig figs. These are defined numbers – such as 1000 m in 1 km, or 100 cm numbers – such as 1000 m in 1 km, or 100 cm in 1 m. They also include numbers of counting in 1 m. They also include numbers of counting objects that you cannot reasonably break down objects that you cannot reasonably break down further – 4 people or 23 pennies.further – 4 people or 23 pennies.

ZeroesZeroes

Do Do NOTNOT count zeroes in front of a number, count zeroes in front of a number, as they are only placeholders.as they are only placeholders.

Example: 0.0224 cm has only 3 sig figs – the 224 Example: 0.0224 cm has only 3 sig figs – the 224 part. We can convert this to 224 um and still have part. We can convert this to 224 um and still have the same value.the same value.

Do Do NOTNOT count zeroes following a number count zeroes following a number unless there is a decimal in the measured unless there is a decimal in the measured value. value.

Example: You ran 3 000 m (1 sf) – but unless you Example: You ran 3 000 m (1 sf) – but unless you measured it exactly on a track at 3 000.0 m (5 sf), measured it exactly on a track at 3 000.0 m (5 sf), you may have gone more or less than 3 000 m…you may have gone more or less than 3 000 m…

Zeroes between numbers count as Zeroes between numbers count as significant.significant.

Example: 204 m has 3 sf; 3 007 km has 4 sf; Example: 204 m has 3 sf; 3 007 km has 4 sf; 0.02030 m has ___ sf0.02030 m has ___ sf

Trailing zeroes do not count, and a bar Trailing zeroes do not count, and a bar indicates the last sig fig of uncertainty.indicates the last sig fig of uncertainty.

Example: 1 200 000 km has 2 sf; 1 400 (bar) 000 Example: 1 200 000 km has 2 sf; 1 400 (bar) 000 000 m has 4 sf (up to the bar)000 m has 4 sf (up to the bar)

Digital Integrity Part 2 – Digital Integrity Part 2 – How Many Digits Can I Have in My How Many Digits Can I Have in My

Answer?Answer?

1. Addition and Subtraction Rule:1. Addition and Subtraction Rule: When adding or subtracting, calculate When adding or subtracting, calculate the answer and then round off to the the answer and then round off to the LEAST number of DECIMAL PLACES LEAST number of DECIMAL PLACES contained in the question. contained in the question.

Please notePlease note: if more than one size of unit : if more than one size of unit is given, generally convert to the is given, generally convert to the larger unit and then do the larger unit and then do the calculation.)calculation.)

2. Multiplication and Division 2. Multiplication and Division RuleRule:: Do the math, and then round Do the math, and then round off to the LEAST number of DIGITS off to the LEAST number of DIGITS (Sig Figs) contained in the question. (Sig Figs) contained in the question.

Please notePlease note - If doing a series of - If doing a series of calculations, round off the final calculations, round off the final answer only. You will end up with a answer only. You will end up with a very inaccurate answer if you round very inaccurate answer if you round off after every step!off after every step!

Exact numbersExact numbers

If using an If using an exact numberexact number, do the math , do the math and then use the least number of and then use the least number of decimal places in the other values of the decimal places in the other values of the question.question.

Example: Example:

4242 horses x 1.25 bales/horse/day = horses x 1.25 bales/horse/day =

52.5 = 52.50 bales/day52.5 = 52.50 bales/day

Rounding Off RuleRounding Off Rule

If the number following the last one you If the number following the last one you can keep is a 6 or more, round it up.can keep is a 6 or more, round it up.

If the number following the last one you If the number following the last one you can keep is a 4 or less, leave it as is.can keep is a 4 or less, leave it as is.

If the number following the last one you If the number following the last one you can keep is a 5, use the ODD/EVEN rule.can keep is a 5, use the ODD/EVEN rule.

Odd/Even Rule…Odd/Even Rule…

Statistically, the number 5 is exactly in the Statistically, the number 5 is exactly in the middle. There are 5 even digits (0,2,4,6,8) middle. There are 5 even digits (0,2,4,6,8) and 5 odd digits (1,3,5,7,9). The reason of and 5 odd digits (1,3,5,7,9). The reason of the odd/even rule is that if you always the odd/even rule is that if you always round digits from 5 – 9 up, then there are round digits from 5 – 9 up, then there are only 1 – 4 times that you don’t round at all. only 1 – 4 times that you don’t round at all.

Therefore, if the number preceding the 5 is Therefore, if the number preceding the 5 is odd, round up; if it is even, leave it.odd, round up; if it is even, leave it.

Scientific NotationScientific Notation

Writing very large or small numbers can be Writing very large or small numbers can be awkward, and difficult to manage in awkward, and difficult to manage in calculations. Besides using the preferred metric calculations. Besides using the preferred metric system of prefixes, scientific notation can also system of prefixes, scientific notation can also make values much easier to work with. make values much easier to work with.

For example, the average wavelength of For example, the average wavelength of gamma rays is 0.000 000 000 064 m. This is 64 gamma rays is 0.000 000 000 064 m. This is 64 trillionth of a metre…try doing calculations with trillionth of a metre…try doing calculations with that! Scientific notation would change this to that! Scientific notation would change this to 6.4 x 106.4 x 10-11-11 m. m.

The coefficient number (ie The coefficient number (ie 1111) is how many ) is how many placeholders (for a power of ten each) are placeholders (for a power of ten each) are present, and if it is negative, you have a present, and if it is negative, you have a decimal value; a positive is a whole decimal value; a positive is a whole number.number.

To put into scientific notation, move the To put into scientific notation, move the decimal until only ONE digit is in front of it. decimal until only ONE digit is in front of it. Count how many spaces you moved, and Count how many spaces you moved, and that is your coefficient for the power of that is your coefficient for the power of ten. ten.

23 000 cm = 2.3 x 1023 000 cm = 2.3 x 1044 cm cm0.000 000 388 m = 3.88 x 100.000 000 388 m = 3.88 x 10-7-7 m m

To expand a number, multiply it by the To expand a number, multiply it by the power of 10…power of 10…

4.3 x 104.3 x 103 3 m = 4 300 mm = 4 300 m5.08 x 105.08 x 10-2-2 cm = 0.0508 cm cm = 0.0508 cm

Metric Rules and Metric Rules and SymbolsSymbols

Symbols are always printed, and Symbols are always printed, and lowercase letters used. The exceptions lowercase letters used. The exceptions are units derived from a proper name, are units derived from a proper name, such as Joule, Watt, Newton, Litre…such as Joule, Watt, Newton, Litre…

Symbols are never pluralized – 12 g, not Symbols are never pluralized – 12 g, not 12 gs12 gs

Do not put a period after a symbol, Do not put a period after a symbol, unless it is at the end of a sentence.unless it is at the end of a sentence.

A full space is between the number and A full space is between the number and the symbol: 25 m, not 25mthe symbol: 25 m, not 25m

Use decimals, not fractions, and put Use decimals, not fractions, and put a zero before the decimal: 0.87, not a zero before the decimal: 0.87, not .87.87

Generally, we use the term Generally, we use the term massmass for the weight of something.for the weight of something.

Temperatures are indicated by the Temperatures are indicated by the Celsius scale.Celsius scale.

Numbers and symbols are used Numbers and symbols are used together, not mixing numbers and together, not mixing numbers and names: 10 km, not 10 kilometres…names: 10 km, not 10 kilometres…