Math in Science

[Page 1 of 5] Physics 1.3 – Math in Science

Accuracy & Precision   • Precision – a description of the level of exactness of a measurement

• Related to the tool used to make the measurement • Accuracy – a description of how close a measurement is to the standard or correct value  

• Measurements can be precise and not accurate • All measurements have some level of uncertainty Scientific Notation • Scientific notation – a method of expressing numbers as a value between one and ten (A) multiplied by a whole number

power of ten (n)

𝐴  ×  10!

• Power of ten represents the number of places the decimal point was moved • Large numbers have positive n • Small numbers have negative n

• Steps for converting a large number into scientific

notation • Step 1 | Count the number of place values from

the ones position to the digit preceding the highest or left-most digit.

• Step 2 | Write the highest place value digit, followed by a decimal point and the following digits that are known to be accurate.

• Step 3 | After the numeric value, write a multiplication symbol times ten raised to the number of place values from Step 1.

• Steps for converting a small number into scientific notation

• Step 1 | Count the number of place values from the tenths position up to and including the first non-zero digit.

• Step 2 | Write the first non-zero digit, followed by a decimal point and the following digits that are known to be accurate.

• Step 3 | After the numeric value, write a multiplication symbol times ten raised to the negative number of place values from Step 1.

Math in Science

[Page 2 of 5] Physics 1.3 – Math in Science

Significant Figures • Significant figures – a method of reporting a measurement based on the digits of the number that are known to be

accurate • Smallest place value digit is not absolutely certain

• Rules to determine which digits of a number are significant

1. Non-zero digits and zeros between non-zero digits are always significant. 2. Leading zeros, or zeros that come before all non-zero digits, are not significant. 3. Zeros to the right of all non-zero digits are only significant if a decimal point is shown. 4. For values written in scientific notation, the digits in the coefficient are significant. 5. In a common logarithm, there are as many digits after the decimal point as there are significant figures in the

original number.

• Rules for adding, subtracting, multiplying, or dividing numbers with significant figures • When adding or subtracting numbers with significant figures, the smallest significant place value in the answer must

be the same as the smallest significant place value of the measurement with the least number of significant figures. • When multiplying or dividing numbers with significant figures, the number of significant figures in the answer must

be the same as the measurement with the least number of significant figures.

• Steps for performing a mathematical operation with measurements in significant figures • Step 1 | Determine the number of significant figures in each measurement • Step 2 | Add, subtract, multiply, or divide the values. • Step 3 | Round the answer to the correct number of significant figures.

• When doing a series of mathematical operations

• Do not round the answer after each step • Keep track of the number of significant figures produced at each step • Round the answer at the end

Math in Science

[Page 3 of 5] Physics 1.3 – Math in Science

SI Units • International System of Units (SI units) – a set of standard units of measure for the fundamental quantities of science

• SI prefixes can be added to a base unit to indicate very small or large measurement

SI Base Units Quantity Unit Symbol Length meter m Mass kilogram kg Time second s

Electric current ampere A Temperature kelvin K

Amount of substance mole mol Luminous intensity candela cd

SI Prefixes

Prefix Symbol Multiplication Factor

exa- E 1,000,000,000,000,000,000 1018

peta- P 1,000,000,000,000,000 1015

tera- T 1,000,000,000,000 1012

giga- G 1,000,000,000 109

mega- M 1,000,000 106 kilo- k 1,000 103

hecto- h 100 102

deka- da 10 101

deci- d 0.1 10–1

centi- c 0.01 10–2

milli- m 0.001 10–3 micro- µ 0.000 001 10–6

nano- n 0.000 000 001 10–9

pico- p 0.000 000 000 001 10–12 femto- f 0.000 000 000 000 001 10–15

atto- a 0.000 000 000 000 000 001 10–18

Math in Science

[Page 4 of 5] Physics 1.3 – Math in Science

Dimensional Analysis • Dimensional analysis – a problem-solving method that determines relationships between quantities by using their units

• Conversion factor – a mathematical ratio between two different units that measure the same quantity

Conversion Factors

Length 1 in = 2.54 cm

1 ft = 12 in 1 mi = 5280 ft

Mass 1 kg = 2.2046 lb 1 lb = 16 oz

Time 1 min = 60 s 1 hr = 60 min 1 day = 24 hr

Volume 1 cm3 = 1 mL = 1 cc 1 m3 = 1000 L

Energy 1 cal = 4.18 J 1 Cal = 1kcal = 1000 cal

Pressure 1 atm = 760 mm Hg = 101.3 kPa

• Steps for using dimensional analysis to convert between two units

• Step 1 | Set up a mathematical expression to multiply the known measurement by the conversion factor such that the current units of the measurement will cancel out with the same units of the conversion factor and replace the original units with the converted units.

• Step 2 | Repeat the process in Step 1 until the measurement is in the correct units. • Step 3 | When the correct units have been reached, multiply the values.

Math in Science

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Scientific Calculators • Scientific calculators are used in many science courses

• Different calculators require different steps to enter a number in scientific notation

• Use a button labeled E, EE, or EXP • Ex) For 3  ×  10!!, type 3 E -5

• Different calculators use different procedures

• Enter value before operation • Enter the operation before value • Ex) To determine the square root of 9

• In this calculator, type 9   𝑥 • Another calculator, type 𝑥  9

• Scientific calculators can perform more functions than they have room for buttons

• Buttons contain more than one function • In this calculator, press INV to access functions above the buttons • Ex) For 3!, type 3  INV   𝑥

• Scientific calculators can simplify complicated processes in math