Lab Math 1Exponents, Scientific Notation and the Metric System

Exponents

• An exponent is used to show that a number has been multiplied by itself a certain number of times.

24 =2 x 2 x 2 x 2 = 16• The number that is multiplied is called the

base and the power to which the base is raised is the exponent.

• By definition, any number raised to the power of 0 is 1.

Manipulation of Exponents

• To multiply two numbers in the same base, add the exponents.

53 x 52 = 55 106 x 10-4 = 102

• To divide two numbers in the same base, subtract the exponents.

53/56 = 5 3-6 = 5-3

Manipulation of Exponents

• To raise an exponential number to a higher power, multiply the two exponents.

(53)2 = 56

(106)-4 = 10-24

Manipulation of Exponents

• To multiply or divide numbers with exponents that have different bases, convert the numbers to their corresponding values without exponents, and then multiply or divide.

52 x 42 = (5 x 5)(4 x 4) = 400

103/2-2 = 1000/-0.25 = 4000

Manipulation of Exponents

• To add or subtract numbers with exponents (whether their bases are the same or not), convert the numbers with exponents to their corresponding values without exponents.

43 + 23 = 64 + 8 = 72

Base 10

• Base 10 is the most commonly used system of exponents.

• Base 10 underlies percentages, and the decimal system, orders of magnitude, scientific notation and logarithms.

• Scientific Units also use Base 10.

Base 10 Numbers > 1:

• The exponent represents the number of places after the number (and before the decimal point).

• The exponent is positive

• The larger the positive exponent, the larger the number.

Use of Exponents in Base 101,000,000 = 106 = one million

100,000 = 105 = one hundred thousand

10,000 = 104 = ten thousand

1,000 = 103 = one thousand

100 = 102 = one hundred

10= 101 = ten

1 = 100 = one

Base 10 Numbers < 1:

• The exponent represents the number of places to the right of the decimal point including the first nonzero number.

• The exponent is negative.• The larger the negative

exponent, the smaller the number.

Use of Exponents in Base 100.1 = 10-1 = one tenth (10%)

0.01= 10-2 = one hundredth (1%)

0.001 = 10-3 = one thousandth (0.1%)

0.0001 = 10-4 = one ten thousandth

0.00001= 10-5 = one hundred thousandth

0.00001= 10-6 = one millionth

Orders of Magnitude

• One order of magnitude is 101 or 10 times.

• A number is said to be orders of magnitude bigger or smaller than another number.

• 102 is two orders of magnitude smaller than 104.

Scientific Notation

• Useful for very small and very large numbers.

• A number written in scientific notation is written as a number between 1 and 10 raised to a power.

• The first part is called the coefficient and the second part is 10 raised to some power.

Conversion to Scientific Notation

• For numbers greater than 10, move the decimal point to the left so there is one nonzero digit to the left of the decimal point. This gives the first part of the notation.

• Count how many places were moved. This is the exponent. It is positive.

5467 = 5.467 x 103

Conversion to Scientific Notation

• For numbers less than 1, move the decimal point to the right so there is one nonzero digit to the left of the decimal point. This gives the first part of the notation.

• Count how many places were moved. This is the exponent. It is negative.

0.5467 = 5.467-3

Scientific Notation

Number Coefficient Exponent

1000 1 X 103

100,000,000 1 X 108

0.0000000000000000000000602 6.02 X 10-23

Multiplication in Scientific Notation

• To multiply numbers in scientific notation, use two steps:– Multiply the coefficients together.– Add the exponents to which 10 is raised.

(2.5 x 102)(3.0 x 103) =

(2.5 x 3.0)(102+3) =

7.5 x 105

Division in Scientific Notation

• To divide numbers in scientific notation, use two steps:

• Divide the coefficients.• Subtract the exponents to which 10 is

raised.(6.0 x 102)/(3.0 x 10-4) =

(6.0 / 3.0)(102-4) =2.0 x 10-2 =

0.02

Addition/Subtraction in Scientific Notation

• If the numbers are the same exponent, just add or subtract the coefficients.

3.0 x 104

+ 4.5 x 104

7.5 x 104

Addition/Subtraction in Scientific Notation

• If the numbers are different exponents, convert both to standard notation and perform the calculation.

(2.05 x 102) – (9.05 x 10-1) =

205 - 0.905 = 204.095

Addition/Subtraction in Scientific Notation

• If the numbers have different exponents, convert one number so they have 10 raised to the same power and perform the calculation.

(2.05 x 102) – (9.05 x 10-1) =

2.05 x 102

-0.00905 x 102

2.04095 x 102

Common Logarithms

• Common logarithms (also called logs or log10) are closely related to scientific notation.

• The common log of a number is the power to which 10 must be raised to give that number.

• The antilog is the number corresponding to a given logarithm.

• pH uses natural logarithms.

Common Logarithms of Powers of Ten

Number Name Power Log

10,000 Ten thousand 104 4

1,000 One thousand 103 3

100 One hundred 102 2

10 Ten 101 1

1 One 100 0

0.1 One tenth 10-1 -1

0.01 One hundredth 10-2 -2

International System of UnitsLength meter m

Mass kilogram kg

Time second s

Electric current ampere A

Temperature     Kelvin K

Amount of substance molemol

Luminous intensity candela cd

Meter is the Unit of Length

• The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

• The meter was intended to equal 10-7 or one ten-millionth of the length of the meridian through Paris from pole to the equator.

• The first prototype was short by 0.2 millimeters because researchers miscalculated the flattening of the earth due to its rotation.

• Platinum Iridium Bar was cast to this length.

Kilogram is the Unit of Mass • A kilogram is equal to the mass of the

international prototype of the kilogram.

• At the end of the 18th century, a kilogram was the mass of a cubic decimeter of water. In 1889, scientists made the international prototype of the kilogram out of platinum-iridium, and declared: This prototype shall henceforth be considered to be the unit of mass.

Liter is a Volume Unit

• A liter (abbreviated either l or L) is equal to 1 dm3 = 10-3 m3

• Liters can be liquid or air.

Time Units

• Minute min 1 min = 60 s

• Hour h 1 h = 60 min = 3600 s

• Day d 1 d = 24 h = 86,400 s

• Second can be abbreviated " (a double tick).

• Minute can be abbreviated ´ (a single tick).

Temperature

• The Kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

• Temperature T, is commonly defined in terms of its difference from the reference temperature T0 = 273.15 K, the ice point.

• This temperature difference is called a Celsius temperature, symbol t, and is defined by the quantity equation

• t= T- T0.

Mole is the Unit of Amount of Substance• A mole is the amount of substance of

a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

• Physicists and chemists have agreed to assign the value 12, exactly, to the "atomic weight of the isotope of carbon with mass number 12 (carbon 12, 12C).

Moles and Avogadro's Number• "Avogadro's Number" is an honorary

name attached to the calculated value of the number of atoms, molecules, etc. in a gram molecule of any chemical substance.

• 12 grams of pure carbon, whose molecular weight is 12, will contain 6.023 x 1023 molecules.

Avocado Avogadro

Moles

• You should specify if you have a mole of atoms, molecules, ions, electrons, or other particles, or specified groups of such particles.

Not a Gram Mole

Metric Prefixes (Big)

• 1024 yotta Y• 1021 zetta Z• 1018 exa E• 1015 peta P• 1012 tera T• 109 giga G• 106 mega M• 103 kilo k• 102 hecto h• 101 deka da

Metric Prefixes (Small)

• 10-1 deci d• 10-2 centi c• 10-3 milli m• 10-6 micro µ• 10-9 nano n• 10-12 pico p• 10-15 femto f• 10-18 atto a• 10-21 zepto z• 10-24 yocto y

Use of the Prefixes for Mass

• Kilogram Kg 103 g

• Gram g 1 g

• Milligram mg 10-3 g

• Microgram µg 10-6 g

• Nanogram ng 10-9 g

• Picogram pg 10-12 g

• Femtogram fg 10-15 g