UNIT 1 Notes: Scientific Notation, Significant Figures ...



UNIT 1 NOTES: Scientific Notation,

Significant Figures & Rounding, Metric Units

(Chapter 3 in your Chemistry Text)

Scientific Notation

When scientists talk about very large or very small numbers, it gets difficult to name the number (e.g. a one followed by 15 zeros is quadrillion or quintillion?) and it’s painstaking to write them down. So scientists prefer to use a form of exponential notation that has come to be known as scientific notation.

Scientific notation consists of two parts:

1) a mantissa (or coefficient), which is a number whose absolute value is between 1 and 10 (that is, -10 to -1 and 1 to 10). The mantissa may be a whole number or carry decimal places.

2) a characteristic (commonly called the exponent), which is a whole number exponent of the base 10. It may be positive, negative, or zero.

EXAMPLE:

32,000,000,000 (Thirty-two billion) = 3.2 x 10 10

|Long-Hand Number |Scientific Notation |

|1000 |1 x 103 |

|4,200,000,000,000,000,000,000,000 |4.2 x 1024 |

|-502,000,000,000,000,000 |-5.02 x 1017 |

|0.0006 |6 x 10 - 4 |

|0.000 000 000 000 000 000 000 310 |3.10 x 10 - 22 |

|-0.000 000 000 000 089 |-8.9 x 10 - 14 |

Notice that numbers with positive exponents are larger than one, and numbers with negative exponents are smaller than one. Often times it is the exponent (characteristic) that is the most meaningful number in scientific notation.

Also see that scientific notation can express both positive and negative numbers. WARNING - Do not confuse the sign on the mantissa with the sign on the characteristic ! The sign on the mantissa describes the number as positive or negative. The sign on the characteristic describes the number as smaller than or larger than one.

A Note on Zeros – When converting long-hand numbers to scientific notation, remember these rules for zeros.

1) Trailing zeros before the decimal point should not be written in scientific notation.

EXAMPLE 34,000 = 3.4 x 104

34,000 ≠ 3.4000 x 104

2) Trailing zeros after the decimal point should be written in scientific notation.

EXAMPLE 0.0007700 = 7.700 x 10-4

0.0007700 ≠ 7.7 x 10-4

3) Leading zeros should not be written in scientific notation.

EXAMPLE 0.000 02 = 2 x 10-5

0.000 02 ≠ 00002 x 10-5

4) Captive zeros (zeros between non-zero numbers) should be written in scientific notation.

EXAMPLE 904,000 = 9.04 x 105

904,000 ≠ 9.4 x 105

PRACTICE 1.1

Convert the following numbers to scientific notation.

a) 17600.0 d) 10.2

b) 0.00135 e) -0.000130

c) -67.30 f) 301.0

PRACTICE 1.2

Expand the following scientific notation to regular notation.

a) 4.96 x 10-2 d) -9.3 x 10-3

b) 5.50 x 10-4 e) 7.01 x 100

c) -8.37 x 104 f) 4.92 x 102

PRACTICE 1.3

Correct the following incorrect scientific notation.

a) 36.7 x 101 d) 851.6 x 10-3

b) 0.0123 x 104 e) -966 x 10-1

c) -0.015 x 10-3 f) 0.047 x 1033

Doing Math in Scientific Notation

Though you will perform most calculations involving scientific notation on your calculator, it’s important to know how the calculator is getting the answers.

Addition/Subtraction 1) Rewrite all the numbers in the operation to have the same exponent as the number with the largest exponent.

(Use incorrect scientific notation)

2) Add/Subtract the mantissas

3) Rewrite the answer in correct scientific notation (if necessary)

EXAMPLES

(3.44 x 103) + (4.33 x 105)

(0.0344 x 105) + (4.33 x 105)

4.3644 x 105

(2.32 x 10-3) – (4.4 x 10-4)

(2.32 x 10-3) – (0.44 x 10-3)

1.88 x 10-3

Multiplication/Division 1) Multiply or Divide the mantissas

2) for Multiplication, ADD the exponents

for Division, SUBTRACT the exponents

3) Rewrite the answer in correct scientific notation (if necessary)

EXAMPLES

(3.44 x 103) x (4.33 x 105)

(3.44 x 4.33) x 10(3+5)

14.9 x 108 = 1.49 x 109

(2.32 x 10-3) ÷ (4.4 x 10-4)

(2.32/4.4) x 10(-3 - -4)

5.27 x 101

PRACTICE 1.4

Perform the following calculations.

a) (2.1 x 104) + (3.5 x 105)

b) (5.2 x 10-5) + (-2.69 x 10-4)

c) (6.0 x 1016) – (1.77 x 1018)

d) (-4.5 x 10-11) – (1.83 x 10-9)

e) (8 x 1015) x (6 x 103)

f) (1.5 x 1026) x (-3.0 x 10-24)

g) (6 x 10-7) ÷ (3 x 10-15)

h) (-5.6 x 10-44) ÷ (2.8 x 10-34)

Now consider performing a longer calculation like,

(5 x 10-5) + (-2 x 10-4) – (1 x 10-3)

(4 x 10-8) x (6 x 103)

Solve this one just like any other mathematical operation by following order of operations (remember PEMDAS?). First solve for the numerator and denominator separately, and then divide through:

(5 x 10-5) + (-2 x 10-4) – (1 x 10-3)

(4 x 10-8) x (6 x 103)

(0.05 x 10-3) + (-0.2 x 10-3) – (1 x 10-3)

(4 x 6) x 10(-8+3)

(-1.15 x 10-3)

(24 x 10-5)

(-1.15 ÷ 24) x 10(-3- -5)

-0.0479 x 102

-4.79 x 100 = -4.79

Note the denominator in the third step is written in incorrect scientific notation. You don’t need to correct scientific notation until you arrive at the final answer.

Using a Calculator with Scientific Notation

Performing the type of calculations as the previous example is time consuming. Additionally the multiple steps required allow more opportunity for careless errors. For these reasons, scientific calculators are invaluable tools for doing math in science. Chances are that your calculator is capable of far more functions than you know how to use, or would even have occasion to use. But if you learn just a few functions, seemingly complex calculations become manageable.

Calculator Lesson 1: Parentheses

Scientific calculators follow order of operations (PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Proper use of parentheses allows you to enter large operations in your calculator all at once.

EXAMPLE Take 4 + 4 – 6

5 – 7 + 3

If you enter this on your calculator without using parentheses, you will get the wrong answer.

4 + 4 – 6 ÷ 5 – 7 + 3 = 2.8

Instead, enter the numerator in parentheses, followed by the denominator in parentheses.

( 4 + 4 – 6 ) ÷ ( 5 – 7 + 3 ) = 2

Calculator Lesson 2: Exponents

Because you will often work with numbers in scientific notation, you ought to learn the most efficient way of entering scientific notation into your calculator.

On many scientific calculators, the exponent key for scientific notation is x10n

If you have a TI-8* calculator, exponents are entered by 2nd EE

You should avoid using the “caret” ^ key. This requires extra keystrokes and the use of parentheses in division, providing more opportunities for careless mistakes.

EXAMPLE You need to enter 6.022 x 1023 in your calculator

6 . 0 2 2 x 1 0 ^ 2 3

6 . 0 2 2 2nd EE 2 3

6 . 0 2 2 x10n 2 3

Calculator Lesson 3: Storing Numbers in Memory

On several occasions, it will be necessary to carry out long computations using large numbers. For example, you will often use Avogadro’s number which is 6.022 x 1023. It gets annoying entering that number every time it appears in a calculation. To reduce keystrokes, you can store numbers in your calculator. Most scientific calculators can store a few numbers in memory. TI-8* calculators can store more numbers than you will need to in this class. Each calculator will have a slightly different method to store numbers. So consult your user’s guide or ask your teacher for help storing numbers in your calculator.

Follow the steps below to store numbers in TI-30XS class set calculators (but remember, the memory of these calculators is cleared before each test).

1) Enter the number in your calculator and press ENTER

2) press the STO> key. The calculator will display Ans->

3) press the x yzt abc key multiple times until the desired variable appears

4) press the ENTER key. The calculator should again display the stored number

5) to use the stored number press the variable key until the desired letter appears

Follow the steps below to store numbers in a TI-8* calculator.

1) Enter the number in your calculator and press ENTER

2) press the STO> key. The calculator will display Ans->

3) press the ALPHA key followed by whichever key corresponds to the letter of the alphabet to which you will assign the number.

4) press the ENTER key. The calculator should again display the stored number.

5) To use stored numbers in calculations, just press the ALPHA key followed by whichever key corresponds to the letter assigned .

NOTE: The calculator will display the assigned letter – not the actual number stored.

PRACTICE 1.5

Use your scientific or graphing calculator to perform the following operations.

a) (1.43 x 10-14)(5.22 x 107)(3.27 x 10-1)

(6.45 x 108)(7.27 x 10-8)(1.16 x 105)

b) [(34.5 + 744.3)4(2.35 x 10-3)]4

[(711.44)(2.89 x 104)3(1.54)6]2

Significant Figures & Rounding

Use of significant figures in science is an attempt to be as precise and accurate about numerical data as possible.

Precision refers to how closely multiple measurements of the same object agree with one another.

Accuracy refers to how closely the measurement is to the true or accepted value.

For example, how long is the Great Wall of China? Say that you and two of your friends set out to answer the question. You split the wall into three sections and each measure one section. You measure the wall in ten-mile increments. One friend measures in one-mile increments, and the other measures in 0.1-mile increments. After a few weeks, you and your friends meet to share the results. You measured your section at 1330 miles. One of your friends measured a section of 1333 miles, and the other measured a section of 1333.3 miles. Would it be correct to say that the Great Wall is 3996.3 miles long? The answer is NO! While you measured your section at 1330 miles long, you didn’t measure it to be exactly 1300.0 miles long. The same goes for the 1333 mile section. So it would be incorrect to report a length of 3996.3 miles, because you and your first friend did not measure the sections precisely to the nearest tenth of a mile. You could only correctly report the Great Wall as being 3990 miles long.

Use of significant figures acknowledges the fact that you can only know a calculated value as precisely as the least precise contributing factor!

To properly round with significant figures you need to know how to identify significant figures, and how to do math and conserve significant figures.

Identifying Significant Figures

When asked to conserve significant figures in a calculation, it is necessary to identify which figures in the calculation are significant. There are five rules for identifying significant figures.

1) Non-zero digits and zeros between non-zero digits are ALWAYS significant.

2) Leading zeros (zeros at the beginning of numbers) are NOT significant.

3) Zeros to the right of all non-zero digits are only significant if the decimal point is shown.

If the decimal is ABSENT start counting significant digits from the first non-zero on the ATLANTIC side of the number.

EXAMPLE: 1200 is 2 significant digits

If the decimal is PRESENT start counting significant digits from first non-zero on the PACIFIC side of the number.

EXAMPLE: 0.0001200 is 4 significant digits

4) For values written in scientific notation, the digits in the coefficient (mantissa) 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.

PRACTICE 1.6

How many significant figures are in each of the following numbers?

a) 114.0 e) 4.50 x 103

b) 733.02 f) -2340

c) 0.000310 g) 6.0040

d) 908010. h) -4.010 x 10-4

Rounding with Significant Figures

There are only two rules that you need to conserve significant figures in calculations. Conserving significant figures means that you may need to round the answer to a problem. Normal rounding rules apply.

Addition/Subtraction Round your answer to the same order as the LEAST PRECISE number in the calculation.

EXAMPLES 67.443 + 3.07 - 2.9 = 67.6

861.2 – 78.554 + 91 = 874

Multiplication/Division Round your answer to have the same number of significant figures as the LEAST NUMBER OF SIGNIFICANT FIGURES in the operation.

EXAMPLES 67.443 x 3.07 ÷ 2.9 = 71

861.2 ÷ 78.554 x 90.1 = 988

PRACTICE 1.7

Report the answers to the following operations to the correct number of significant figures.

a) 23.5732 + 4.08 – 245.21 + 75.6 =

b) 180.00 – 76.33 – 33.456 =

c) 1 + 0.023 – 0.046 =

d) 23.5732 x 4.08 ÷ 245.21 =

e) 180 ÷ 76.33 x 33.5 =

f) 30.59 + 28.205 =

13.3 x 0.023

Metric Units

The English system of measures that Americans use is antiquated and cumbersome. The U.S.A. is one of only two countries in the world still using this system! Everybody else in the world uses the metric system (even England). That America still uses the English system, is a testament to the tenacity of tradition. Hopefully, America will make the switch in your lifetime. While the metric system is no more accurate than the English system, it is much easier to work with. The English system has about 88 different units. While the metric system uses only 3 units with about 10 prefixes.

All scientists around the world use the metric system of measures. Below are a table of metric units and a table of metric prefixes. One way to think of the prefixes is like scientific notation. The prefix tells you by what factor of ten to multiply a given value. NOTE – The temperature scale does not use prefixes.

|Metric Units |

|Length or Distance |meter (m) |

|Mass |gram (g) |

|Volume |liter (l) or (L) |

|Temperature |Celsius (°C) also Kelvin (K) |

|Common Metric Prefixes |

|Prefix |Meaning |10n |Decimal |

|pico (p) |trillionth |10-12 |0.000 000 000 001 |

|nano (n) |billionth |10-9 |0.000 000 001 |

|micro (μ) |millionth |10-6 |0.000 001 |

|milli (m) |thousandth |10-3 |0.001 |

|centi (c) |hundredth |10-2 |0.01 |

|deci (d) |tenth |10-1 |0.1 |

|deca (da) |ten |101 |10 |

|hecto (h) |hundred |102 |100 |

|kilo (k) |thousand |103 |1000 |

|mega (M) |million |106 |1 000 000 |

|giga (G) |billion |109 |1 000 000 000 |

|tera (T) |trillion |1012 |1 000 000 000 000 |

EXAMPLES 5 mm = 5 x 10-3 m = 0.005 m = 5 thousandths of a meter

6 kg = 6 x 103 g = 6000 g = 6 thousand grams

7 GL = 7 x 109 L = 7 000 000 000 L = 7 billion liters

also

1 000 000 μL = 1000 mL = 1 L = 0.001 kL = 0.000 000 001 GL

PRACTICE 1.8

Give the metric abbreviation for each of the following numbers.

a) one millionth of a gram g) 1 x 103 grams

b) two billion liters h) 3.2 x 106 liters

c) five hundredths of a liter i) 9300 meters

d) six thousand grams j) -1.4 x 10-12 gram

e) nine billionths of a meter k) 3/10 of a liter

f) twelve thousand meters l) 7/100 000 of a gram

Using Dimensional Analysis for Unit Conversion

Because the U.S.A. still uses the English system of measures, you need to learn how to convert units between systems of measures. There is more than one way to convert units, but this class will learn a system known as “dimensional analysis,” or “factor-label method.” This method is useful for converting between all sorts of units. Mastery of dimensional analysis is essential for success in chemistry class.

Dimensional analysis is a method of conversion based on equivalencies. The best way to learn dimensional analysis is to practice it. Here is the method:

1) Determine all necessary conversion factors.

2) Set up a fraction. Write what you are given in the numerator.

Include all units!

3) Write fractions of equivalent units as multipliers to the given unit. Include all units!

4) Make sure all unwanted units cancel.

5) Solve: Multiply across numerator and denominator separately, then divide numerator by denominator.

EXAMPLE You need to convert 8 ounces to grams.

1) Find the conversion for ounces to grams. (1 oz = 28.3 g)

2) Set up a fraction.

8 oz

1

3) Write fractions of equivalent units as multipliers.

8 oz | 28.3 g

1 | 1 oz

4) Make sure all unwanted units cancel.

8 oz | 28.3 g

1 | 1 oz

5) Solve.

8 x 28.3 g = 226 g

PRACTICE 1.9

Complete the following conversions. Conserve significant figures!

|1 inch = |2.54 centimeters |

|1 pound = |454 grams |

|1 gallon = |3.785 liters |

a) Convert 5.2 gallons to liters.

b) Convert 16 tons to kilograms.

c) Convert 3.92 yards to picometers.

d) Convert 1.75 x 106 grams to pounds.

e) Convert 25.6 tons to mg.

f) Convert 70. mph to km/hr.

g) Convert 1.023 x 107 km/day to feet/sec.

h) Convert 45.8 pounds/gallon to ng/μL.

Answers to Practice Problems:

PRACTICE 1.1

a) 1.76000 x 104

b) 1.35 x 10-3

c) -6.730 x 10-1-

d) 1.02 x 101

e) -1.30 x 10-4

f) 3.010 x 102

PRACTICE 1.2

a) 0.0496

b) 0.000550

c) -83700

d) -0.0093

e) 7.01

f) 492

PRACTICE 1.3

a) 3.67 x 102

b) 1.23 x 102

c) -1.5 x 10-5

d) 8.516 x 10-1

e) -9.66 x 101

f) 4.7 x 1031

PRACTICE 1.4

a) 3.71 x 105

b) 3.21 x 10-4

c) -1.71 x 1018

d) -1.875 x 10-9

e) 4.8 x 1019

f) -4.5 x 102

g) 2 x 108

h) -2 x 10-10

PRACTICE 1.5

a) 4.49 x 10-14

b) 10.6

PRACTICE 1.6

a) 4

b) 5

c) 3

d) 6

e) 3

f) 3

g) 5

h) 4

PRACTICE 1.7

a) -142.0

b) 70.21

c) 1

d) 0.392

e) 79

f) 190

PRACTICE 1.8

a) 1 μg

b) 2 GL

c) 5 cL

d) 6 kg

e) 9 nm

f) 12 km

g) 1 kg

h) 3.2 ML

i) 9.3 km

j) -1.4 pg

k) 3 dL

l) 70 μg

PRACTICE 1.9

a) 20 L

b) 15,000 kg

c) 3.58 x 109 pm

d) 3.85 x 103 lbs

e) 2.32 x 1010 mg

f) 110 km/hr

g) 3.88 x 105 ft/sec

h) 5.49 x 106 ng/μL

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mantissa

characteristic

Wrong

Right

Wrong

Right

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