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The importance of significant figures lies in their application to fundamental computation. In addition and subtraction, the sum or difference should contain as many digits to the right of the decimal as that in the least certain of the numbers used in the computation (indicated by underscoring in the following example).

Addition and subtraction with significant figures

Add 4.383 g and 0.0023 g.

Solution

4.38 3 _ g 0.002 3 _ g 4.38 5 _ g
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In multiplication and division, the product or quotient should contain no more digits than that in the factor containing the least number of significant figures.

Multiplication and division with significant figures

Multiply 0.6238 by 6.6.

Solution

0.623 8 _ × 6. 6 _ = 4. 1 _
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When rounding numbers, increase the retained digit by 1 if it is followed by a number larger than 5 (“round up”). Do not change the retained digit if the digits that follow are less than 5 (“round down”). If the retained digit is followed by 5, round up if the retained digit is odd, or round down if it is even (after rounding, the retained digit will thus always be even).

The use of logarithms and exponential numbers

The common logarithm of a number (log) is the power to which 10 must be raised to equal that number. For example, the common logarithm of 100 is 2, because 10 must be raised to the second power to equal 100. Additional examples follow.

Logarithms and Exponential Numbers
Number Number Expressed Exponentially Common Logarithm
1000 10 3 3
10 10 1 1
1 10 0 0
0.1 10 −1 −1
0.001 10 −3 −3

What is the common logarithm of 60? Because 60 lies between 10 and 100, which have logarithms of 1 and 2, respectively, the logarithm of 60 is 1.7782; that is,

60 = 10 1 .7782

The common logarithm of a number less than 1 has a negative value. The logarithm of 0.03918 is −1.4069, or

0.03918 = 10 1.4069 = 1 10 1.4069

To obtain the common logarithm of a number, use the log button on your calculator. To calculate a number from its logarithm, take the inverse log of the logarithm, or calculate 10 x (where x is the logarithm of the number).

The natural logarithm of a number (ln) is the power to which e must be raised to equal the number; e is the constant 2.7182818. For example, the natural logarithm of 10 is 2.303; that is,

10 = e 2 .303 = 2 .7182818 2 .303

To obtain the natural logarithm of a number, use the ln button on your calculator. To calculate a number from its natural logarithm, enter the natural logarithm and take the inverse ln of the natural logarithm, or calculate e x (where x is the natural logarithm of the number).

Logarithms are exponents; thus, operations involving logarithms follow the same rules as operations involving exponents.

  1. The logarithm of a product of two numbers is the sum of the logarithms of the two numbers.
    log x y = log x + log y , and ln x y = ln x + ln y
  2. The logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers.
    log x y = log x log y , and ln x y = ln x ln y
  3. The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number.
    log x n = n log x and ln x n = n ln x

The solution of quadratic equations

Mathematical functions of this form are known as second-order polynomials or, more commonly, quadratic functions.

a x 2 + b x + c = 0

The solution or roots for any quadratic equation can be calculated using the following formula:

x = b ± b 2 4 a c 2 a

Solving quadratic equations

Solve the quadratic equation 3 x 2 + 13 x − 10 = 0.

Solution

Substituting the values a = 3, b = 13, c = −10 in the formula, we obtain

x = 13 ± ( 13 ) 2 4 × 3 × ( −10 ) 2 × 3
x = 13 ± 169 + 120 6 = 13 ± 289 6 = 13 ± 17 6

The two roots are therefore

x = 13 + 17 6 = 2 3 and x = 13 17 6 = −5
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Quadratic equations constructed on physical data always have real roots, and of these real roots, often only those having positive values are of any significance.

Two-dimensional ( x - y ) graphing

The relationship between any two properties of a system can be represented graphically by a two-dimensional data plot. Such a graph has two axes: a horizontal one corresponding to the independent variable, or the variable whose value is being controlled ( x ), and a vertical axis corresponding to the dependent variable, or the variable whose value is being observed or measured ( y ).

When the value of y is changing as a function of x (that is, different values of x correspond to different values of y ), a graph of this change can be plotted or sketched. The graph can be produced by using specific values for ( x , y ) data pairs.

Graphing the dependence of y On x

x y
1 5
2 10
3 7
4 14

This table contains the following points: (1,5), (2,10), (3,7), and (4,14). Each of these points can be plotted on a graph and connected to produce a graphical representation of the dependence of y on x .

A graph is titled “Dependency of Y on X.” The x-axis ranges from 0 to 4.5. The y-axis ranges from 0 to 16. Four points are plotted as a line graph; the points are 1 and 5, 2 and 10, 3 and 7, and 4 and 14.
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If the function that describes the dependence of y on x is known, it may be used to compute x,y data pairs that may subsequently be plotted.

Plotting data pairs

If we know that y = x 2 + 2, we can produce a table of a few ( x , y ) values and then plot the line based on the data shown here.

x y = x 2 + 2
1 3
2 6
3 11
4 18
A graph is titled “Y equals x superscript 2 plus 2.” The x-axis ranges from 0 to 4.5. The y-axis ranges from 0 to 20. Four points are plotted as a line graph; the points are 1 and 3, 2 and 6, 3 and 11, and 4 and 18.
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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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