James Richard Fromm
Modern science makes extensive use of mathematics. In introductory university chemistry it is necessary to use scientific notation, logarithms, algebra, and to a limited extent the differential and integral calculus. We now address each of these areas in turn. To any scientist, mathematical operations are tools to be applied to scientific problems. It is the use of these tools on chemical problems at which you are expected to become proficient and for that limited purpose, and at the introductory level, this discussion suffices.
Experimental measurements in science may include very large multiples of units and very small submultiples of them. Since measurements are expressed as a product of a number and a unit, this means that scientists have had to express very large and very small numbers. For example, one copper Canadian penny contains 564,240,000,000,000,000,000,000 atoms of copper. This is a clumsy notation, and it implies that we actually know that the number is not 564,240,000,000,000,000,000,005 atoms. We do not, and cannot, know this. It is therefore both more convenient, and by implication more honest, to express the number of atoms of copper in a copper penny as 56.424 x 10+23 atoms in this exponential notation. A number is written in exponential notation as the product of a real number (with a decimal point) multiplied by ten to some integral power (the exponent). Alternatively, but less conveniently, we could explicitly give the precision of the measurement. On an accurate analytical balance, the number of copper atoms in a copper penny might be 56.424 +/- 0.005 x 10+23 atoms.
Scientists usually write numbers in a form of exponential notation called scientific notation, which means that the number is written with one non-zero digit to the left of the decimal point and an integer exponent or power of ten. The number of atoms of copper in a copper penny would be written as 5.6424 x 10+24 atoms in scientific notation.
Another form of exponential notation called engineering notation is also convenient. In engineering notation the number is written with one, two, or three digits to the left of the decimal point and the integer exponent is always expressed as a number divisible by three. For example, 5.6424 x 10+24 atoms is in both scientific notation and engineering notation while 56.424 x 10+23 atoms is in neither, although both represent exactly the same number in exponential notation. Engineering notation is particularly convenient in the International System of Units (SI) because many powers of ten which are evenly divisible by three have a named prefix with an easily identifiable symbol.
The expression of a number as a power of ten is convenient because our number system is decimal. We call the power to which ten is raised an exponent of ten, and exponents are normally written as superscripts. Thus 10+2 = 10 x 10 = 100 and 10+3 = 10 x 10 x 10 = 1000.
A number which is raised to a power is called, in mathematics, a base. Numbers other than ten can be used as bases. For example, 2+3 is base two to exponent three, more often described as two to the third power, and equals 2 x 2 x 2 = 8; 2+4 = 16, and so on. Use of base ten is convenient because shifting the decimal point one place to the right and increasing the exponent by one are equivalent operations, as are shifting the decimal point one place to the left and decreasing the exponent by one. If either of these operations are repeated until the exponent is zero, the exponential part disappears because 100 = 1, and the number is again in ordinary non-exponential notation.
All forms of exponential notation are particularly convenient when products, quotients, powers, and roots must be calculated. Multiplications of two numbers in exponential form involves addition of their exponents, while division of two numbers in exponential form involves subtraction of their exponents.
Example. 274.35 x 1032.1 = 2.7435 x 102 x 1.0321 x 103 = 2.8316 x 105.
Example. 204.2/1946.3 = (2.042/1.9463) x 10+2-3 = 1.0492 x 10-1 = 0.10492
For powers, the exponent is multiplied, and for roots, the exponent is divided. Taking of a root is easier if the exponent of the number whose root is to be taken is evenly divisible by the desired root.
Example. (124.6 x 10+2)3 = (1.246 x 10+4)3 = 1.2463 x 10+12 = 3.931 x 10+12.
Example. 3(the square root of)(19.42 x 10+7) = 3(the square root of)(194.2 x 10+6) = 10+2 x 3(the square root of)(194.2) = 5.791 x 10+2 = 579.1.
For the purpose of the discussion which follows it is useful to have a brief table of common (base ten) logarithms, which is given below. All modern scientific calculators are equipped with logarithm functions, both log (logarithm base ten) and ln (logarithm base e), as well as their respective inverse functions, the exponential 10x and ex, which are considerably more accurate than this table.
No. log No. log No. log No. log No. log 1.0 .00 3.0 .48 5.0 .70 7.0 .85 9.0 .95 1.1 .04 3.1 .49 5.1 .71 7.1 .85 9.1 .96 1.2 .08 3.2 .51 5.2 .72 7.2 .86 9.2 .96 1.3 .11 3.3 .52 5.3 .72 7.3 .86 9.3 .97 1.4 .15 3.4 .53 5.4 .73 7.4 .87 9.4 .97 1.5 .18 3.5 .54 5.5 .74 7.5 .88 9.5 .98 1.6 .20 3.6 .56 5.6 .75 7.6 .88 9.6 .98 1.7 .23 3.7 .57 5.7 .76 7.7 .89 9.7 .99 1.8 .26 3.8 .58 5.8 .76 7.8 .89 9.8 .99 1.9 .28 3.9 .59 5.9 .77 7.9 .90 9.9 .00 2.0 .30 4.0 .60 6.0 .78 8.0 .90 1.00 .00 2.1 .32 4.1 .61 6.1 .79 8.1 .91 1.01 .00 2.2 .34 4.2 .62 6.2 .79 8.2 .91 1.02 .01 2.3 .36 4.3 .63 6.3 .80 8.3 .92 1.03 .01 2.4 .38 4.4 .64 6.4 .81 8.4 .92 1.04 .02 2.5 .40 4.5 .65 6.5 .81 8.5 .93 1.05 .02 2.6 .42 4.6 .66 6.6 .82 8.6 .93 1.06 .03 2.7 .43 4.7 .67 6.7 .83 8.7 .94 1.07 .03 2.8 .45 4.8 .68 6.8 .83 8.8 .94 1.08 .03 2.9 .46 4.9 .69 6.9 .84 8.9 .95 1.09 .04
The convenience of exponential notation can be greatly extended by the use of non-integral powers. These powers, applied to appropriate bases, constitute logarithms of numbers. Logarithms to base two are used by digital computers, but only two logarithms are used by human beings with any frequency. These are logarithms to base 10 and logarithms to base e. Logarithms using base 10 are called common logarithms and are the usual logarithms employed in calculations with decimal numbers. The common logarithm of x is usually written as log x; log x is simply the power to which ten must be raised to obtain x and may be either a positive or a negative number:
10log x = x
Logarithms using base e are called natural logarithms and often arise as a result of calculus operations. The natural logarithm of x is usually written as ln x. The quantity ln x is the power to which e, an irrational endless number whose value is 2.71828..., must be raised to obtain x and again may be either a positive or negative number:
eln x = x
Natural logarithms can easily be converted to common logarithms:
ln x = 2.30258 ...log x
The form ln x = 2.303 log x is sufficiently accurate for most calculations. Modern hand-held electronic calculators often have both common (log x) and natural (ln x) logarithm functions, as well as the antilogarithm functions for common logarithms (10x) and for natural logarithms (ex, exp x, or exp).
A logarithm consists of two parts called the characteristic and the mantissa. The characteristic of the logarithm is that part preceding the decimal point and the mantissa is that part of the logarithm which follows the decimal point. In most cases the mantissa is an irrational (endless) number.
Example. log 60 = 1.78...; the characteristic is 1 and the mantissa is 0.78...
Decimal numbers expressed in scientific notation have, or upon proper shifting of the decimal point can have, values ranging only between zero and one multiplied by an integer exponent of ten. This integer is the characteristic of the logarithm of the number, which can then easily be obtained by inspection. The mantissa can be obtained by looking it up in an appropriate table or using a calculating device (manual calculation is too tedious to be useful). Since mantissas are irrational endless numbers, a useful table can only list them to a reasonable number of places. Tables are commonly available giving values to two, three, four, five and six places. Electronic calculators can be used to obtain values to more places. A two-place table of common logarithms is given in the Table above. Interpolation between table entries can be used if desired. This table, like most tables of common logarithms, leaves the characteristic to be determined by inspection. It may be used to obtain either logarithms or antilogarithms. An antilogarithm is the number obtained when ten is raised to a power. It is not possible to obtain the logarithm of a negative number, but it is quite possible to obtain the antilogarithm of a negative power.
Example. To calculate the antilogarithm of 23.46, we write x = (antilog 0.46)(antilog 23). Then x = (antilog 0.46) x 10+23 = 2.9 x 10+23
Calculation of the antilogarithm of a negative number requires an additional step. Since the antilogarithm of a negative mantissa cannot be taken directly, the negative number is rewritten as a positive mantissa and a negative characteristic.
Example. To calculate the antilogarithm of -19.41, we write -19.41 = +0.59 - 20; x = (antilog 0.59)(antilog -20); x = (antilog 0.59) x 10-20 = 3.9 x 10-20
The result of calculations using logarithms are the same as those obtained by using other forms of exponentiation. The parallel rules are given below.
An x Am = An + m
log AB = log A + log B
An/Am = An - m
log (A/B) = log A - log B
(An)1/m = An/m
log (A1/n) = (log A)/n
The two latter forms are used for powers and roots, since Cm is C to the power m while C1/m is the mth root of C.
Example. To multiply 2.46 by 27.9 using logarithms, obtain the logarithms, add, and
take the antilogarithm:
0.391 + 1.446 = 1.837; antilog 1.837 = 68.7
Example. To multiply 5.71 by 0.00349 using logarithms, obtain the logarithm, add, and
take the antilogarithm:
0.757 + (-2.457) = -1.700; antilog -1.700 = antilog (0.300 - 2) = 2.0 x 10-2
Example. To divide 9.62 by 4.32, obtain the logarithms, subtract them, and take the
0.983 - 0.635 = 0.348; antilog 0.348 = 2.23.
Example. To divide 5.21 x 10-3 by 5.14 x 10+2, obtain the
logarithms, subtract, and take the antilogarithm:
(0.717 - 3) - (2.711) = -4.994 = 0.006 - 5; antilog (-4.994) = antilog (0.006) x 10-5 = 1.01 x 10-5
Example. To raise 9.26 x 10-5 to the fifth power, obtain the logarithm,
multiply by five, and take the antilogarithm:
(0.967 - 5) x 5 = 4.833 - 25 = 0.833 - 21; antilog (0.833 - 21) = 6.81 x 10-21
Example. To take the fourth root of 97.43, obtain the logarithm, divide by four, and
take the antilogarithm:
log 97.43 = 1.989; 1.989/4 = 0.497, antilog 0.497 = 3.14