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This book uses the notational conventions that are presented in this appendix.
The term, unitless value, means the numeric value of a dimensioned entity without its accompanying units of measure. Normally, when a symbol represents a dimensioned entity, the symbol includes a numeric value followed by a combination of factors, which are its units of measure.
For example, the symbol, h, means: 6.6 × 10-34 kg·m2·s-1. We know that the value, 6.6 × 10-34, is expressed in SI units because SI units follow the number. However, during the manipulation of an equation, we might want to cancel out the units in h with other units in the equation leaving h without its SI units. To specify this, we delimit the symbol, h, with vertical bars, as follows:
|h| = 6.6 × 10-34. However, in the SE of unit measures: h = 1. To distinguish between the two, we specify the system of unit measures as a subscript following the last vertical bar. Therefore:
|h|m = 6.6 × 10-34 (212
and
|h|e = 1 (213
This notational convention is useful when we manipulate the force equations.
This book displays fractions in-line rather than vertically-built, which precludes the necessity to create graphics boxes. A superscript suffix, -1, rather than a forward-slash prefix, /, specifies a denominator. An example is:
a b c (d e f)-1
which reads:
the product of a, b, and c divided by the product of d, e, and f.
Equations use a space to separate either two adjacent factors or one factor and an adjacent unit of measure. These factors are either a dimensionless number, such as 4 or pi, or a FUPCON, such as h, c, or ε0, that implicitly includes units of measure.
Equations use a centered dot to separate two adjacent units of measure. Examples are:
For SI units: s2·C2·kg-1·m-3 and kg·m2·s-1
For SE units: me·λe·qe-2 and λe3·me-1·te-2
For SG units: λg3·mg-1·tg-2
The metric system of unit measures is intended to measure physical phenomena that are on a human scale. However, if we use the Système International of units to measure magnitudes on either a microscopic or a celestial scale, very small or very large values can result. For instance: The distance to a remote galaxy might be 534,000,000,000,000,000,000,000,000.0 meters, or the diameter of an atom might be 0.000,000,000,215 meters.
These values display in fixed-point notation. The decimal point is fixed in a particular position, no matter the value. It is located between the units- and tenths-digit positions; therefore, the very large and very small numbers require the inclusion of many zeros. This method of representing these types of numbers is cumbersome because, when performing arithmetic operations, you must continually tally decimal-point-offset values one digit location at a time.
If we can keep track of the number of leading or trailing zeros, we need not display them. We can float the decimal point left or right until it is to the right of the most-significant digit in the number.
The significant digits are the string of contiguous digits that excludes leading and trailing zeros. In exponential notation, this string is the mantissa. The most-significant digit is the left-most non-zero digit in the string; the least-significant, the right-most.
Each time the decimal point moves left by one digit position, the value of the number is, in effect, divided by ten. Conversely, each move of the decimal point to the right multiplies the number by ten. To keep track of the number of moves and maintain the same value for the number, we include a suffix factor of a power of ten as part of the number.
Some examples of powers of ten are:
| Fractional | Decimal | Exponential |
| 1,000 | 1,000.0 | 103 |
| 100 | 100.0 | 102 |
| 10 | 10.0 | 101 |
| 1 | 1.0 | 100 |
| 1/10 | 0.1 | 10-1 |
| 1/100 | 0.01 | 10-2 |
| 1/1,000 | 0.001 | 10-3 |
We use exponential (scientific) notation to maintain the value of the number when we move the decimal point to either the left or the right by enough digit locations to place it to the right of the most-significant digit.
Each decimal-point offset changes the value of the number by a factor of ten . . . to the left, ten-times less . . . to the right, ten-times more. However, we must maintain the number's value by multiplying the significant digits by an appropriate power of ten. In essence, the value of this power of ten indicates by how many digit locations the decimal point is offset. A positive integer indicates that left offsets occur, and a negative integer, right offsets.
The exponential format is more economical at high powers of ten. For positive powers of ten, the magnitude of the exponent indicates the number of zeros to the left of the decimal point. For negative powers of ten, the magnitude of the exponent indicates one more than the number of zeros to the right of the decimal point.
The following examples clarify the notation:
Example Fixed-Point Floating-Point
1 -3450000000 = -3.45 × 109
2 +0.0000345 = +3.45 × 10-5
3 3.14159 = 3.14159 × 100
An exponential-notation number contains, from left to right, six components:
1) Sign of the mantissa (+ or -)
2) Value of the mantissa (significant figures)
3) Multiplication sign (×)
4) Base number (usually 10)
5) Sign of the characteristic (or exponent) (+ or -)
6) Magnitude of the characteristic (or exponent).
In the first example, above, the sign of the mantissa is negative (-). It indicates the sign of the complete number. In the second and third examples, the mantissa sign is positive (+). If no sign exists, the mantissa is positive by implication.
The value of the mantissa is expressed as a value from one to, but not including, ten. The accuracy of a number is a function of the number of digits in the mantissa. In the above examples, the values of the mantissas are: 3.45, 3.45, and 3.14159.
The base number for the exponent in exponential notation is ten because it must match the base number of the mantissa, which, as a decimal number, also uses ten as its base number.
The sign of the characteristic ( or exponent) indicates the direction that the decimal point floats from its fixed-point location. It does not indicate the sign of the complete number. A positive sign indicates a float to the left; a negative sign, to the right. In the above examples, the exponent signs are: positive, negative, and positive.
The value of the characteristic (or exponent) is a whole number and indicates the power to which the base number is raised or by how many digit locations that the decimal point is floated away from its fixed-point location.
Some examples of exponential notation are, as follows:
The mass of the electron in exponential format is equal to:
me = 9.109,389 ... × 10-31 kg
and its threshold temperature:
ke = 5.929,861 ... × 109 K
The value of the circumference-to-diameter ratio of a circle in fixed-point format is:
π = 3.141,592,653 ...
In exponential format, it is:
π = 3.141,592,653 ... × 100
where the factor, 100, is equal to one. When a factor is equal to one, you can remove it from the calculation without changing its value. Therefore, the value of the ratio π need not be expressed in exponential format.
In equations that pertain to physics, we encounter factors more often than we do terms. Therefore, we multiply and divide more often than add and subtract. Exponential notation is particularly powerful when we multiply and divide numbers. In essence, we add and subtract when multiplying and dividing powers of ten.
The following examples demonstrate the ease of manipulating a string of factors that are composed of powers of ten:
Powers of Ten Value
10-34 × 10-3 × 10-7 × 10-5 = 10-49
1034 × 10-3 × 10-7 × 105 = 1029
1034 × 103 × 107 × 105 = 1049
10-34 × 103 × 107 × 1024 = 1.
To date, the Internet's HTML format does not recognize some mathematical symbols. When viewing this book on the Internet, these symbols are substituted by other characters as shown in the following table:
| Constant or Mathematical Symbol | Usual Symbol | Substitute Symbol |
| Planck's constant / 2π | h bar | hbar |
| infinity | infinity symbol | î |
| proportional | proportional symbol | Þ |
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