Subscripts, Superscripts, Exponents for Physics Help

Introduction

Subscripts are used to modify the meanings of units, constants, and variables. A subscript is placed to the right of the main character (without spacing), is set in smaller type than the main character, and is set below the baseline.

Superscripts almost always represent exponents (the raising of a base quantity to a power). Italicized lowercase English letters from the second half of the alphabet ( n through z ) denote variable exponents. A superscript is placed to the right of the main character (without spacing), is set in smaller type than the main character, and is set above the baseline.

Examples Of Subscripts

Numeric subscripts are never italicized, but alphabetic subscripts sometimes are. Here are three examples of subscripted quantities:

Ordinary numbers are rarely, if ever, modified with subscripts. You are not likely to see expressions like this:

3 ₅

−9.7755π

−16 _x

Constants and variables, however, can come in many “flavors.” Some physical constants are assigned subscripts by convention. An example is m _e , representing the mass of an electron at rest. You might want to represent points in three-dimensional space by using ordered triples like ( x ₁ , x ₂ , x ₃ ) rather than ( x, y, z ). This subscripting scheme becomes especially convenient if you’re talking about points in a higher-dimensional space, for example, ( x ₁ , x ₂ , x ₃ , ..., x ₁₁ ) in cartesian 11-space. Some cosmologists believe that there are as many as 11 dimensions in our universe, and perhaps more, so such applications of subscripts have real-world uses.

Examples Of Superscripts

Numeric superscripts are never italicized, but alphabetic superscripts usually are. Examples of superscripted quantities are

There is a significant difference between 2 ³ and 2! There is also a difference, both quantitative and qualitative, between the expression e that symbolizes the natural-logarithm base (approximately 2.71828) and e ^x , which can represent e raised to a variable power and which is sometimes used in place of the words exponential function .

The “times Sign”

The multiplication sign in a power-of-10 expression can be denoted in various ways. Most scientists in America use the cross symbol (X), as in the preceding examples. However, a small dot raised above the baseline (·) is sometimes used to represent multiplication in power-of-10 notation. When written this way, the preceding numbers look like this in the standard form:

2.56 · 10 ⁶

8.0773 · 10 ⁻¹⁸

1.000 · 10 ⁰

This small dot should not be confused with a radix point, as in the expression

m.n · 10 ^z

in which the dot between m and n is a radix point and lies along the baseline, whereas the dot between n and 10 ^z is a multiplication symbol and lies above the baseline. The dot symbol is preferred when multiplication is required to express the dimensions of a physical unit. An example is kilogram-meter per second squared, which is symbolized kg · m/s ² or kg · m · s ⁻² .

When using an old-fashioned typewriter or a word processor that lacks a good repertoire of symbols, the lowercase nonitalicized letter x can be used to indicate multiplication. But this can cause confusion because it’s easy to mistake this letter x for a variable. Thus, in general, it’s a bad idea to use the letter x as a “times sign.” An alternative in this situation is to use an asterisk (*). This is why occasionally you will see numbers written like this:

2.56* 10 ⁶

8.0773* 10 ⁻¹⁸

1.000*10 ⁰

Plain-text Exponents

Once in awhile you will have to express numbers in power-of-10 notation using plain, unformatted text. This is the case, for example, when transmitting information within the body of an e-mail message (rather than as an attachment). Some calculators and computers use this system. The uppercase letter E indicates 10 raised to the power of the number that follows. In this format, the preceding quantities are written

2.56E6

8.0773E − 18

1.000E0

Sometimes the exponent is always written with two numerals and always includes a plus sign or a minus sign, so the preceding expressions appear as

2.56E + 06

8.0773E − 18

1.000E + 00

Another alternative is to use an asterisk to indicate multiplication, and the symbol ^∧ to indicate a superscript, so the expressions look like this:

2.56* 10 ^∧ 6

8.0773*10 ^∧ − 18

1.000* 10 ^∧ 0

In all these examples, the numerical values represented are identical. Respectively, if written out in full, they are

2,560,000

0.0000000000000000080773

1.000 

Practice problems for these concepts can be found at: Scientific Notation for Physics Practice Test