(M-14) Powers of Numbers

Raising a numbers to the power which is a positive whole number

The concept of logarithms arose from that of powers of numbers. If the properties of powers are familiar to you, you may quickly skim through the material below. If not--well, here are the details. Powers of a number are obtained by multiplying it by itself. For instance .

2.2 can be written 2²	"Two squared" or "2 to the 2nd power"
2.2.2 = 2³	."Two cubed" or "2 to the 3rd power""
2.2.2.2 = 2⁴	"Two to the 4th power" or simply "2 to the 4th""
2.2.2.2.2 = 2⁵	"Two to the 5th power" or simply "2 to the 5th""
2.2.2.2.2.2 = 2⁶	"Two to the 6th power" or simply "2 to the 6th""

and so on...

The number in the superscript is known as an "exponent." The special names for "squared" and "cubed" come because a square of side 2 has area 2² and a cube of side 2 has volume 2³. Similarly, a square of side 16.3 has area (16.3)² and a cube of side 9.25 has volume (9.25)³. Note the use of parentheses--they are not absolutely needed, but they help make clear what is raised to the second or 3rd power.

Quick Quiz:

The Greek Pythagoras showed (about 500 BC) that if (a,b,c) are lengths of the sides of a right-angled triangle, with c the longest, then a² + b² = c² In a right angles triangle, a = 12, b = 5. Can you guess c?
Which is larger--2³ or 3²? 2⁷ or 5³?
A slight modification of an old riddle goes:
It all involves powers of 7:
Total count: 1 + 7 + 343 + 2401 = 2752
As noted, this is slightly modified from the original riddle, which asks "how many were going to St. Ives?" The answer is of course just one, the person telling the riddle. Many listeners however are distracted by the many details given, miss the difference and perform the above calculation. Their answer is then wrong!
The famous Indian mathematician Ramanujan was sick in a hospital (tuberculosis, probably) when he was visited by his friend the mathematician G.H Hardy, who had earlier invited him to England. Hardy later told:
Cubes are third powers. What are they, in this example? Try guessing, choices are limited.

Multiplying powers

Note that (2³).(2²) = 2⁵
since the first term contributes three factors of 2 and the second term contributes two--together, 5 multiplications by 2. The same will hold if "2" is replaced by any number. So, if that number is represented by "x" we get (x³).(x²) = x⁵
and in general (since there is nothing special about 2 and 3 which will not hold for other whole numbers) (x^a).(x^b) = x^(a+b) where a and b are any whole numbers.
The most widely used powers by whole numbers, for users of the decimal system, are of course those of 10

10¹ = 10 ("ten")
10² = 100 ("hundred")
10³ = 1000 ("thousand")
10⁴ = 10,000 ("ten thousand"))
10⁵ = 100,000 ("a hundred thousand")
10⁶ = 1,000,000 ("a million")

Note that here the "power index" also gives the number of zeros. For larger numbers, it used to be that in the US 10⁹ = 1,000,000,000 was called "a billion" while in Europe it was called a "milliard" and one had to advance to 10¹² to reach a "billion." These days the US convention is gaining ground, but the world remains divided between nations where the comma denotes what we call "the decimal point", while the point divides large numbers, e.g. 10⁹ = 1.000.000.000 (in the US commas would be used).

It also should be noted that some cultures have assigned names to some other powers of 10--e.g. the Greeks used "myriad" for 10,000 while the Hebrew Bible named it "r'vavah," and in India "Lakh" still means 100,000, while "crore" is 10,000,000. A 9-year old in 1920 coined the name "Googol" for 10¹⁰⁰, but the word found little use beyond inspiring the name of a search engine on the world-wide web.

Dividing one power by another

In a manner very similar to the above, we can write (2⁵) / (2²) = 2³ since dividing a power of 2 by some smaller power means canceling from the numerator a number of factors equal to those in the denominator. Writing it out in detail

(2.2.2.2.2) / (2.2) = 2.2.2
Here too the number raised to higher power need not be 2--again, denote it by x--and the powers need not be 5 and 2, but can be any two whole numbers, say a and b:

(x^a) / (x^b) = x^(a–b)
Here however a new twist is added, because subtraction can also yield zero, or even negative numbers. Before exploring that direction, it helps outline a general course to follow.

Expanding the meaning of "Number"

Back at the dim beginnings of humanity, "numbers" simply meant positive whole numbers ("integers"): one apple, two apples, three apples...

Simple fractions were also found useful--1/2, 1/3 and so on.

Then zero was added, originally from India.

Then negative numbers were given full status--rather than view subtraction as a separate operation, it was re-interpreted as addition of a negative number.

Similarly, to every integer x there corresponded an "inverse" number (1/x) (many calculators have a 1/x button too). In ancient Egypt, 5000 years ago, these were the only fractions recognized, and they are therefore still sometimes called "Egyptian fractions." When an Egyptian of that time wanted to express 3/4, it was presented as (1/2 + 1/4). Sometimes long expressions were needed, e.g

99/100 = 1/2 + 1/4 + 1/5 + 1/25 but it always worked.

The ancient Greeks went further and defined as "rational number" (or "logical" numbers--"rational" comes from Latin) any multiple of such an inverse, for instance 4/13, 22/7 or 355/113. Rational numbers are dense: no matter how close two of them are to each other, one could always place another rational number between them--for instance, half their sum is one choice out of many. Decimal fractions which stop at some length are rational numbers too, though decimal fractions having infinite length but with a repeating pattern (0.33333..., 0.575757... etc.) can always be expressed as rational fractions.

Greek philosophers in the early days of mathematics were therefore surprised to find that in spite of that density, some extra numbers could still "hide" between rational ones, and could not be represented by any rational number. For instance, √2 is of this class, the number whose square equals 2. Most square roots and solutions of equations are also of this kind, as is π, the ratio between the circumference of a circle and its diameter (denoted by the Greek letter "pi"). Pi has a fair approximation in 22/7 and a much better one in 355/113, but its exact value can never be represented by any fraction. Mathematicians view all the preceding types of number as a single class of "real numbers".

Logarithms of positive numbers are real numbers, too. When one writes

2 = 10^0.3010299.. so that 0.3010299.. = log 2
(the dots represent an irregular continuation) one views it as 10 raised to a power which is some real number. Earlier, powers were integers, denoting the number of times some number was multiplied by itself. To make the above expression meaningful, it is therefore necessary to generalize the concept of raising a number to some power to where any real number can be the power index.

Logarithms of powers of 10

These are all whole numbers:

10¹ = 10	so log 10 = 1
10² = 100	so log 100 = 2
10³ = 1000	so log 1000 = 3
10⁴ = 10,000	so log10,000 = 4
10⁵ = 100,000	so log 100,000 = 5
10⁶ = 1,000,000	so log 1,000,000 = 6

These logarithms also satisfy the rules we found

(x^a).(x^b) = x^(a+b)

So if x=10
U = (10^a) V = (10^b) W = (10^(a+b)) = U.V
then since
a = log U b = logV (a+b) = log W
we have
logV + log U = log (U.V)

This relation holds whenever U and V are powers of 10:

The logarithm of the product is the sum of the logarithms
of the multiplied numbers.

As the concept of logarithm is broadened, that property always remains. That is what originally made logarithms useful: converting multiplication into addition. Instead of having to multiply U and V, we only need add their logarithms and then look for the number whose logarithm equals that sum: that will be the product (U.V).

Similarly,
(x^a) / (x^b) = x^(a–b)
so if x=10,
U = (10^a) V = (10^b) W = (10^(a–b)) = U/V

then in the division we have logU – log V = log (U/V)
or "the logarithm of the quotient is the difference between the logarithms of the divided numbers," e.g. 10⁷ / 10⁴ = 10³ which agrees with 7 – 4 = 3. Division, though, opens up a new territory: by the same rule, for instance

10⁴⁰ / 10⁴³ = 10^–3 = 0.001
And
10⁴ / 10⁴ = 10⁰ = 1

since a number is being divided by itself must equal 1. Indeed, this is consistent with the rule, the adding or subtracting 1 to the logarithm moves its number one decimal to the right of left. Earlier

10⁶ = 1,000,000	so log 1,000,000 = 6
10⁵ = 100,000	so log 100,000 = 5
10⁴ = 10,000	so log10,000 = 4
10³ = 1000	so log 1000 = 3
10² = 100	so log 100 = 2
10¹ = 10	so log 10 = 1

and now this can be extended, dividing by 10 at each step

10⁰ = 1	so log 1 = 0
10^–1 = 0.1	so log 0.1 = –1
10^–2 = 0.01	so log 0.01 = –2
10^–3 = 0.001	so log 0.001 = –3
10^{– 4} = 0.000 1	so log 0.000 1 = –4
10^–5 = 0.000 01	so log 0.000 01 = –5
10^–6 = 0.000 001	so log 0.000 001 = –6

The above demonstrates another property of logarithms: Log (V^Q) = Q log V For the special case V = 10, logV = 1

Scientific Notation

The quantities with which scientists work are sometimes very small or very large. It is then convenient (for calculation, and also for applying logarithms) to separate the number into two parts--a number from 1 to 10, giving its structure, and a power of 10, giving the magnitude.

Electric charge, for instance, is measured in coulombs: about one coulomb flows each second through a 100-watt lightbulb. That current is carried by a huge number of tiny negative particles, found in any atom and known as electrons. Each electron carries a charge of

q = 1.60219 10^–19 coulomb
If this were to be written as a decimal fraction, the expression would take about half a line, with 18 zeros following the decimal point in front of the significant digits--and a quick look at it would not give much information, one still would have had to count the zeros. The mass of the electron is similarly small

m = 9.1095 10^–29 kg
Scientific notation simplifies writing such numbers. Yet another example is the speed of light, as decimal number (accuracy to 6 figures) 299,792,000, in scientific notation

c = 2.99792 10⁸ meter/second
Scientific notation also makes multiplication and division easier and less error prone. One multiplies or divides separately the numerical factors, each between 1 and 10, and usually sees at a glance if the result is of the right range of magnitude. Separately, one adds together all power exponents of multiplied factors, and subtracts those of divided ones, to get the appropriate power of 10 which then appears in scientific notation.

Of course, in any calculation, one must use consistent units--it would not do to mix meters and inches, or pounds and grams (such inconsistent use apparently led to an error which caused a space probe to Mars to miss the planet and get lost). The most common consistent system in physics and technology is the MKS system, measuring distance in meters, mass in kilograms and time in seconds. All other units are determined by the choice of these three standards, and as long as one stays in the MKS system, results conform to units of that system too (e.g. if energy is being calculated, it always comes out in joules).

An example

Electrons of the polar aurora ("northern lights") move at about 1/5 the velocity of light, in a magnetic field B which near the ground is about 5 10^–5 Tesla (the Tesla is the MKS unit of magnetic field: at the pole of an iron magnet you get about 1 Tesla). The magnetic field causes an electron to spiral around the direction of the magnetic force ("magnetic field line") with a radius of

r = mv/(qB)
where v is the part of the velocity perpendicular to the direction of B. If the component perpendicular to B is half the total velocity (i.e c/10), what is r?
We have

^–29

⁷

^–19

^–5

Collecting all numerical factors, and rounding off to 3 decimals (9.11).(3.00)/[(1.6).(5)] = 3.42 Collecting all exponents (– 29+7) – (–19 – 5) = (– 22) – (– 24) = +2 The radius is therefore 3.42 10² meters or 342 meters. That is the order of the radius of a very thin auroral ray, seen from the ground. Considering that the ground from which aurora is usually viewed is 100 kilometers below the aurora, such a ray must appear as very thin indeed.

Next section: (M-15) Raising one Power to Another

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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .

Updated 9 November 2007, edited 28 October 2016

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