Maths/Powers and logarithms

Growth by a constant factor

After thiswhat you will be able to doExpand powers into repeated multiplication, extend exponents to zero, negative, and half counts, and define e from increasingly fine compounding.

Questionwhat this lesson answersRepeated multiplying eventually outruns repeated adding. What must the shorthand mean when its count is not whole, and where does e come from?

Not coveredwhat this lesson leaves outWe do not prove that repeated finer crediting settles toward one number. We do not use calculus to study e. We do not locate the first overtaking step.

Suppose a colony begins with one cell. After each round, every cell has become two cells. For comparison, start another record at one and add two after every round.

The first record reads longhand as repeated multiplication:

The second record reads longhand as repeated addition:

The totals now sit beside one another:

Both records repeatedly use the number 2. The first multiplies by it. A number used in multiplication is called a factor. The second adds it. By round five, the first record has reached 32 and the second has reached 11. The separation keeps widening.

A raised count counts factors

Writing every factor soon becomes awkward. The raised count stores how many factors are present. ${2^5}$ means five twos multiplied together.

It does not mean five twos added. It does not mean two times five. The repeated number is called the base. The raised count is called the exponent.

Joining factor lists adds their counts

Start with three factors of 2, then multiply by four more factors of 2.

Three factors followed by four factors make seven factors. Let ${b}$ name any positive base. Let ${a}$ and ${c}$ name two positive whole counts. The same counting gives the addition law:

This law follows from counting the factors. It is not an extra rule chosen afterward.

The addition law extends the definition

So far, an exponent has counted a positive whole number of factors. The addition law tells us what other exponents must mean if that law keeps working.

For a zero count, the law requires

Only multiplying by 1 leaves ${2^3}$ unchanged. Therefore ${2^0 = 1}$, and the same reasoning gives ${b^0 = 1}$ for every positive base.

A negative count must undo a positive count:

Therefore ${2^{-3}}$ must be one divided by ${2^3}$, which is ${1/8}$. In general, ${b^{-n} = 1/b^n}$. One divided by a number is called its reciprocal.

A half count must combine with another half count to make one full count:

So ${2^{1/2}}$ is the positive number that gives 2 when multiplied by itself. It is about 1.414214. Likewise, ${b^{1/2}}$ is the positive number whose square is ${b}$.

The notation began as a definition for whole counts. The addition law decides what other counts must mean if the law remains consistent. This extends a definition. It does not discover a new physical fact.

Crediting more often approaches one number

Start with a balance of 1. Suppose it gains 100 percent over one year. Credit the gain once, and the final balance is 2. Split the credit into equal periods, and each period changes the current balance before the next period begins. This repeated crediting of gains on earlier gains is called compounding.

Ten thousand periods give about 2.718146. The totals rise, but the added amount keeps shrinking. They remain below 2.718282 and settle toward that number. More multiplications occur, but each multiplication is gentler. Those two changes balance rather than making the total run away.

Let ${n}$ name the number of equal periods. Each period multiplies by ${1 + 1/n}$, and there are ${n}$ such factors. The final total is therefore ${(1 + 1/n)^n}$.

An ordered list of numbers like these totals is called a sequence. A number approached by a sequence is called its limit. We define ${e}$ as the limit of these compounding totals. Its value begins ${2.718282}$. The definition names the destination. Proving that the sequence really has one destination is separate mathematics.

No arbitrary growth base was selected. The number appeared by splitting one whole increase into ever finer equal parts. This compounding construction is why ${e}$ is called the natural base for growth.

Changing the base multiplies every count

The same total can use different repeated factors. A base of 4 contains two factors of 2:

A count of three in base 4 becomes six in base 2. A count of five becomes ten. Changing from base 4 to base 2 always multiplies the count by the fixed number 2. Base 8 would multiply it by 3. Other base changes can need a non-whole multiplier, which the extended notation now permits. The multiplier depends on the two bases, not on the original count.

What power notation does not tell you

Growth made by adding one fixed amount is called linear. Growth made by multiplying by one fixed factor is called exponential. For every factor above 1, the exponential quantity eventually outruns any fixed multiple of the step count.

The word “eventually” can hide a long wait. At step 1,000, ${1.01^{1000}}$ is about 20,959. That is still below 100 times the step count, which is 100,000. Power notation alone supplies no first step where one quantity overtakes the other. That step is called the crossover. An eventual comparison needs an argument that finds a step after which the result keeps holding.

The notation also makes no promise about the world. A real colony can run out of food. A balance follows the credit rules its institution actually uses. Whether any measured quantity keeps one factor must be answered by observation. That makes it an empirical question, even when the power arithmetic is exact.