Exponentiation is Repeated Division just as Exponentiation is Repeated Multiplication

As a child, I liked patterns, yet it seemed math didn't like me. How could a child like patterns yet not the Science of Patterns? (I'll answer in a moment...)

The American mathematician, Lynn Arthur Steen, wrote:

Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. (1)

The British mathematician, G. H. Hardy, wrote:

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. (2)

Despite all this talk of patterns, from what I see, it seems half our arithmetic is missing!

Yin and Yang (Negative and Positive)
OK, China was comfortable with positive and negative numbers more than 2000 years ago. To the Chinese, the idea of Yin and Yang, or opposing yet interconnected forces, fits beautifully with arithmetic.

In 263 CE, Liu Hui wrote a commentary on the ancient The Nine Chapters on the Mathematical Art, (九章算术 Jiǔzhāng Suànshù circa 100 CE). Liu Hui said:

I read the Nine Chapters as a boy, and studied it in full detail when I was older. I observed the division between the dual natures of Yin and Yang [the negative and positive negative aspects] which sum up the fundamentals of mathematics. (3)

In another article, I wrote about the Sanskrit of Brahmagupta, who in 628 CE defined zero to be the sum of a positive number and negative number of equal magnitude, सम-ऐक्यम् खम् (Brāhma Sphuta-siddhānta, Chapter 18:30a).

To me, a secret, seemingly lost, is that Yin and Yang can embed the concept of positive and negative into the unit.

Let me explain...

Assume the population of a town is 100. We can count births (positive) and we can count deaths (negative). Assume every year there are 10 births and 10 deaths. The number of 10 births equals the number of 10 deaths and so there is ZERO change.

There are 20 events a year, yet consistent with the complementary, interconnected roles of life and death, the opposing forces (births and deaths) cancel one another out.

So what happens when we take away the 'positive' yang events of birth?

Population start 100 + zero births + 10 deaths = population end 90. Change = 10 less.

And what happens when we take away the 'negative' yin events of death?

Population start 100 + 10 births + zero deaths = population end 110. Change = 10 more.

Taking away deaths, with no change in births results in an increase in population.

Subtracting a negative results in a positive.

Taking away debt results in an increase in wealth.
North/South, East/West, Up/Down, Debts/Assets, Births/Deaths are all dual concepts that require their 'additive inverse' in order to have meaning.

1 Yin debt + 1 Yang asset = zero net change in wealth

Taking away cold results in an increase in temperature.
1 Yin cold + 1 Yang heat = zero net change in temperature

So the opposing forces of Yin and Yang are part of our universe. Zero can be considered the sum of all numbers on our bi-directional number line. Just as multiplication can involve repeated addition from zero, it can also involve repeated subtraction from zero. (YinYang)

So if the idea of negative numbers has gone missing in the West, (it has) then maybe that can explain why half our arithmetic is missing!

The identity elements are one and zero. Any number multiplied by one, or divided by one, keeps its identity and remains unchanged. Any number that has zero added to it, or zero subtracted from it, keeps its identity and remains unchanged.

Returning the identity elements to arithmetic reveals patterns not often seen.

We'll start with simple integral multiplication.

a × ⁺4 = 0 + a + a + a + a
a × ⁺3 = 0 + a + a + a
a × ⁺2 = 0 + a + a
a × ⁺1 = 0 + a
a ×   0 = 0

The pattern, row by row, is as each integral multiplier reduces by one, we subtract an 'a'. So let's continue the pattern of integral multipliers reducing by one, line by line.

a × ⁺4 = 0 + a + a + a + a
a × ⁺3 = 0 + a + a + a
a × ⁺2 = 0 + a + a
a × ⁺1 = 0 + a
a ×   0 = 0
a × ^–1 = 0 – a
a × ^–2 = 0 – a – a
a × ^–3 = 0 – a – a – a
a × ^–4 = 0 – a – a – a – a

So as I revealed here, integral multiplication involves either repeated addition or repeated subtraction, depending on the sign of the multiplier.

OK, we reintroduced the identity element zero into the pattern of multiplication, so what if we reintroduced the identity element one into the pattern of exponentiation?

Everybody knows exponentiation is repeated multiplication. There are about 29,900 results for the Google search phrase "exponentiation is repeated multiplication" as can be seen when you click on this Google search link.

So just as there were pretty much zero Google search results for the phrase "multiplication is repeated subtraction", as I type this, there are zero results for the Google search phrase "exponentiation is repeated division" as can be seen when you click on this Google search link.

There are 29,900 results for the Google search phrase "exponentiation is repeated multiplication" and zero results for the Google search phrase "exponentiation is repeated division"

Let's see the simple pattern of exponentiation once we reintroduce the identity element one.

As we can't type superscript characters here, 'a to the power of two' or 'a squared' will be written a^2, where 'a' is the base and the '2' is the exponent.

a⁺⁴ = 1 × a × a × a × a
a⁺³ = 1 × a × a × a
a⁺² = 1 × a × a
a⁺¹ = 1 × a
a ⁰  = 1

The pattern, row by row, is as each integral exponent reduces by one, we divide by 'a'. So let's continue the pattern of integral exponents reducing by one, line by line.

a⁺⁴ = 1 × a × a × a × a
a⁺³ = 1 × a × a × a
a⁺² = 1 × a × a
a⁺¹ = 1 × a
a ⁰  = 1
a^–1 = 1 ÷ a
a^–2 = 1 ÷ a ÷ a
a^–3 = 1 ÷ a ÷ a ÷ a
a^–4 = 1 ÷ a ÷ a ÷ a ÷ a

The English language definition ab = a added to itself b times quoted by mathematicians for centuries is wrong because of an incorrect translation in 1570 of Euclid's definition of multiplication, that dates back to 300 BCE.

The Collins dictionary says an exponent is "a number or variable placed as a superscript to the right of another number or quantity indicating the number of times the number or quantity is to be multiplied by itself." Source: www.collinsdictionary.com/dictionary/english/exponent

The English language definition of exponentiation
a^b = a multiplied by itself b times is also wrong

In a^b, when b = 1, we are told that a^1 equals a multiplied by itself one time, or a × a. However, a × a is 'a squared' or ' a to the power of two', or a⁺² .

An exponentiation definition you won't see is a^b = one multiplied by a, b times in a row.

Yet rather than just correct a defective definition of exponentiation (I also correct a 447-year-old defective definition of multiplication), let's extend it from the naturals to the integers.

Look again and notice the patterns...

a⁺⁴ = 1 × a × a × a × a
a⁺³ = 1 × a × a × a
a⁺² = 1 × a × a
a⁺¹ = 1 × a
a ⁰  = 1
a^–1 = 1 ÷ a
a^–2 = 1 ÷ a ÷ a
a^–3 = 1 ÷ a ÷ a ÷ a
a^–4 = 1 ÷ a ÷ a ÷ a ÷ a

Exponentiation such as a⁺³ is 1 × a × a × a and exponentiation like a^–3 = 1 ÷ a ÷ a ÷ a, so integral exponentiation is either repeated multiplication or repeated division, as per the sign of the exponent!

To quote again from G H Hardy's "A Mathematician's Apology":

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.

Yes, multiplication involves repeated subtraction, as well as repeated addition and exponentiation, involves repeated division as well as repeated multiplication. Examples of authors discussing math as a science of patterns whose notes may now need updating, include Mathematics as a science of patterns, Michael D Resnik, Mathematics, the science of patterns, Keith Devlin and Mathematics as the Science of Patterns, Michael N. Fried.

I didn't like mathematics as a child because what I was seeing often didn't match what I was hearing from my teachers. So these articles are notes of my observations.

Exponentiation IS repeated division just as exponentiation IS repeated multiplication.

Yin and Yang. Go figure...

REFERENCES

1) Science, 1988, Vol.240, p.611-16
2) A mathematician's Apology, Cambridge University Press, 1940.
3) The Nine Chapters on the Mathematical Art: Companion and Commentary, Shen Kangshen, John N. Crossley and Anthony W. C. Lun, Oxford University Press, 2000.

Jonathan Crabtree
Mathematics Researcher:
The Evolution of Elementary Pedagogies
Melbourne Australia