Jul 14

Is this why are Chinese kids better at math?

Is this why Chinese kids are better at mathematics? China ignored Henry the Haberdasher, the first translator of Euclid's Elements from Greek to English.
Did Albert Einstein say, "If you can't explain it to a six year old, you don't understand it yourself"?

I began researching the history of elementary mathematics in 1983.

Yet I have never been more excited than in recent days.

In 1968, my teacher said ‘two multiplied by three’, is “two added to itself three times”. Yet as ‘two added to one three times’, is ‘1 + 2 + 2 + 2’, in reply to, “Who can tell me what two added to itself three times equals?”, I said, “Eight!”

Whilst right, I was left nonplussed.

Two multiplied by three, or ‘two added to itself three times’, cannot be six because ‘two added to one three times’ is seven.

I asked why 2 + 2 + 2 wasn’t ‘two added to itself two times’. My teacher paused, then said, “The first two is positive, so an optional positive sign at the start is the third addition sign.”

Thus, Euclid’s definition of multiplication was not to be successfully challenged by a boy yet to master his times tables.

Yet Euclid’s definition of multiplication was incorrectly translated in 1570 by a London haberdasher.Strangely, it has remained both in use and defective, for the past 444 years.
(I reveal this at www.jonathancrabtree.com/euclid/elements_book_VII_definitions.html)

ab, defined as ‘a added to itself b times’, is linguistically and mathematically incorrect.

a × 0 or 0, cannot equal a added to itself zero times, because a added to itself zero times equals a, not 0.

a × 1 or a, cannot equal a added to itself one time, because a added to itself one time equals 2a, not a.

Yet we keep reading "to multiply a by positive integral b is to add a to itself b times."
Source: www.collinsdictionary.com/dictionary/english/multiplication

English speaking countries have followed a faulty definition of multiplication, that in 300 BCE correctly meant, in the modern sense, a multiplied by b is simply 'a put together b times'.

One a becomes many a's. How many? As many a's as there are units in b.

So why do we follow a definition of multiplication, that even correctly translated, still predates the idea of 0 and 1 as numbers?

Any 21st century discrete multiplication definition needs updating for China’s idea of negative number and India’s idea of zero, and that is what I have done.

English Language Definition of Integral Multiplication

For the non-negative Naturals (N), a × b = a added to itself b times is to be recalled.

For the non-negative Naturals (N), a × b = a added to zero b times.

The above updated definition on the naturals is extended to the Integers (Z), as a × b = a either added to or subtracted from zero, b times, as per the sign of b.

The [long form] words below are implicit in the definition of integer multiplication on Z.
a × (+b) = [the total of] a added to zero b times [in succession]
a × (-b) = [the total of] a subtracted from zero b times [in succession]

ab, defined as ‘a added to itself b times’, is linguistically and mathematically incorrect.

a × 0 or 0, cannot equal a added to itself zero times, because a added to itself zero times equals a, not 0.

a × 1 or a, cannot equal a added to itself one time, because a added to itself one time equals 2a, not a.

Teachers, mathematicians, administrators and publishers are requested to undertake action to ensure a London haberdasher's definition of multiplication from February 1570 is recalled and updated in 21st century classrooms to reflect China's concept of negatives and India's concept of zero.

Further information is available at www.j.mp/multiplication-product-recall-notice


NOTE: I wrote about the updated multiplication definition for Z at point 3) above, in
'Why Multiplication 'IS' Repeated Subtraction Just as Multiplication 'IS' Repeated Addition'.

Which brings me to the Chinese Connection.
The English language edition of Euclid's Elements used for the Chinese edition of Books VII- XV in 1857 was that of London haberdasher, Henry Billingsley. It was Henry Billingsley that incorrectly translated Euclid's definition of multiplication, to read:

“A number is sayd to multiply a number, when the number multiplyed, is so oftentimes added to it selfe, as there are in the number multiplying unities: and an other number is produced.” Source: www.jonathancrabtree.com/mathematics/henry-billingsleys-definition-multiplication-euclids-elements/

The bolded phrase added to it selfe was never in Euclid's Greek. That was Henry the haberdasher's idea, and yet today, mathematics professors unknowingly quote Henry and not Euclid! (Examples below)

China was comfortable with negative numbers about 2000 years before English speaking countries. China also embraced India's idea of zero as a number and from the 14th century had no particular fondness of any mathematics in which 1 and 0 were not considered numbers.

Notably, despite it being repeated again and again in English editions of Euclid's Elements, Henry the Haberdasher's buggy definition of multiplication was NOT reproduced in the 1857 traditional Chinese translation of Alexander Wylie and LI Shan-lan.

Neither the word 'unit' nor the invented phrase 'added to itself' is in this Chinese translation of Euclid's definition of multiplication.


From the 16th century, the English just kept parroting a silly illogical definition of multiplication in the mistaken belief it was Euclid's words. Euclid NEVER defined multiplication as repeated addition!

Now, we have professors in the west who keep saying, in effect, 1 x 1 is 1 added to itself one time, or 2!

Yet China's utilitarian logic saw them reject such stupidity.

Our concept of number starts with zero and yet the west keeps using the non-mathematical word 'itself' and have long been mucking up the elements of arithmetic that originated in India.

Of course this isn't the only reason Chinese kids are good at math and there are other factors others have discussed.

Yet it's heartening to discover the Chinese didn't blindly follow a London haberdasher as so many mathematicians have done in the west.

There was some sniggling in my 1968 classroom when I said the answer to two added to itself three times was eight. Others who knew their times tables just accepted six as the answer.

And yet, there it is, in the Chinese 算法原本, Suan fa yuan ben (Elements of Calculation), "...adding two to itself three times must be equal to eight;..."

Source: The emperor's new mathematics : western learning and imperial authority during the Kangxi Reign (1662-1722) P.183, Professor Catherine Jami. Oxford University Press, 2012.

I was laughed at as a seven year old. Later I failed mathematics and repeated a year of school. Yet now it appears China is enjoying the last laugh in mathematics at the west's expense.

The tears in my eyes as I type are no longer from sadness, but from joy. (I wasn't so stupid after all...)

(They know mathematics, yet their words do not make mathematical sense.)

Example 1
Professor of applied mathematics, Dr. Steven Strogatz.
“Does seven times three, mean seven added to itself three times, or three added to itself seven times?”

This question, in simpler form, reveals a paradox.

"Does two times one, mean two added to itself one time, or one added to itself two times?" We can paraphrase this as, "Does two mean four or three?"

Source: The Joy of X: A guided tour of math, from one to infinity. Steven Strogatz P. 23. Houghton Mifflin Harcourt, New York 2012.
Excerpt available online at: www.scientificamerican.com/article/commuting-strogatz-excerpt

Example 2
Emeritus Professor of Mathematics, Dr. Hung-Hsi Wu.
“We call attention to the fact that mk, the multiplication of k by m, is a shorthand notation for adding k to itself m times, no more and no less. Please be sure to impress this fact on your students.” https://web.archive.org/web/20140412052638/http://math.berkeley.edu/~wu/EMI1c.pdf

Yet when m = 1 then mk = k + k = 2k according to the above, and yet (1)k = k.

For centuries, western mathematicians have been saying 3 multiplied by 2 is three added to itself twice which equals six.Yet adding a number to itself is doubling and three added to itself once equals six.

The irony is as a seven year-old I never understood how adding a positive number to itself twice could equal the same positive number added to itself once. I was told multiplication is only confusing if you think about it. Later I heard "Minus time minus results in a plus. The reason for this we need not discuss." Then later again I heard, "Ours is not to reason why, just invert and multiply."

Many mathematicians who themselves, do not understand the elements of arithmetic, have told their students to obey and not to think! And they say a benefit of studying math is it helps develop critical thinking...

That's enough therapy for one day!

Thank you for reading this post.

Best wishes,

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