The Big X - Numberphile

Summary

In this engaging video script, the presenter explores 'The Ground of Arts,' a pioneering arithmetic book by Robert Record, published in 1542. It aimed to bring England's arithmetic skills in line with the rest of Europe by introducing the Indo-Arabic numeral system, including the revolutionary concept of 'cipher' or zero. The script also delves into the historical significance of the cipher, its impact on arithmetic, and its playful references in Shakespeare's works. The presenter speculates on the origin of the multiplication symbol and its possible connection to Record's times table technique, offering a unique perspective on the evolution of mathematical symbols.

Takeaways

• 📚 'The Ground of Arts' is a significant historical arithmetic book written by Robert Record in 1542, aimed at the general public in English to explain arithmetic concepts.
• 🌐 The book was a response to England's lag in arithmetic knowledge compared to the rest of Europe, particularly due to the country's break from the Catholic Church and the Renaissance's influence.
• 🔢 It introduced the Indo-Arabic numeral system to the English public, including the concept of place value and the revolutionary 'cipher' or zero, which simplified arithmetic significantly.
• 🎲 Robert Record made learning arithmetic more approachable by incorporating verses and a friendly tone, demystifying the subject for the general populace.
• 📖 The term 'cipher' for zero was not universally used; it was a term that emerged later, and at the time, people referred to it as 'nothing' or 'nil'.
• 🤔 The script ponders the origin of the multiplication symbol, suggesting it might have been inspired by the 'X' shape of the times table technique introduced in 'The Ground of Arts'.
• 🎨 Shakespeare, who was part of the first generation to learn this new arithmetic, made references to it in his plays, indicating the impact of the cipher and arithmetic on culture.
• 🧩 The script highlights the ingenuity of the times table technique for tricky multiplications, such as 8 * 7, and demonstrates how it works with an algebraic explanation.
• 📐 It also discusses the practical application of arithmetic in the book, including long multiplication and word problems that reflect the everyday life of the Elizabethan era.
• 🤓 The video script speculates on the influence of 'The Ground of Arts' on the development of mathematical symbols and the cultural significance of arithmetic during the English Renaissance.
• 🔍 The script encourages viewers interested in the intersection of mathematics and culture to explore further resources, such as Rob's book on numbers and the Objectivity channel.

Q & A

What is the book 'The Ground of Arts' about?

'The Ground of Arts' is an arithmetic book first written in 1542 by Robert Record. It was the first book aimed at the general public in England to explain how to do arithmetic using the new Indo-Arabic numeral system.

Why was 'The Ground of Arts' significant in England during its time?

The book was significant because it helped England catch up with the rest of Europe in arithmetic, which was important since England had fallen behind due to the Reformation and the break from the Catholic Church.

What numeral system does 'The Ground of Arts' introduce?

The book introduces the new Indo-Arabic numeral system, which included the use of zero, referred to as 'cipher' in the book.

How does Robert Record explain the concept of zero in his book?

Robert Record refers to zero as 'cipher,' a term derived from Arabic. He explains that this symbol, despite representing nothing, is crucial in making numbers large and revolutionizing arithmetic.

What multiplication technique does Robert Record introduce in 'The Ground of Arts'?

Record introduces a technique for doing times tables, which involves drawing diagonals and using number bonds to make multiplication easier. This method was particularly useful for multiplying single-digit numbers.

Does Robert Record's multiplication technique work for numbers outside the range of 6 to 9?

Yes, the technique works for numbers outside the range of 6 to 9, though it may seem unnecessarily complex for such cases. The algebra behind the method shows that it is valid for any numbers.

How were long multiplication problems addressed in 'The Ground of Arts'?

The book includes practical word problems that require long multiplication. Record demonstrates how to solve these problems without the modern multiplication symbol, which had not been invented yet.

Who is credited with inventing the multiplication symbol, and how might Robert Record's work have influenced this?

William Oughtred is credited with inventing the multiplication symbol around 1630. It's suggested that he may have been inspired by Record's technique of drawing diagonals, which resembles an 'X'.

How does Shakespeare reference arithmetic in his plays, and what might this indicate?

Shakespeare refers to arithmetic in his plays, such as 'Romeo and Juliet' and 'Henry V'. This indicates that he was likely familiar with arithmetic concepts from 'The Ground of Arts', which was published shortly before his time.

What is a notable mathematical reference in Shakespeare's 'Henry V'?

In 'Henry V', the chorus refers to himself as a 'crooked figure a one' and the other actors as 'ciphers'. This reflects the arithmetic concept introduced by Record that small symbols, like zero, can significantly expand numbers.

Why is the term 'cipher' significant in the context of Robert Record's work and Shakespeare's plays?

The term 'cipher' is significant because it represents the concept of zero, which was a new and revolutionary idea in arithmetic. Shakespeare's references to 'cipher' in his plays suggest that he found this concept intriguing and important.

Outlines

00:00

📚 Introduction to 'The Ground of Arts' Arithmetic Book

The video script introduces 'The Ground of Arts,' an arithmetic book written by Robert Record in 1542. It was the first of its kind to explain arithmetic to the general English-speaking public. The book played a role in helping England catch up with other parts of Europe that were already adept at arithmetic. England had been somewhat behind due to its break from the Catholic Church and the Renaissance's delayed impact. The script highlights the book's unique approach, including the use of verses to make learning fun and the introduction of the Indo-Arabic numeral system, which was revolutionary at the time, especially the concept of 'zero' or 'cipher,' which was a new and somewhat mystical concept in mathematics.

05:00

🔢 The Innovation of the Indo-Arabic Numeral System

This paragraph delves into the specifics of the Indo-Arabic numeral system introduced in 'The Ground of Arts.' It discusses the significance of the cipher (zero) and its representation, which was a major innovation in arithmetic. The script also explains Robert Record's method for teaching multiplication, particularly for numbers 6 through 9, using a clever technique involving diagonal lines and simple arithmetic. The method is demonstrated through the multiplication of 8 by 7 and then 8 by 9, showing how this system simplifies the process. The script also touches on the broader implications of the cipher's introduction, noting that it allowed for more efficient arithmetic without the need for complex Roman numerals.

10:01

🎭 The Influence of Arithmetic on Shakespeare and the Multiplication Symbol

The final paragraph explores the impact of 'The Ground of Arts' and the new arithmetic concepts on English literature and society, particularly on William Shakespeare. It suggests that the book may have influenced Shakespeare's use of arithmetic in his plays, such as the reference to 'the book of arithmetic' in 'Romeo and Juliet.' The script also speculates on the origin of the multiplication symbol, proposing that it might have been inspired by Record's times table technique. The paragraph discusses the broader cultural significance of the cipher and arithmetic during Shakespeare's time, indicating that these mathematical concepts were not only educational but also found their way into the creative arts and everyday language.

Keywords

💡The Ground of Arts

The Ground of Arts is the title of a book first written in 1542 by Robert Recorde. It is significant because it was the first book aimed at the general public to explain arithmetic in English. The book is central to the video's theme as it represents a historical milestone in making arithmetic accessible to the English public, and it is the main subject of the video's discussion.

💡Arithmetic

Arithmetic is the branch of mathematics dealing with the properties and manipulation of numbers. In the context of the video, arithmetic is the subject matter of 'The Ground of Arts,' and the video discusses how the book introduced and simplified arithmetic concepts for the general public in England during the 16th century.

💡Indo-Arabic numeral system

The Indo-Arabic numeral system is the base-10 number system that includes the digits from 0 to 9, which is the most common system used for numerical calculations worldwide. The video explains how Robert Recorde's book introduced this system to the English public, emphasizing the importance of the numeral system in advancing arithmetic education.

💡Place value

Place value is the numerical value assigned to a digit based on its position in a number. The video mentions how Recorde's book illustrated place value, showing the significance of each digit's position in forming numbers, which is a fundamental concept in arithmetic and the Indo-Arabic numeral system.

💡Cipher

In the context of the video, 'cipher' refers to the numeral '0', which was a new concept introduced in 'The Ground of Arts'. The term 'cipher' is used to explain the historical term for zero and its importance in revolutionizing arithmetic by allowing for a more efficient representation of numbers.

💡Times tables

Times tables are mathematical tables used to easily multiply numbers together. The video describes how Robert Recorde introduced a method for memorizing and calculating times tables, particularly focusing on the trickier multiplications, which is a key part of the arithmetic education presented in his book.

💡Long multiplication

Long multiplication is a method for multiplying two numbers together, typically used for larger numbers. The video shows an example from 'The Ground of Arts' where long multiplication is used to solve a practical problem, illustrating how the book taught this technique to the public.

💡Multiplication symbol

The multiplication symbol, represented by '×', is used to denote the operation of multiplication. The video discusses the absence of a standard multiplication symbol in Recorde's time and suggests a possible origin for the symbol inspired by the times table technique in 'The Ground of Arts'.

💡Shakespeare

William Shakespeare is mentioned in the video as the first generation to learn arithmetic with the new numeral system and concepts introduced by books like 'The Ground of Arts'. The video suggests that Shakespeare's works contain references to arithmetic and the new concepts of numbers, indicating the cultural impact of arithmetic education at the time.

💡Renaissance

The Renaissance was a period of cultural, artistic, political, and economic rebirth in Europe. The video refers to the English Renaissance and how England was somewhat behind other parts of Europe in terms of arithmetic education, which 'The Ground of Arts' helped to address.

💡Robert Recorde

Robert Recorde was a Welsh physician and mathematician who authored 'The Ground of Arts'. The video highlights his contribution to making arithmetic understandable to the English public and discusses his innovative teaching methods and the historical context of his work.

Highlights

The Ground of Arts is a math book written in 1542 by Robert Record, aimed at the general public to explain arithmetic in English.

The book was written to help England catch up with the rest of the world in arithmetic, as England was behind due to breaking away from the Catholic Church and missing the Renaissance.

Robert Record made math fun by adding little verse bits to make the book engaging.

The book introduces the new Indo-Arabic numeral system, showing place value and the concept of zero.

The word for zero at the time was 'cipher', derived from the French word 'chiffre' which means a digit.

Record's book includes a method for doing times tables using a unique technique involving diagonal lines and number bonds.

This method works by finding the number bonds to 10, multiplying these numbers, and then using diagonals to simplify the calculation.

The method is demonstrated with examples like 8 x 7 and 8 x 9, and an algebraic explanation of why it works is provided.

The technique even works with numbers outside the 6 to 9 range, as demonstrated with the example 5 x 13.

Robert Record's book also teaches long multiplication through practical word problems, such as calculating the total number of lambs owned by 264 men.

At the time of the book's writing, there was no multiplication symbol, which was later introduced by William Oughtred around 1630.

It's speculated that the multiplication symbol may have been inspired by Record's technique of drawing diagonals.

Shakespeare, who lived around the same time as the book's publication, made several references to arithmetic and the concept of zero in his plays.

In Romeo and Juliet, a character refers to fighting by the book of arithmetic, which might be a nod to Record's book.

In Henry V, Shakespeare uses the concept of cipher to explain how a small number of actors can represent a large army.

The transcript suggests that Shakespeare was excited by the new arithmetic concepts, using them as sophisticated references in his works.

Transcripts

so this is a book called The Ground of

arts and it's actually a math book and

more specifically it's an arithmetic

book it's a book first written in 1542

by Robert record and it's the first book

aimed at the general public to explain

in English how to do arithmetic so we

can catch up in England with the rest of

the world who always already pretty good

at doing arithmetic was England behind

the curve was it behind the curve yeah

because of breaking away from the

Catholic Church and all that it was like

we were behind the Renaissance that

happened in like Italy and was working

its way through Europe so this is just

before the English Renaissance and it's

quite a fun book he he adds little verse

bits in here and there just to make it

fun you know Matt is fun it's not too

scary and so on ground of Arts is a

strange name for a maths book but uh

Arts and Sciences and maths obviously

the same thing to him let's turn to an

important page because what he is

introducing people to

is the new indo-arabic numeral system

that had had come in as he got a little

table here so uh he's showing place

value so we've got uh nine units nine

tens hundreds whatever he's just written

the nine all the way across uh and says

this is nine which is a word that they

would have known eights can be placed in

any place value up to millions and tens

of millions 7 6 5 4 3 2 uh 1 and then

here is the big innovation this circle

what we now called zero do you know what

that word is going to be underneath

underneath here it's not zero because

that's a French word that appeared in

like 1650 in in England really any

guess no idea Rob okay be n or nil or

something no yes yeah that would be I

would I would have guessed yeah like

null or something yeah yeah good guess

but wrong sorry about that because when

it says cth uh which is thought for the

word Cipher this is the cipher we think

these days of ciphers as like code and

mysticism and so on and I think there's

an element of this was a mystical symbol

because it was like it means nothing and

yet it can make numbers big as well so

it is the cipher and we still have the

French word shifa which is uh means a

digit and this is the digit the most

important one because it's what

revolutionized arithmetic um it meant

you could add things up uh without

having to have lots of different symbols

like the Romans did so here we are but

he's spelling it out literally was

everyone in England calling it C or is

that just something that this author has

opted for cuz he had nothing else uh

well everyone was calling it Cipher not

the cipher not CF but no that was it

there was no other word for it other

than nothing nil shakespare refers to

the O which is the shape of it but

that's just him being sort of figurative

no it it was the cipher there was no

zero in the late 1500s here is something

else that Robert record is introducing

people to which is how to do your time

tables especially those tricky ones like

8 * 7 so here you can see the eight and

the Seven first of all you draw this

these two diagonal lines to sort of join

the corners then for each of the numbers

so eight and two you you you basically

find the number bond of that so what's

the number you have to add to eight to

make 10 and it's two okay seven plus

what is 10 three so that's where the

three comes from so that's stage two of

this technique next in your head or

using your fingers not using this method

you multiply the two numbers in the

right hand column so 2 * 3 is six so six

is entered there and then finally you

pick a diagonal and it doesn't matter

which diagonal let's take the bottom

left to top right s and two take that

number from this one 7 takeway two is

five so 8 * 7 is

56 I mean what could be simpler than

that um do you want to try it for

another uh uh another thing we can we

can uh we can we can do it and then I

can show you how it works as well all

right let's do it do you want to give me

a nice safe times table to do uh did you

say do you want single that were single

digits and yeah let's do ones in the

range six to n because that's what

Robert record intended this for um oh

okay so could I do uh 8 * 9 okay let's

do 8 * 9

so8 * 9 let's just draw those diagonals

in like that lovely and then eight plus

two there very very nice yeah excellent

now we multiply these two numbers

together let's draw a line underneath

two ones are two and pick a diagonal do

you have a preference for diagonal do

you like this one I like the one you

used yeah that one okay so nine takeway

two is 7 72 how wonderfully elegant is

that but of course I'm sure everyone on

number file is asking why does it work

and uh it's rather a nice little algebra

exercise actually so we're multiplying

together two numbers let's call them

for sake of Simplicity A and B let's

draw the diagonals so this is 10 minus

a and this is 10 - B let's draw the line

underneath we next multiply these two

numbers together so it's 10 - A * 10 - B

ooh ouch okay so that is 100 minus 10 a

- 10

B plus a there we go so that's that's

the units column and then the 10's

column is this take away this so it's

B minus 10 -

A Time 10 because these are the 10 so

it's 10 times

B which is equal

to 10

B plus 10 a because the negative cancel

out minus

100 okay so that is what a * B is is

this horrible lot well if we add this

lot together the hundreds cancel out the

10 A's cancel out the 10 BS cancel out

and all you're left with is a which is

what we're looking for a * B is equal to

a * B so that's why it works ingenious

the question you and everyone watching

is asking is can we break this by making

numbers not in the range 6 to 9 okay so

let's see if it still works so do you

want to give me two other numbers

outside that range to multiply together

we'll see if I can get this right let's

do uh 5 * 13 5 * 13 okay let's let's see

how it works under this stress test so 5

* 13 okay so 5 plus 5 that's that's easy

enough makes 10 13 +us 3 makes 10 okay

so in the units column we've got

-5 and then let's pick a diagonal let's

go this one it's pretty change 5 takeway

minus three is eight now Robert record

and the uh English folk of the time

would not have been into uh subtracting

negatives it wasn't part of their

general math vocabulary but anyway we've

got 80 take away 15 because this 8

represents 80 and 80 take away 15

is equal

to 65 which I believe is what 5 * 13 is

so it works it works for any numbers

just you got to ask why would anyone

want to do that that's just just your

time tables what about big long

multiplication people didn't know about

it they need to learn about long

multiplication and there's word problems

through this book and there's one where

it's uh there are 264 men who have 29

Lambs each how many lambs are there in

total one of those entirely practical

everyday Elizabeth life problems to work

out just as an opportunity to do some

long multiplication and uh here we've

got it 264 * 29 for some reason record

has little lines through it's like you

cross out each of the numbers maybe to

check you've done them all but anyway

just as we do today 49's a 36 carry the

three except I don't see any three being

carried so you're keeping it in your

head uh and do it all that so 9 * it is

2376 then 20 * it is 528 no no there's

no Cipher has been put there but you

don't need it it's just nice practice

and we get

7,656 as the answer now I said a

Cipher's missing there's something else

that's missing there as well which is

there is no multiplication symbol for

the simple reason when he wrote this

book it hadn't been invented yet all

these symbols didn't just appear out of

nowhere they had to be invented and the

multiplication symbol is always credited

to a an English guy called William out R

Who sometime around 1630 just used it

for the first time but as far as I know

no one has ever said where he got the

idea for it he just kind of came up with

it well I think it seems to me pretty

clear everybody would have had and known

about Robert record's book and they'd be

very familiar with that little trick for

doing your times tables well there's a

there's a very obvious big X happening

there why not represent multiplication

by having a miniature version of Robert

records times tables technique and use

that as a multiplication symbol so like

is it are you suggesting that that

little diagram there on page 71 of This

Book Is the is the birth of the

multiplication symbol or I I have I've

not had anyone challenge me on this um

and perhaps it's been known by people

for years and just no one's made it made

it popular until number file did but uh

you know perhaps every day we're all

just reminding ourselves of that quirky

Little Tim table technique of Robert

record so that's not like it's not like

that's not like the official story on

Wikipedia as to where the the symbol

came from not not yet it isn't no but

it's just a matter you know anyone can

change Wikipedia um but um but no I I

had genuinely not seen any source of it

and and it could and it could not be as

well it could not be as well and and or

indeed it could be on Wikipedia um but

um no it was just in reading through

this book and and discovering there was

no symbol it and and using this this big

X method lots of times it just seemed so

obvious that uh and it probably when you

first saw it you probably thought is

that is that a multiplication symbol so

it it it does kind of jump out of the

page so I mentioned earlier the the

cipher and I think this was a new idea

the cipher and uh Shakespeare I think

was excited by this first of all this is

his vintage is it this is his vintage

this is basically it came out it was

first published about 20 years before

Shakespeare was born so um hence his dad

would only ever if he used Roman

numerals cuz this didn't catch on very

much so Shakespeare was actually the

first generation really who' have been

learning this kind of arithmetic um he

refers to arithmetic quite a bit in his

plays and in Romeo and Juliet there's a

fight between mausio and tibol where

mikio says TI you fight by the book of

arithmetic well which book of arithmetic

is it well I think it could well be this

one um and we saw earlier about this new

symbol the the cipher that can make

numbers big well in the opening scene of

Henry V which is the Battle of ainor and

the actors are playing this massive Army

but there's only let's say seven of them

the the character comes on stage he's

called the chorus and he says to the

audience you can have to use your

imagination guys because uh there's not

many of us and we're representing a

massive Army but that's okay if you look

at me as being a crooked figure a one

and you'll see elsewhere in in records

one was very much a crooked figure in

1599 when that play was first performed

um so I'm the one and if you treat the

other actors as ciphers and there're six

of them that makes a million so that

Shakespeare playfully using this notion

of you can expand numbers really big

with small number of

symbols if if one didn't know about this

the the the use of Cipher cph is

zero that like that would just make no

sense to to to Modern viewers that line

in sh must just be completely bizarre I

think it's not actually even doesn't

even get used in the Henry VI movie I

looked for it perhaps that's the reason

because people and I wonder whether I

mean it's quite a sophisticated

reference even then frankly and Sh does

this a few times with Cipher uh which

convinces me he thought it was a really

cool idea that was sweeping through

England that these simple symbols can do

so much for you and and uh in one case

in King Leah he talks about uh

I think he might use Cipher or or or

refers to the symbol AS basically saying

you know without a figure in front of

you you're nothing you're like a you're

like a a 10 without the one in other

words you are zero if this crossover

between mathematics and shakespear is

your kind of thing you really need to

check out Rob's new book much ad do

about numbers it's not just a great

cover it's a cracking read and I'll

include a link in the video description

he was not just a words guy he was also

quite a numbers guy as well and if if

you love pouring over these ancient

documents and old objects and you like

that sciency mathy bent as well you

really might like my channel objectivity

have you ever watched these videos

possibly a lock of Z mutants here I'll

also include a link in the video

description so you can go and look at

them too there's hundreds you should

check out what I do know is that this is

one of the most juicy r that has ever

happened in history of science