History of

quadratics

John Westwell

Why are we still studying quadratics?

Quadratic equations in parliament – 26/06/2003

“I put this matter on the agenda today because I have been troubled since the president of a teachers' union suggested a couple of months ago that mathematics might be dropped as a compulsory subject by pupils at the age of 14. Mr. Bladen of the National Association of Schoolmasters and Union of Women Teachers was given a lengthy slot on the "Today" programme to present his views. He cited the quadratic equation as an example of the sort of irrelevant topic that pupils study. I had hoped that the Government would make a robust rebuttal, but there was no defence either of mathematics in general or the quadratic equation in particular.” Mr Tony McWalter (MP for Hemel Hempstead) Adjournment debate (Hansard Columns 1259-1269, 2003)

Back to the Babylonians

Imagine you have a rectangular field of area 96 square units and with semi-perimeter (length + width) 20 units, what is the length and the width?

96

length

wid

th

Let’s try some rhetorical algebra

1. Divide your semi-perimeter by 2 2. Square you answer 3. Subtract your area from Answer 2 4. Work out the square root 5. Add this to Answer 1 for the length 6. Subtract this from Answer 1 for the width

Al-Khwarizmi

• Abu Ja'far Muhammad ibn

Musa al-Khwarizmi

• Persian mathematician

• ~780-850 AD

• Hisab al-Jabr wa-al-Muqabala

1. Squares equal to roots.

2. Squares equal to numbers.

3. Roots equal to numbers.

4. Squares and roots equal to numbers, e.g. x2 + 10x = 39.

5. Squares and numbers equal to roots, e.g. x2 + 21 = 10x.

6. Roots and numbers equal to squares, e.g. 3x + 4 = x2.

Six types of mercantile or inheritance problems

…….a square and 10 roots are equal to 39 units. The question

therefore in this type of equation is about as follows: what is the

square which combined with ten of its roots will give a sum total

of 39? The manner of solving this type of equation is to take one-

half of the roots just mentioned. Now the roots in the problem

before us are 10. Therefore take 5, which multiplied by itself

gives 25, an amount which you add to 39 giving 64. Having taken

then the square root of this which is 8, subtract from it half the

roots, 5 leaving 3. The number three therefore represents one root

of this square, which itself, of course is 9. Nine therefore gives the

square.

The compendious book on calculation by completion and balancing

Completing the square

Dusting off the old formula

Whose formula? Bhaskara’s formula

Bhaskaracharya

• Indian

• Mathematician/astronomer

• 1114-1185 AD

• Bijaganita

“Triumphant is the illustrious Bhaskaracharya whose feats are

revered by both the wise and the learned. A poet endowed with fame

and religious merit, he is like the crest on a peacock.”

Let’s sing a song

https://www.youtube.com/watch?v=E2eVZFy9yzk

Brahmagupta

• 598-670 AD

• Indian mathematician

• Astronomer

• Brahmasphutasiddhanta

Zero and negatives

Thomas Harriot

• ~1560-1621 AD

• Oxford educated

• Mathematician/astronomer

• Worked for Walter Raleigh

• “English school” of algebra

• Artis Analyticae Praxis (1631)

An epoch-making principle

“This article deals with the role

of zero in that branch of algebra

which is known as Theory of

Equations. To be specific, we are

concerned here with a procedure

which consists in transposing all

terms of an equation to one side

of the equality sign and writing it

in the form P(x) = 0, where P(x)

is a polynomial. I call this

procedure Harriot’s Principle.”

Tobias Dantzig, 1930, Number, the

language of science

“Conversation between Sir Charles Cavendish and Roberval following publication of Descartes’ “La Geometrie” in 1637”

English versus French

Back to Tony McWalter

“Someone who thinks that the quadratic equation is an empty manipulation, devoid of any other significance, is someone who is content with leaving the many in ignorance. I believe also that he or she is also pleading for the lowering of standards. A quadratic equation is not like a bleak room, devoid of furniture, in which one is asked to squat. It is a door to a room full of the unparalleled riches of human intellectual achievement. If you do not go through that door—or if it is said that it is an uninteresting thing to do—much that passes for human wisdom will be forever denied you.” Mr Tony McWalter (MP for Hemel Hempstead) Adjournment debate (Hansard Columns 1259-1269, 2003)