Quadratic Equations Starting with the

1
Quadratic EquationsStarting with the

• Chinese
• in
• 2000 BC

2
What they knew

• The area of a square is the square of the lengths
  of its sides.
• If they multiplied the lengths of the sides of a
  square hay loft by 3 the area would be multiplied
  by 9.
• How to work out the area of compound shapes such
  As rectangles and T shapes.

3
What they didnt know

• How to work out the lengths of the sides of a
  square if they knew its area
• So if somebody went to the builder knowing what
  area he needed to store his hay the builder
  couldnt work out how long to make the sides of
  the loft

4
The Egyptians 1500 BC

• The Egyptians did not have a formula for working
  out the lengths of the sides if they knew its
  area.
• They made a table which showed the area for all
  the possible outcomes. If someone wanted a space
  with a certain area, they would look in their
  table

5
What the Egyptian area table might have looked
like
6
THE BABYLONIANS

• The Babylonians used a base 60 number system
  which meant that they could check their
  calculations and produce more accurate tables
  than the Egyptians.
• By 400BC they had found a method called
  completing the square to solve problems
  involving areas.

7
Mediterranean information

• Pythagoras (500bc in Italy)and Euclid(300bc in
  Egypt) used geometry to solve the Quadratic
  equations.
• Pythagoras noted that the ratio of the area of a
  square and the length of its side (the square
  root) was not always a whole number but he
  refused to accept any that were not rational.
• Euclid however, accepted the existence of
  irrational numbers

8
The architects, builders and engineers of the day
were still looking for a way to find the square
root of any number so that they could make their
buildings the right size. Euclids big work
called Elements gave the theory but he didnt use
the same notation as we do today and it still
wasnt possible to find a square root.
9
Mediterranean's and Quadratic equations300bc
10
Indian maths

• Hindu maths has used the decimal system since
  600AD. Hindu maths was strongly influenced by the
  commercial world and the average Hindu merchant
  was quite fast at simple maths. The numbers would
  be negative if people had debts and the positive
  if someone had credits. Zero is an important
  number in mathematics and the Hindus were amongst
  the first to accept its existence.

11
Indian maths continued

• Around 700AD the general solution for the
  quadratic equation (using numbers) was devised,
  by a Hindu mathematician called Brahmagupta, who
  used irrational numbers. He also recognised 2
  roots in the solution. The final complete
  solution, that we know, came around 1100AD by
  another Hindu mathematician called Baskara. He
  was the first to recognise that any positive
  number has 2 square roots.

12
Persian Mathematics

• This part of our presentation is to inform of the
  exciting history of Persian quadratics and maths.

13
The History of the Persian Empire

• Persia is now known as Iran and is situated next
  to Iraq and Afghanistan. At the peak of its time,
  its empire stretched from Greece to Egypt and
  India.

14
Persian Mathematicians

• Around 820AD, near Baghdad, Mohammad bin Musa
  Al-Khwarismi, a famous Islamic mathematician and
  the father of algebra also derived the
  quadratic equation. But he rejected negative
  solutions.
• This derivation of the quadratic equation was
  brought to Europe by a Jewish mathematician
  called Abraham bar Hiyya. He lived in Barcelona
  in around 1100AD.

15
1500AD The renaissance in Europe By 1545 Girolamo
Cardano blended Al-Khwarismis solution with
Euclidean Geometry to come up with a solution for
a quadratic equation. He allowed for the
existence of complex or imaginary numbers which
involve the square root of negative numbers. In
1637 when Rene Descartes published La Geometrie,
modern mathematics was born and the quadratic
formula appeared in the form that we know today!