Something Less Than
Nothing?
Negative Numbers
By: Rebecca Krumrine and
Kristina Yost
Introduction
 Negative
numbers were not generally
accepted until a few hundred years
ago.
 Negative
numbers first appeared
when people began to solve
equations.
Lets try a problem…
I
am 18 years old and my sister is 11.
When will I be exactly twice as old as
my sister?
 How
would you react to that answer if
you did not know about negative
numbers?
Main Topics
 Development
numbers in…





China
Greece
India
Middle East
Europe
of concepts of negative
China 100BCE – 50BCE
 In
the “Nine Chapters of Mathematical Art”
they used red rods as positive coefficients
and black rods for negative coefficients to
explain methods for finding area of figures.
 The Nine Chapters also included rules for
dealing with negative numbers.
Greece 570BCE – 300BCE
 Greeks
ignored negative numbers
completely.
 Aristotle made a distinction between
numbers and magnitude, but gave no
indications of the concept of negative
numbers.
 Euclid continued this distinction in his
work Elements.
Greece 3rd century CE
 Diophantus
did not deal with negative
numbers but he was aware of rules for
multiplying with the minus and solving
equations.
 In book V of his Arithmetica, he
encounters the equation 4x+20 = 4

He believes that this problem is absurd,
since to him 4x + 20 meant adding
something to 20 to equal 4.
India 7th century CE

Brahmagupta recognized and worked with
negative numbers.


Positive numbers were possessions and negative
numbers were debts
Stated rules for adding, subtracting,
multiplying, and dividing negative numbers in
his work Correct Astronomical System of
Brahma.
 Expanded on Diophantus concepts of the
quadratic equations (ax2 + bx = c, bx + c =
ax2, ax2 + c = bx) using negative numbers
forming the general form of the quadratic
equations.
India 12th century CE
 Bhaskara
gives negative roots, but rejects
the negative root since it was
inappropriate in the context of the
problem.

“…For people have no clear understanding in
the case of a negative quantity”
Middle East 9th century CE

Arabs were familiar with negative numbers
from the work of India mathematicians, but
they rejected them.


Muhammad Ibn Musa Al-Khqarizimi did not use
negative numbers or negative coefficients in his
two books.
Knew how to expand products such as
(x – a)(x – b), but they only used this concept
when the problems involved subtractions
whose answers are positive.
Europe 16th Century

Negative numbers were still being ignored
and considered as “fictitious solutions.”
 Mathematicians of this time still resisted
negative numbers and thought of them as
“fictitious” or “absurd.”
 Some of the mathematicians of this time
were:



Cardano from Italy
Stifel from Germany
Viete from France
Europe 17th Century
 Negative
numbers started to become
accepted.
 Along with the acceptance, came the
realization of problems with negative
numbers.

I.e. square roots of negatives
 Rene
Descartes partially accepted
these numbers.
Question:
 When
taking the square root of a
negative number, we refer to the
result as….?
IMAGINARY!!
 Rene
Descartes was the mathematician
who called these results imaginary!
17th century continued…
 Many
mathematicians who started
accepting negatives didn’t know where
they belonged in relation to positives.

One math guy, John Wallis, thought that
negatives were larger than infinity.
 Isaac
Newton wrote a book in 1707
called Universal Arithmetick. In this
book he states, “Quantities are either
Affirmative or greater than nothing, or
Negative, or less than nothing.”
Questions for thought…
 How
can a quantity of something be
negative and less than nothing?
 Can
you have a negative quantity of
books, food, clothing, or money?
 It
was hard for people to grasp the
concept of negative numbers being
debt.
Europe Middle 18th century

Negatives are officially accepted as real
numbers!!
 Euler was fine with negatives during the
writing of his book Elements of Algebra.
 Even though negative numbers were known
and used, it was common for people to still
ignore them as results to equation systems.
 It was still common practice to ignore a
negative results in any system of equations.
Europe 19th century
 Negatives
finally become important
enough to not be ignored.
 The works of Gauss, Galois, and
Abel really had a big impact on
equation systems with negative
results.
 Doubts of negative numbers finally
disappear.
Summary
 Although
negative numbers were
“discovered” in BCE, negative numbers
were not completely accepted until the
1800’s.
 Still, generally, mathematicians used
negative numbers in computations, but did
not understand the concept of them.
Timeline






4th century BCE- Aristotle made a distinction
between numbers and magnitude.
100 BCE- In the Nine Chapters of Mathematical
Art, the Chinese used negative numbers in
solving systems of equations.
3rd century CE- Diophantus solved equations with
negative numbers in Arithmetica, but then
rejected the equation itself.
7th century CE- Indians used negative numbers
to represent debt.
9th century CE – Arabs were familiar with
negative numbers, but rejected them.
12th century CE- Bhaskara (Indian) gives
negative roots for quadratic equations, but rejects
the roots because people do not approve of
negative roots.
Timeline continued…
16th Century CE- European Mathematicians
thought of negative numbers as “fictitious” or
“absurd.”
 17th Century CE- Rene Descartes claims the
result of negative square roots as “imaginary.”
 18th Century CE- Negatives start to become
accepted in Europe even though they are still
commonly ignored.
 19th Century CE- Doubts of negative numbers
finally disappear and negatives are known
now as real numbers.

References

Berlinghoff, William P. , and Fernando Q. Gouvea.
Math through the Ages A Gentle History for Teachers
and Others. 1st ed. Farmington, Maine: Oxton House
Publishers, 2002.
 Katz, Victor J.. A History of Mathematics. New York:
Pearson/Addison Wesley, 2004.
 Negative and non-negative numbers." Wikipedia.
2006. 7 Sep 2006
<http://en.wikipedia.org/wiki/Negative_numbers>.
 "Number." Wikipedia. 2006. 7 Sep 2006
<http://en.wikipedia.org/wiki/Number>.
 Smith, Martha K.. "History of Negative Numbers."
University of Texas at Austin. 19 Feb 2001. 9 Sep
2006
<http://www.ma.utexas.edu/users/mks/326K/Negnos.
html>.