WHY ARE THERE NEGATIVE NUMBERS?
 NEGATIVE NUMBERS ARE NECESSARY TO
DESCRIBE VALUES ON A SCALE THAT GOES
BELOW ZERO.
Consider the following situations
 Temperatures falling below 0˚C on a
thermometer
 Elevators in a building that have
floors below ground level
 Bank statements that show funds
spent and funds deposited
 Coordinates on a grid with all four
quadrants
 Bookkeeping for businesses
If negative numbers didn’t exist, how
would these be represented?
So…Why do negative numbers “get
bigger?”
 Look at the number line showing negative numbers, they seem to get 'bigger'.
However, as the negative number gets bigger, the value gets lower. -10 is a larger number than
-5, so it is further below zero. As you travel to the left on a number line the values of the
numbers are less.
when did the concept of negative
numbers come into play?
• In 200 BCE the Chinese used number rods to
represent positive (red) and negative
(black) numbers. However they did not
accept negative numbers as solutions to
problems.
• In the 7th century, Brahmagupta used the
idea of positive and negative numbers to
represent fortunes and debts. He is
accredited for setting the rules for
dividing integers.
• The western world could not accept
negative numbers. As late as the 1500’s
Europeans said “Zero signifies ‘nothing’,
and it’s impossible for anything to be less
than nothing.”
• By publishing the solution to cubic
equations In the sixteenth century,
Girolamo Cardano helped negative gain
acceptance. However, he called them
“ficticious”.
• Finally in the 19th century Caspar Wessel,
Jean Argand, Augustus De morgan, George
Peacock, William Hamilton and others
began working of the “logic” of algebra
and the definition of negative numbers
became clearer.
• Now negative numbers are built into
mathematical models, Physical science,
engineering and the commercial world.
Why is a negative multiplied or
divided by a positive a negative?
• First let’s remember that Multiplication is repeated addition and
division is the inverse of multiplication.
• If we want to multiply 3(4), we are adding 3 sets of 4 giving us 12.
So if we multiply 3(-4), we are adding 3 sets of -4 giving us -12.
This can be shown by drawing out the symbols.
- - - - - - - - - - - - = -12
• When multiplying -3(4), we need to think of this as “take away” 3
sets of 4. This leaves just the negatives so the answer is -12. To
show this, we need to have sets of positive and negatives which
when paired are zero. A positive plus a negative is zero. Without
the pairs we would have nothing to take away.
+ + + + + + + + + + + + - ---
----
----
Why is a negative multiplied or
divided by a negative positive?
• Think back to when we multiplied -3(4) and got -12 because we
took away 3 sets of positive 4. We can show -3(-4) is the same
way. We will again start with pairs of zeros. Now instead of
taking away 3 sets of positive 4, we will take away 3 sets of -4
leaving us with 12 positives.
+ + + + + + + + + + + + -­‐ -­‐ -­‐ -­‐
-­‐ -­‐ -­‐ -­‐
-­‐ -­‐ -­‐ -­‐
-­‐-­‐
-­‐-­‐
-­‐-­‐
BIBLIOGRAPHY
http://www.bbc.co.uk/skillswise/numbers/wholenumbers/whatarenumbers/negativenumbers/facts
heet.shtml
http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers
http://www.mathpages.com/home/kmath298.htm
http://mathforum.org/library/drmath/sets/select/dm_pos_neg.html
http://www.und.edu/instruct/lgeller/negnum.html
http://mathforum.org/library/drmath/view/57864.html
http://nrich.maths.org/public/viewer.php?obj_id=5961
http://www.basic-mathematics.com/history-of-negative-numbers.html
http://www.ma.utexas.edu/users/mks/326K/Negnos.html
Seife, Charles. ZERO,The Biography of a Dangerous Idea. New York, NY. Penguin Books,
2000.
Coyner, Elizabeth, Brockhoff,Beverly.
CPM Educational Program 2002.
Foundations for Algebra: Year 1. Sacramento, CA.
Ellerman, David P. The mathematics of double entry bookkeeping. Math. Mag. 58 (1985), no. 4, 226--233.
(Reviewer: D. J. Struik.) SC: 90C99 (01A99 20G99), MR: 87a:90151.