Logical Reasoning
The quadratic functions unit we did earlier leads to a study of math
Called CALCULUS. The statistics unit leads to a study called STATISTICS,
Trigonometry is a part of GEOMETRY
Logical Reasoning fits into a branch of mathematics called
PURE MATHEMATICS
Science and Engineering are fields that rely heavily on
EXPERIMENTS and OBSERVATION
The experiments lead to theories and are used to predict
the behaviour of the world around us; however, we
cannot PROVE anything with this type of thinking.
This type of thinking is called INDUCTIVE Reasoning
Inductive Reasoning – drawing a general
conclusion by observing patterns and
Identifying properties from examples
Since it is based on examples, you cannot sit in a room and do
Inductive reasoning in your mind. You have to experiment
Example: You notice that the sun rises around the same time each day
so you draw the general conclusion that the sun will rise tomorrow
Although this type of reasoning is dominant
In our society; there are alternatives!
A big part of inductive reasoning is coming
up with a conjecture
Conjecture – testable expression based on
Evidence but not proven
“A general opinion based on incomplete
information”
Example: Consider the following information
[See diagram on precipitation]
What conjecture can you make?
Conjecture: the most precipitation occurs in January.
To support this conjecture: we show evidence by summing the
Precipitation for january
150.5 + 249.6 + 283.6 + 181.4 + 137.6 = 1002.7
Is this enough to support the conjecture?
NO. Because you claim that it is the MOST, you
should compare it to some other months.
Example: Make a conjecture about the
product of two ODD integers
Conjecture: The product of two ODD
integers is ODD
To support this conjecture, I provide some evidence:
3 * 3 = 9 (odd)
5 * 5 = 25 (odd)
-3 * -5 = 15 (odd)
Is this a convincing conjecture?
NO. It is only based on evidence
and not even that much evidence
BUT. Even if we did find 1000 examples,
It still wouldn’t be that convincing
Because finding one that didn’t work
Would ruin all of the 1000 examples
No matter how many examples you find… you are NOT proving
anything.
Mathematical PROOFS are NOT based on evidence but instead
are based on smaller claims that you know “are always true”
Since you don’t need evidence, you can indeed sit in a room by yourself
and prove that something is true just by thinking about it
Prove that the product of two ODD integers
is always ODD.
Claim 1: you can represent the product of two numbers as a grid
2x3
4x4
This is always true
Consider multiplying any number by an EVEN number
Think about the grids that we created previously, since the number of columns is
EVEN, we can pair the columns together
7x4
I can pair the columns this way without any squares
left over.
This means that the PRODUCT which is the total
number of squares is EVEN
However, we want to show that two ODD numbers will give
us an ODD product… or an ODD number of squares
We start with EVEN times ODD
2x5
Claim: If we pair up the columns that we have,
We will always have exactly ONE column left over
(remember that the number of columns here is the ODD
Number we are multiplying by)
ALWAYS one column left over
However, we want to show that two ODD numbers will give
us an ODD product… or an ODD number of squares
We start with ODD times ODD
3x5
Claim: In our left over column, there will always be an ODD
number of squares
I know this is always true because the number of rows is ODD
(based on what we multiplied)
Now, we can pair up the squares of
The last column… and we will ALWAYS
STILL one column left over Have ONE SQUARE left over
This implies we have paired up ALL of the
Squares this way except for one which proves
The number of squares in total is ODD.
End of proof