Transcript Common mistakes about mathematics and use of mathematics in

Common mistakes about
mathematics and use of
mathematics in everyday life
By : Dr. Nitin Oke
Our own feelings - - - !
• As a student or teacher of mathematics
many a times we feel that why to teach
problems like –
– A tank is filled in 3 hr by a tab and
empties in 5 hours then if both tabs are
open in how much time tank will be filled?
– Instead of calculating this time close the
other tab why to keep it open?
Our own feelings - - - !
• As a student or teacher of mathematics
many a times, we feel that; why to teach
problems like –
– A work is done in ten days by one worker
if five workers are doing the same job
the work will be done in how many days?
– How many of us really think that after
increasing number of workers the work will
be done with same speed?
Our own feelings - - - !
• As a student or teacher of mathematics
many a times, we feel that; why to teach
problems like –
– A table and a four chairs cost 2000 Rs
and two tables and four chairs will cost
3000 then cost of a table is ?
– Just imagine such conversation at a shop
when you wish to buy a table and chair. You
will decide to go to other shop immidiatly.
Our own feelings - - - !
• As a student or teacher of mathematics
many a times, we feel that; why to teach
problems like –
– Sum of my age and my father’s age is 60
years and my fathers age is double that
of my age then my age is ?
– Just imagine such conversation when you
asked age of your student and he answers
like this.
Our own feelings - - - !
• As a student or teacher of mathematics
many a times, we feel that; why to teach
problems like –
– The last digit of 107
is ?
– At this moment we start thinking is it
really going to make any difference to my
life?
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Status of Mathematics is
• Even if we call mathematics as mother
of all sciences it itself is NOT science.
• Well the reason is it never tries to revel
the laws behind the facts in nature on
the other hand it provides the language
to do so, it provides methods to do so in
short mathematics can live without
science but science can not.
What Mathematics is?
• Well it is not just study of numbers.
• Neither it is study of figures.
• Mathematics; for science is
– method of proof : Logic, a statement to
prove, an implication to prove , note the
difference between statement and its
converse
– language of expression : F = m.a,
a = (v – u)/t or F = d(P)/dt or a = dv/dt
In short
• Mathematics is not science itself, so we can not
expect to have direct applications of
mathematics. This is something like you can not
expect; to travel with a can of a 20 liter of petrol ,
but you can travel by a car using this petrol.
• So now mathematics turns out a process to train
human brain
– to think
– to relate
– to predict.
Some technical mistakes during our
teaching or learning
• During very early school days the process of
subtraction is carried by considering “ carry”
and then we use to return with thanks this is
not scientific.
Concept of rules of division
• These are not hypothesis these are results so
need to have proof. You only need place values
for this for example
• A number is divisible by 3 if sum of all digits is
divisible by 3 let the number be abc ( actually it
can be ( a1a2a3a4----an) so
abc = 100a + 10b +c = 99a +9b + a + b + c
3 divides 99a and 9b under any case hence if
3 divides a + b + c then will divide abc and hence
the result.
Concept of rules of division
• These are not hypothesis these are results so
need to have proof. You only need place values
for this for example
• A number is divisible by 11 if sum of all digits at
even place and sum of all digits at odd place
differ by a number divisible by 11 let the number
be abcd ( actually it can be ( a1a2a3a4----an) so
abcd = 1000a + 100b + 10c + d
= 1001a - a + 99b+ b + 11c – c + d
Remaining part you can get on your own
Euclidian geometry most easy to
understand most difficult to explain
• Concept of point – segment – any figure seen
( ?) as set of points.
• A for apple why not elephant (?)
• Future – Nature - ______
• Charges of travel
• a = b ------ 1 = 2
Rational – irrational and real number
2.34555535353467282543846252673748589
5963374364464647484888888888888888888
8888888888888888888888888888888888888
8888888888888888888888866455555555555
5555555555555555555555555555555555555
5555512345678901234567890123456789012
3456789012345678901234567890123456789
0123456789012345678901234567890123456
7890123456789012345678901234567890122
Rational – irrational and real number
• You can be confirm that number is rational if it is
expressed in form of
– fraction of two integers ( with nonzero denominator)
– Terminating or recurring decimal expansion
• If the number is not in form of fraction and
decimal expansion is there and there and there
then because of finite life span we are not in
position to tell whether number is rational or not
• Be careful about difference in surd and  or e
Difference between countable and
uncountable infinity
• Concept of infinity is very difficult ( not
possible) to explain.
• Is it not really difficult to explain that you
remove half the objects and remaining
number is same as before.
a
• Please note that  
0
Trigonometry
• We define trigonometric ratio with the help of
triangle but when it comes to sin (0o) or
cos(0o) (can’t even think of cos(180o))then it is
meaningless by right angle triangle. You need
other approach, for the same, as polar co
ordinate system and Cartesian system.
Trigonometry
• We define trigonometric ratio with the help of
polar co ordinate system and Cartesian system
cartesian x
• cos(polar ) =
polar r
• sin(polar ) =
cartesian Y
polar r
A(r,)
A(x,y)
cartesian x
cos() 
polar r

cartesian Y
sin( ) 
polar r
mathematics in our every day life
• Two eyes and two years are for estimating
distance of source. Using trigonometry.