Purpose of
this Lesson
After learning this lesson you should be
able to do the following things.
1) Write long numbers in short form in order to save your time and
space in the paper. So that it would look neat. (Writing in Scientific
Notation)
2)
Convert a number to Scientific Notation and the vice versa.
3)
Rounding off a number to the nearest power of 10.
4)
Rounding off a Decimal number according to purpose.
Now we can simple state that the purpose of
this lesson is to simplify our work. So we will be able to work more
efficiently without wasting time. Let’s get started.
Scientific
Notation
Can you understand the meaning of the title?
Mathematics is a branch of Science. We use it to solve problems in Science. So
that the ‘Notation’ we use in ‘Science’ to write numbers simple is
named “Scientific Notation”.
How
Scientific Notation works?
125,600,000,000,000,000,000,000,000,000,000
526,460,000,000,000,000,000,000,000,000,000,000
123,000,000,000,000,000,000,000,000
879,000,000,000,000,000,000,000,000,000,000,000,000,000,000
389,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Assignment
No 1
Take a piece of paper (A book would be
better to follow all assignments) and write down all the numbers above. (There
are five numbers in total) After you finish return to the text below.
How did you feel when you were writing all
those numbers? Were you bored or else scolding me for instructing you to do
such a foolish thing? It seems foolish and certainly is. I agree. If I instructed a foolish act how to act
intelligently?
You
should learn to write the numbers above shortly. Learn
the system below to write the five
numbers above shortly and quickly. Remember, even if you write the number in
short form the both numbers should be the same without any difference.
How else can you write the numbers above?
(Don’t think of short of long, but the possibilities) After thinking look at
how I write the first number above in different ways. At this point I assume
that you have previous knowledge on Indices. (Writing a number in power form)
125,600,000,000,000,000,000,000,000,000,000
A)
First set of ways
= 12,560,000,000,000,000,000,000,000,000,000 × 10 to the power 1 (Remember 10 to the power one equal 10)
= 1,256,000,000,000,000,000,000,000,000,000 × 10 to the power 2 (10 to the power 2 equal 100)
= 125,600,000,000,000,000,000,000,000,000 × 10 to the power 3 (10 to the power three equal 1000)
We can go on writing all possible methods
by multiplying by 10 to the power 4, 10 to the
power 5 and so on. Likewise we can write in the methods below too.
B)
Second set of ways
=
1256 × 10
to the power 29
= 125.6 × 10 to the power 30 (At this point you need
to be well familiar with the previous knowledge of decimals and fractions)
= 12.56 × 10 to the power 31
= 1.256 ×10 to the power 32
=
0.1256 × 10 to the power 33
As I said earlier we can continue to write
in numerous ways. (Infinitive – that means it has no end)
What
is the difference that you noticed between the first and the second set of
different ways that we wrote the number in?
You had difficulty writing down the first
set of numbers, but what happened in case of the second set of numbers? Surely,
that would have been easy. Didn’t you feel so?
But can we use all the methods in step B?
There should be a common one to be accepted internationally. In that case 1.256 ×10 to the power 32 is accepted. To
understand more clearly look below.
General Form of Scientific Notation
a × 10 to the power n
In this equation “a” and “n” are the
unknown terms. (They are unknown since we don’t know their value. Since “10” in the equation is always TEN we
call it constant)
Remember! Whenever there is an unknown
you should define it. So let’s define both the unknowns.
1 ≤ a < 10
(It denoted that a should always be equal to OR greater than one, but less than
10 – so it can’t take the value 10)
n is an Integer.(It can be positive as well as negative)
Now check whether the short form that we
wrote for the example above is correct. The “a” in the equation should satisfy the condition of “a” in the general form.
Now you have understood all about a
Scientific Notation. But think what if “a”
takes the value 1. Look for the example below.
1,000,000,000,000,000,000
When writing down this in Scientific
Notation, you will get (1 × 10 to
the power 18) as the answer.
Hence the use of ONE in place of “a” also is apparent.
Assignment
2
In the first assignment I told you to write down five long numbers. We explored what Scientific Notation is all about using the first number. Write down the other four numbers in scientific Notation and send me an email or else check it with your Teacher or someone who is familiar with Mathematics.
In the first assignment I told you to write down five long numbers. We explored what Scientific Notation is all about using the first number. Write down the other four numbers in scientific Notation and send me an email or else check it with your Teacher or someone who is familiar with Mathematics.
Writing
Scientific Notation with Negative Indices
Don’t get too confused looking at the word “NEGATIVE”. This means that in the
general equation for Scientific Notation (Mentioned Above) n takes a negative value.
(Since we mentioned n as an integer it can be positive as well as negative)
Write the number ONE in scientific
notation.
1 = 1 × 10 to the power zero. (Any Base number to the power zero equal 1)
So far we had been dealing with positive “n” value. But now it has gone to zero
when we chose to write number one in scientific notation.
Let us write the numbers below in
scientific Notation which are less than ONE
0.3 = 3/10 = 3 × 10 to the power (-1)
0.45
= 45/100 = 4.5/10 = 4.5 × 10
to the power (-1)
0.0056
= 56/10000 = 5.6/1000 = 5.6 × 10
to the power (-3)
Note:
Every
time the decimal point is moved to the left side by one point it means that the
number is divided by 10. If the decimal point is moved to the right by one
point means the number is multiplied by 10.
So for example when the decimal point is moved to the left two times, it means first it is divided by 10 and again by another 10. We can derive conclusion that the number is divided by 100.
For example take the number 100. Divide it by 10.
100/10
= 10
Now divide by another 10.
10/10
= 1
Let us try it in short by simply moving two decimal points to the left of 100. When a number is written normally remember there is always a decimal point after it. Since there is no purpose to use it we are not writing it to all numbers. But we write it when we get numbers which are not whole numbers.
100.
Move decimal point by two places to the
left. It results in 1.
So practice these methods by doing a lot
of sums to keep the whole in Mind. After a considerable amount of practice
figuring out these sums will become second nature to you.
Rounding Off
“Rounding
off” simply means to approximate a number to the
nearest value in order to make our work easy. For example
2.9996 × 4.00001 × 3
Get the answer to this sum. Once you see
this itself you become a little bit fed up of having to figure out a number
with so many decimal places. Hmmmmmmm!
So now let us approximately solve the
sum.
2.9996
is almost equal to 3.
4.00001
is almost equal to 4.
So now the approximated sum is 3 × 4 × 3
= 36
What
a relief really to solve that sum so easily. This is known as “Rounding Off”. But
the only thing you want to learn here is to round off to the nearest what? It
can be to the nearest 10, nearest 100, nearest 1000, etc.
Look at these examples where the numbers
are rounded off to the nearest 10.
32 ~ 30
46
~ 50
65
~ 70
In the above examples the process of the
first two numbers shouldn’t be confusing. But the number 65 is equally close to 60 and 70. So why should I write 70?
The answer:
When you go to a shop and purchase two items and ended up with 65
Rupees total value. Can you give 60 Rupees and come out of the shop. No in that
case your approximate value should be 70 Rupees.
So when the distances are
equal between the values (the number 5 comes) you should always approximate or
round off to the higher value (In the above example we have rounded off 65 as
70 to the nearest 10).
The same procedures apply when rounding off
to the nearest 100, nearest 1000 and
so on.
Numbers rounded off to the nearest 100
102 ~ 100
289 ~ 300
350 ~ 400
(I explained the procedure using number 65 in the example above)
950 ~ 1000
Numbers rounded off to the nearest 1000
200 ~ 0
560 ~ 1000
3500 ~ 4000
7001 ~ 7000
41,200 ~
41,000
Rounding
Off Decimals
When rounding off decimals you may be asked to
do it in two ways.
1)
To
the nearest whole number
2)
To
the nearest decimal place(May be to the first decimal place, second decimal
place, etc)
Rounding off decimals to the nearest whole
number
2.4 ~ 2
2.7 ~ 3(In both these examples the dividing mark is 2.5 – when it
is less than 2.5 answer is 2, when it is more than 2.5 answer is 3)
4.7 ~ 5
2.897 ~ 3
Rounding off decimals to the nearest first decimal
place
When
asked to round off to the nearest first decimal place, you should look at the
second decimal place to determine the rounded off value – the dividing mark
will be in the 2nd decimal place.
2.56
~ 2.6 (2.55 is the diving mark, since 2.56 is greater than that we get the
answer 2.6)
2.54
~ 2.5
3.55
~ 3.6
Rounding off decimals to the nearest second
decimal place
In this time you should look for the 3rd decimal place to determine the equally dividing value.
2.264 ~ 2.26
2.267 ~ 2.27 (For both these numbers the equally dividing value is
2.265)
6.651 ~ 6.65
This procedure continues to whatever decimal place we round off the
number to. Please do the assignment below.
Assignment 3
Do
all the exercises regarding Scientific Notation and Rounding Off from any
textbook that you may have. Then take a short note of the key ideas from my
content and if you want type it and email it to me. I can review it and reply
you. Finally create a set of questions by yourself in each of the portion that
I covered and send it to me. So I can put your questions on my site to
facilitate students like you.
If
you have any doubts regarding the content, technical terms or anything else
please email me and clarify.
