Transcript Lesson 1- Basics

Lesson 1- Basics
Objectives :
To know how to approx numbers to a required
accuracy by 2-3
1.
2.
3.
4.
5.
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Basic Number Types
Decimal Places
Significant Figures
Writing numbers in Standard Form
Writing numbers in Engineering notation
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Number Types
There are two types of numbers (Scientist)
- Exact -> Amount of money in your pocket
- Approximate -> Measurements like weight height
Mathematicians have more definitions of numbers...........
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Number Types
Counting
Numbers
Natural
Numbers
Integers
Positive Whole Numbers 1, 2, 3, 4, 5……
N
Z
Rational
Numbers
Q
Real Numbers
R
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Counting numbers and
zero
All positive and negative
whole numbers
Numbers which can be
m
written as a fraction
where m and n are n
integers
All rational and irrational
numbers
0, 1, 2, 3, 4……
…-2, -1, 0, 1, 2…
-1, 0, ½ , 2¾ ,
-1, 2¾ , π,
Rational Numbers
Most real numbers can be written as a fraction
m
n
in its lowest form
Example:
Express 0.123123123123......... 123
x  0.123123123
1000 x  123.123123123
999x  123.00000000
 x  123
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999
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as a fraction
Trick x 1000 to get rid of decimals
Subtract to get rid of decimals
 x  41
333
Irrational Numbers
─ But some numbers can not be expressed as fractions
─ Examples include
2 
e
These are numbers where the patterns in the decimals do not repeat
  3.141592654....
We can not express numbers like this in faction form.
The irrational number set is much smaller than the set of rational numbers
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Proof that
2
is irrational
MethodWe will assume that it is rational and then we will contradict this assumption
2
m
n
m2
2  2
n
m and n are integers and the fraction can not
be simplified further (i.e lowest form)
 2n 2  m 2
So m2 is an even number
m2 – even this implies that m is even
so “m” can be written as “2 × a” (as m even)
4a 2
2
2
so 2  2  n  2a (so n is even too!!)
n
So
m m2
1 2
2 4
3 9
4 16
5 25
m
even
is
 Both numerator and denominator are divisible by 2 and
even
therefore m is not in lowest form and can be simplified
n
n
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Contradiction!!
Starter
− You need to buy some carpet for your bedroom
− You measure the width and length of your room as
7.22m x 6.58m
− You do not have a calculator or a pen and you have to
estimate the area quickly in your head!
− How do you estimate the area?
− What values did you use for the length and width?
− The carpet cost £5.80 per square meter, consider how
much money you should take to the shop?
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Area is 7.22m x 6.58m
The area must be smaller then 8m x 7m
=>
56m
The area must be larger then 7m x 6m
=>
42m
=>
49m
42 m < Area < 56m
But a better guess might be 7m x 7m
These workings are all to 1 significant figure (sf)
Obviously taking more (sf) will result in a more accurate answer
How much money should you take? It is easy to how much exactly if you are
good with mental aritmetic or have a calculator, but in principal if you take
more than you need you cant go wrong!!
If bad with numbers take 60 x 6 = £360
Significant Figures
Consider the Real number
37.500
All the digits to the left of the decimal point are important
Only the 5 to the right of the decimal point is important as
37.5 is the same as 37.500
Consider 37.5001, then all of the digits are important
SIGNIFICANT FIGUREs (SF) means IMPORTANT DIGITS
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Sig Figs
37.5
3 -> This is the 1st Sig Fig
7 -> This is the 2nd Sig Fig
5 -> This is the 3rd Sig Fig
The significance of numbers decreases from left to right
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Rounding to Sig Figs
Example Approximate 37.5 to 1 significant figures
Look at the next most significant number
30
37
Round down < 35
37.5
is
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40 (1 sf)
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40
Round up if ≥ 35
40 to 1 significant figures
we write
(2nd number)
Rounding to Sig Figs
Example Approximate 37.5 to 2 significant figures
Look at the next most significant number (this is now 3rd No.)
37
37.5
Round down < 37.5
37.5
is
Round up if ≥ 37.5
38 to 2 significant figures
we write
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38
38 (2 sf)
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Rounding to Sig Figs
Example : Approximate 37.5 to 3 significant figures
This is just 37.5 (because there are only 3 digits)
Significant figures (sf) are counted from the left of a number. Always
begin counting from the first number that is not zero.
9 4 6 0 3. 5 8
1st 2nd 3rd 4th 5th 6th 7th
significant figure
0. 0 0 0 0 0 1 4 9 0 2 0 7
Notice that a zero can be significant if it is in the middle of a number.
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Example
Write the following to 3 sf
a) 12.455
b) 0.013026
c) 0.1005
d) 13445.698
e) 0.1999
Round down <
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Round up if ≥
Find the following
801296
to
1 sf
801296
to
3 sf
-52.9000
to
3 sf
-52.9001
to
4 sf
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Decimal Places
-This is another way numbers are approximated or rounded
-The principal is the same as for sig figs but we are only
interested in the numbers to the right of the decimal place
3.14159
Example :
Interested in these
numbers
Express π (Pi) to 1 decimal place
π = 3.1415926535897932384
π is 3.1 (1 dp)
3.1
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Round down
< -3.5
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3.2
Round up if ≥ 3.5
Decimal Places
Example :
Express π (Pi) to 2 decimal places
π = 3.1415926535897932384
This is < 5 so do not round up
π = 3.14 (2 dp)
Example :
Express π (Pi) to 6 decimal places
π = 3.1415926535897932384
This is ≥ 5 so round up
π = 3.141593 (6 dp)
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Scientific Notation
A short-hand way of writing large or small numbers
without writing all of the zeros
Example :
x
The Distance From the Sun to the Earth
93,000,000
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Step 1
 Move decimal left
 Leave only one number in front of decimal
Step 2
• Write number without zeros
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Step 3
 Count how many places you moved decimal
 Make that your power of ten
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Scientific Notation
Example:
Partial pressure of CO2 in atmosphere  0.000356 atm.
This number has 3 sig. figs, but leading zeros are only placekeepers and can cause some confusion.
So expressed in scientific notation this is
-4
3.56 x 10 atm
This is much less ambiguous, as the 3 sig. figs. are clearly
shown.
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Engineering Notation
This is the same as scientific notation except the POWER
is replaced by the letter E
Examples
Number
Scientific Notation
100
Engineering Notation
1.x102
1.E2
1000
(1 sig fig)
1. x 103
1.E3
1000
(2 dec pl)
1.00x 103
1.00E3
-0.00123
-1.23x 10-3
-1.23E-3
1007
1.007x103
1.007E3
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Summary
1- Significant figures are of more general use as they don’t
depend on units used
e.g.
2,301.2 m (1d.p.) = 2.3012 km (4 d.p.)
2- Answers which are money should usually be given to 2
decimal places, so,
the nearest penny 3 ×£23.57895= £70.73685
= £70.74 to the nearest penny
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3- You must use at least one more s.f. in working than in
your answer
-To give an answer to 3 s.f. you generally need to use at
least 4 s.f. in working.
-To give an answer to 4 s.f. you generally need to use at
least 5 s.f. in working.
Example
Calculate 3.7545 x 8.91235 to 3 sig fig
You should at least use 3.754 x 8.912 but I would use all the
digits on the calculator unless otherwise stated.
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4- When calculating with numbers that have been measured to
different levels of accuracy, it makes sense to work the
calculation to the lowest level of measurement
“Treat Like with Like”
Example
If a cars speed has been measured as 40 to
(1 sig fig)
The distance travelled is measured as 10.91325 km (7 sig fig)
It makes some sense to estimate the time (=dist x speed) as :
40 (1 sf) x 10 (1 sf) = 400 sec
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If a cars velocity has been measured as 40.012
(5 sig fig)
The distance travelled is measured as 10.91325 km (7 sig fig)
It makes sense to estimate the time (=dist x speed) as
40.012 (5 sf) x 10.913 (5 sf)
or
= 436.6501 sec
= 436.65 (5 sig fig)
= 436.65 (2 dec pl)
Try to work to at least one digit higher accuracy.
Try to measure numbers to a sensible order of accuracy
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