Lesson 1- Basics
Objectives : To know how to approx numbers to a required accuracy by 2-3
1. Basic Number Types2. Decimal Places3. Significant Figures4. Writing numbers in Standard Form 5. Writing numbers in Engineering notation
Oct 2011 Foundation - L1
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...........
Oct 2011 Foundation - L1
Number TypesCounting Numbers
Positive Whole Numbers 1, 2, 3, 4, 5……
Natural Numbers
NCounting numbers and zero
0, 1, 2, 3, 4……
Integers ZAll positive and negative whole numbers
…-2, -1, 0, 1, 2…
Rational Numbers
Q
Numbers which can be written as a fraction where m and n are integers
-1, 0, ½ , 2¾ ,
Real Numbers RAll rational and irrational numbers
-1, 2¾ , π,
n
m
Oct 2011 Foundation - L1
Rational NumbersMost real numbers can be written as a fraction in its lowest form
n
m
Example:
Express 0.123123123123......... 123 as a fraction
312312312.0 x312312312.1231000 x Trick x 1000 to get rid of decimals
00000000.123999 x Subtract to get rid of decimals
999123 x 333
41 x
Oct 2011 Foundation - L1
Irrational Numbers─ But some numbers can not be expressed as fractions
─ Examples include e2
These are numbers where the patterns in the decimals do not repeat
....141592654.3We can not express numbers like this in faction form.
The irrational number set is much smaller than the set of rational numbers
Oct 2011 Foundation - L1
Proof that is irrational 2Method-
We will assume that it is rational and then we will contradict this assumption
n
m2
m and n are integers and the fraction can not be simplified further (i.e lowest form)
222
2
22 mnn
m So m2 is an even number
m2 – even this implies that m is even
m m2
1 22 43 94 165 25
so “m” can be written as “2 × a” (as m even)
so
2
242
n
a 22 2an (so n is even too!!)
So is n
m
even
even Both numerator and denominator are divisible by 2 and therefore is not in lowest form and can be simplified
n
m
Contradiction!!Oct 2011 Foundation - L1
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?
Oct 2011 Foundation - L1
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
42 m < Area < 56m
But a better guess might be 7m x 7m => 49m
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 FiguresConsider 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
Oct 2011 Foundation - L1
Sig Figs37.5
3 -> This is the 1st Sig Fig7 -> This is the 2nd Sig Fig5 -> This is the 3rd Sig Fig
The significance of numbers decreases from left to right
Oct 2011 Foundation - L1
Rounding to Sig FigsExample Approximate 37.5 to 1 significant figures Look at the next most significant number (2nd number)
30 4037
Round up if ≥ 35Round down < 35
37.5 is 40 to 1 significant figures
we write 40 (1 sf) Oct 2011 Foundation - L1
Rounding to Sig FigsExample Approximate 37.5 to 2 significant figures Look at the next most significant number (this is now 3rd No.)
37 3837.5
Round up if ≥ 37.5Round down < 37.5
37.5 is 38 to 2 significant figures
we write 38 (2 sf) Oct 2011 Foundation - L1
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.
Oct 2011 Foundation - L1
Example
Round up if ≥ Round down <
Round up if ≥ Round down <
Write the following to 3 sf
a) 12.455b) 0.013026c) 0.1005d) 13445.698e) 0.1999
Oct 2011 Foundation - L1
Find the following
801296 to 1 sf
801296 to 3 sf
-52.9000 to 3 sf
-52.9001 to 4 sf
Oct 2011 Foundation - L1
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
3.14159Interested in these numbers
Example : Express π (Pi) to 1 decimal place
π = 3.1415926535897932384
Round up if ≥ Round down < 3.1 3.2
3.5 3.5
π is 3.1 (1 dp)
Oct 2011 Foundation - L1
Decimal PlacesExample : Express π (Pi) to 2 decimal places
π = 3.1415926535897932384
This is < 5 so do not round upπ = 3.14 (2 dp)
Example : Express π (Pi) to 6 decimal places
π = 3.1415926535897932384
This is ≥ 5 so round upπ = 3.141593 (6 dp)
Oct 2011 Foundation - L1
Scientific Notation
A short-hand way of writing large or small numbers without writing all of the zeros
xExample :The Distance From the Sun to the Earth
93,000,000
Oct 2011 Foundation - L1
Step 1Move decimal leftLeave only one number in front of decimal
Step 2• Write number without zeros
Oct 2011 Foundation - L1
Step 3Count how many places you moved decimalMake that your power of ten
Oct 2011 Foundation - L1
Scientific NotationExample: Partial pressure of CO2 in atmosphere 0.000356 atm.
This number has 3 sig. figs, but leading zeros are only place-keepers and can cause some confusion.
So expressed in scientific notation this is
3.56 x 10-4 atm
This is much less ambiguous, as the 3 sig. figs. are clearly shown.Oct 2011 Foundation - L1
Engineering NotationThis is the same as scientific notation except the POWER is replaced by the letter E
Examples
Number Scientific Notation Engineering Notation
100 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
Oct 2011 Foundation - L1
Summary1- 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
Oct 2011 Foundation - L1
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.
Oct 2011 Foundation - L1
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
Oct 2011 Foundation - L1
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) = 436.6501 sec = 436.65 (5 sig fig) or = 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
Oct 2011 Foundation - L1