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Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1. Basic Number Types 2. Decimal Places 3. Significant Figures 4. Writing numbers in Standard Form 5. Writing numbers in Engineering notation Oct 2011 Foundation - L1

Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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Page 1: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 2: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 3: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 4: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 5: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 6: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 7: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 8: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 9: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 10: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 11: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 12: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 13: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 14: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 15: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 16: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 17: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 18: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 19: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

Step 1Move decimal leftLeave only one number in front of decimal

Step 2• Write number without zeros

Oct 2011 Foundation - L1

Page 20: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

Step 3Count how many places you moved decimalMake that your power of ten

Oct 2011 Foundation - L1

Page 21: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 22: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 23: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 24: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 25: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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

Page 26: Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing

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