Mathematics 1 11 May, 2015M L1 MH. your maths teacher for Maths 1 Dr Michael Hughes (Mike) [email protected] The required textbook A2 Pure Mathematics

Mathematics

118 April 2023 M L1 MH

your maths teacherfor Maths 1

Dr Michael Hughes (Mike)

[email protected]

The required textbook

A2 Pure Mathematics C3/C4



Objectives : - Scientific Notation - Error estimation - Surds recap - Algebraic expression recap


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

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


Step 2• Write number without zeros

Step 3

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



Example: 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 standard form this is

3.56 x 10-4 atm

This is much less ambiguous, as the 3 s.f. are clearly shown.


This is the same as scientific notation except the POWER is replaced by the letter E

Examples

Number Scientific Notation/

Standard Form 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

Rational/Irrational Numbers

Rational numbers can be expressed as a fraction with no common factors

Irrational numbers can not be expressed as a fraction in its lowest terms

Surds are irrational numbers like π, √2

They have a non repeating infinite pattern of decimal places.


Rules for Surds

Try not to be lazy and therefore express them in their lowest form

Example


38

364364192

baba

b

a

b

a

baba

Surd Rules

Rationalise the denominator

If you have the following

Rationalise it by multiplying by 1


3

1

a

ba

ba

15

10

Example

Exercise 5a page 130

Errors

Suppose a cars petrol tank holds 50 litres of petrol and you think the car does 12km for each litre of petrol.

Is it safe to travel 600 km on a full tank of Petrol?

Solution

In practice the car may travel as little as 10km / ltr

or as much as 12.5 km/ltr

Therefore one might be able to drive anywhere between

500 ≤ distance ≤ 650


Example

If we say a piece of wood is 5.0 m long

We are implying that it is 4.95 ≤ length ≤ 5.05

if we say a piece of wood is 5.23 m long

We are implying that it is 5.225 ≤ length ≤ 5.235


Relative and absolute error

A lawn is said to be 12m x 22m

(a) Between what bounds does the area lie

The true Area is 272.55m2 and the householder measured the area as 264m2

(b) What is the absolute error

(c) What is the relative (Percentage) error


Solution

Max Area is 12.5 x 22.5 = 281.25Min area is 11.5 x 21.5 = 247.25

247.25 ≤ Area ≤ 247.25

Absolute error is |272.55-264| = 8.5m2

Relative error is |272.55-264| = absolute error = 3.1%

272.55 true value


Question

Exercise

Find the percentage error when π is given the following approximate values:

(i) 3 (ii) (iii) 3.14 (iv) √10

Take the true value of π to be the number stored on your calculator.


7

22


Algebraic expressions

Adding


xyz

y

yz

x

3

20

13

102

Subtracting


xyz

y

yz

x

3

20

13

102

Multiplying and Dividing

Remember our index rules here


10

20

49

102

4 xyz

yzx

x

Summary

We have recapped on the following topics

- Scientific Notation

- Rational Numbers and Surds

- Absolute and Relative error

- Algebraic Expressions


Documents

Mathematics 1 11 May, 2015M L1 MH. your maths teacher for Maths 1 Dr Michael Hughes (Mike) [email protected] The required textbook A2 Pure Mathematics