Module :MA0001NIFoundation Mathematics

Lecture Week 1

Agenda• Module Introduction

Your Module Leader Your Lecturer and tutorsModule ObjectiveTutorial/ workshopModule outcomes after successful completionModule Assessments and Syllabus SummaryRecommended booklist

Module Leaders’ Roles

• Every module has two module leaders:

•Creates the main lecture/tutorial notes•Writes coursework and examinations•Moderates the coursework and examinations results•Serves as a lecturer for module (usually in London)

•Writes localised lecture/tutorial notes•Marks the coursework and examinations (lecturers/tutors might also be involved in marking)•Serves as a lecturer for that module

Your Module Leaders Are…

(Islington CollegeLecturer)

Mr.David Brown(LondoN Metropolitan

University)Mr.Ashok Dhungana

Your Lecturer/Tutor

Email:

ashok.dhungana@islingtoncollege.edu.np

Phone # 977 (1) 4420054 ext. 26

977(1) 4412929 ext. 26

Mr. Ashok Dhungana

(MSc IT, TU, Nepal)

Module Assessments• Assessments: 40% CW1(Course Work ) 60% CW2 (EXAM )

Note:- Students should obtain 40% pass on aggregate from examination and coursework.

Syllabus• Number

Fractions, decimals, percentages, ratio, proportion, scientific notation, estimation, calculator use

• Basic AlgebraAlgebraic notation, manipulation of algebraic expressions. Transposition and evaluation of algebraic formulae. Formulation of problems in algebraic form. Solution of linear equations, simultaneous equations and quadratic equations.

• GraphsPlotting linear and non-linear graphs. Gradient and intercept.

• Indices and LogarithmsSimple indices, Exponentials and Logarithms

Classifying numbers

Integers

Calculating with integers

Multiples, factors and primes

Prime factor decomposition

LCM and HCF

Classifying numbers• Natural numbers

Positive whole numbers 0, 1, 2, 3, 4 …• Integers

Positive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, …

• Rational numbersNumbers that can be expressed in the form n/m, where n and m are

integers. All fractions and all terminating and recurring decimals

are rational numbers, for example, ¾, –0.63, 0.2.

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Classifying numbers• Irrational numbers

• Numbers that cannot be expressed in the form n/m, where n and m are integers. Examples of

irrational numbers are and 2.• Even numbers are numbers that are exactly

divisible by 2.

The nth even number can be written as E(n) =2n.

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Classifying numbers• Odd numbers leave a remainder of 1 when

divided by 2.The nth odd number can be written as U(n) = 2n –1• Triangular numbers are numbers that can be

written as the sum of consecutive whole numbers starting with 1.

For example, 15 is a triangular number. It can be written as15 = 1 + 2 + 3 + 4 + 5

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Classifying numbers• So, for any triangular number T(n) T(n) =n(n + 1)/ 2We can now use this rule to find the value of the

50th triangular number. T(50) =50(50 + 1)/2 T(50) = 1275 T(100)=???

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Contents

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N1.2 Calculating with integers

Integers

N1.3 Multiples, factors and primes

N1.4 Prime factor decomposition

N1.5 LCM and HCF

N1.1 Classifying numbers

Adding integersWe can use a number line to help us add positive and negative integers.

–2 + 5 =

-2 3

= 3

To add a positive integer we move forwards up the number line.

5-3

Subtracting integersWe can use a number line to help us subtract positive and negative integers.

5 – 8 == –3

To subtract a positive integer we move backwards down the number line.

5 – 8 is the same as 5 – +8

Adding and subtracting integers

To add a positive integer we move forwards up the number line.

To add a negative integer we move backwards down the number line.

To subtract a positive integer we move backwards down the number line.

To subtract a negative integer we move forwards up the number line.

a + –b is the same as a – b.

a – –b is the same as a + b.

When multiplying negative numbers remember:

Rules for multiplying and dividing

Dividing is the inverse operation to multiplying.

When we are dividing negative numbers similar rules apply:

+ × + = +

–+ × = –

–+× =–

– +× =–

+ ÷ + = +

–+ ÷ = –

–+÷ =–

– +÷ =–

Contents

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N1.3 Multiples, factors and primes

Integers

N1.4 Prime factor decomposition

N1.5 LCM and HCF

N1.2 Calculating with integers

N1.1 Classifying numbers

Multiples

A multiple of a number is found by multiplying the number by any whole number.

What are the first six multiples of 7?

To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5 and 6 in turn to get:

7, 14, 21, 28, 35 and 42.

Any given number has infinitely many multiples.

Factors

A factor (or divisor) of a number is a whole number that divides into it exactly.

Factors come in pairs. For example,

What are the factors of 30?

1 and 30, 2 and 15, 3 and 10, 5 and 6.

So, in order, the factors of 30 are:

1, 2, 3, 5, 6, 10, 15 and 30.

Prime numbersIf a whole number has two, and only two, factors it is called a prime number.

For example, the number 17 has only two factors, 1 and 17.

Therefore, 17 is a prime number.

The number 1 has only one factor, 1.

Therefore, 1 is not a prime number.

There is only one even prime number. What is it?

2 is the only even prime number.

Contents

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N1.4 Prime factor decomposition

Integers

N1.5 LCM and HCF

N1.3 Multiples, factors and primes

N1.2 Calculating with integers

N1.1 Classifying numbers

A prime factor is a factor that is a prime number.

For example,

What are the prime factors of 70?

The factors of 70 are:

1 2 5 7 10 14 35 70

The prime factors of 70 are 2, 5, and 7.

Prime factors

The prime factor decompositionWhen we write a number as a product of prime factors it is called the prime factor decomposition or prime factor form.

For example,

The prime factor decomposition of 100 is:

There are two methods of finding the prime factor decomposition of a number.

100 = 2 × 2 × 5 × 5 = 22 × 52

36

4 9

2 2 3 3

36 = 2 × 2 × 3 × 3

= 22 × 32

Factor trees

962

482

242

122

62

33

1

2

2

2

2

2

3

96 = 2 × 2 × 2 × 2 × 2 × 3

= 25 × 3

Dividing by prime numbers

Prime factor decompositionUse the prime factor form of 324 to show that it is a square number.

3242

1622

813

273

93

33

1

2

2

3

3

3

3

324 = 2 × 2 × 3 × 3 × 3 × 3

= 22 × 34

This can be written as:

(2 × 32) × (2 × 32)

or (2 × 32)2

If all the indices in the prime factor decomposition of a number are even, then the number is a square number.

If all the indices in the prime factor decomposition of a number are even, then the number is a square number.

Using the prime factor decompositionUse the prime factor form of 3375 to show that it is a cube number.

33753

11253

3753

1255

255

55

1

3

3

3

5

5

5

3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53

This can be written as:

(3 × 5) × (3 × 5) × (3 × 5)

or (3 × 5)3

If all the indices in the prime factor decomposition of a number are multiples of 3, then the number is a cube number.

If all the indices in the prime factor decomposition of a number are multiples of 3, then the number is a cube number.

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N1.5 LCM and HCF

Integers

N1.4 Prime factor decomposition

N1.3 Multiples, factors and primes

N1.2 Calculating with integers

N1.1 Classifying numbers

The lowest common multipleThe lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.

For small numbers we can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.

For example,

Multiples of 20 are : 20, 40, 60, 80, 100, 120, . . .

Multiples of 25 are : 25, 50, 75, 100, 125, . . .

The LCM of 20 and 25 is 100.

The highest common factorThe highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.

We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.

For example,

Factors of 36 are : 1, 2, 3, 4, 6, 9, , 12 18, 36.

Factors of 45 are : 1, 3, 5, 9, 15, 45.

The HCF of 36 and 45 is 9.

Using prime factors to find the HCF and LCM

We can use the prime factor decomposition to find the HCF and LCM of larger numbers.

For example,

Find the HCF and the LCM of 60 and 294.

602302153551

60 = 2 × 2 × 3 × 5

29421473497771

294 = 2 × 3 × 7 × 7

60 294

60 = 2 × 2 × 3 × 5

294 = 2 × 3 × 7 × 7

22

35

7

7

HCF of 60 and 294 = 2 × 3 = 6

LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940

Using prime factors to find the HCF and LCM

The LCM of co-prime numbersIf two numbers have a highest common factor (or HCF) of 1 then they are called co-prime or relatively prime numbers.

For two whole numbers a and b we can write:

If two whole numbers a and b are co-prime then:

For example, the numbers 8 and 9 do not share any common multiples other than 1. They are co-prime.

Therefore, LCM(8, 9) = 8 × 9 = 72

a and b are co-prime if HCF(a, b) = 1a and b are co-prime if HCF(a, b) = 1

LCM(a, b) = abLCM(a, b) = ab

The LCM of numbers that are not co-prime

If two numbers are not co-prime then their highest common factor is greater than 1.

If two numbers a and b are not co-prime then their lowest common multiple is equal to the product of the two numbers divided by their highest common factor.

We can write this as:

For example,

LCM(a, b) = ab

HCF(a, b)

LCM(8, 12) =8 × 12

HCF(8, 12)=

964

= 24