Handbook of mathematics for engineers and scienteists part 6 ppsx

All integers as well as even numbers and odd numbers form infinite countable sets, which means that the elements of these sets can be enumerated using the natural numbers 1,2,3,.. A prim

Arithmetic and Elementary Algebra

1.1 Real Numbers

1.1.1 Integer Numbers

1.1.1-1 Natural, integer, even, and odd numbers

Natural numbers: 1,2,3, (all positive whole numbers).

Integer numbers (or simply integers): 0, 1, 2, 3,

Even numbers: 0, 2,4, (all nonnegative integers that can be divided evenly by 2).

An even number can generally be represented as n =2k , where k =0,1,2,

Remark 1 Sometimes all integers that are multiples of 2, such as 0 , 2 , 4, , are considered to be

even numbers.

Odd numbers: 1,3,5, (all natural numbers that cannot be divided evenly by 2) An odd number can generally be represented as n =2k+1, where k =0,1,2,

Remark 2 Sometimes all integers that are not multiples of 2, such as 1 , 3 , 5, , are considered to

be odd numbers.

All integers as well as even numbers and odd numbers form infinite countable sets,

which means that the elements of these sets can be enumerated using the natural numbers

1,2,3,

1.1.1-2 Prime and composite numbers

A prime number is a positive integer that is greater than 1 and has no positive integer

divisors other than 1 and itself The prime numbers form an infinite countable set The first ten prime numbers are: 2,3,5,7,11,13,17,19,23,29,

A composite number is a positive integer that is greater than 1 and is not prime, i.e.,

has factors other than 1 and itself Any composite number can be uniquely factored into

a product of prime numbers The following numbers are composite: 4=2 × 2,6=2 × 3,

8=23,9=32,10=2 × 5,12=22× 3,

The number 1 is a special case that is considered to be neither composite nor prime

1.1.1-3 Divisibility tests

Below are some simple rules helping to determine if an integer is divisible by another integer

All integers are divisible by1

Divisibility by2: last digit is divisible by2

Divisibility by3: sum of digits is divisible by3

Divisibility by4: two last digits form a number divisible by4

Divisibility by5: last digit is either0or5

3

Divisibility by6: divisible by both2and3.

Divisibility by9: sum of digits is divisible by9

Divisibility by10: last digit is0

Divisibility by11: the difference between the sum of the odd numbered digits (1st, 3rd, 5th, etc.) and the sum of the even numbered digits (2nd, 4th, etc.) is divisible by 11

Example 1 Let us show that the number80729 is divisible by 11

The sum of the odd numbered digits is Σ 1 = 8 + 7 + 9 = 24 The sum of the even numbered digits is

Σ 2 = 0 + 2 = 2 The difference between them is Σ 1 – Σ 2 = 22 and is divisible by 11 Consequently, the original number is also divisible by 11

1.1.1-4 Greatest common divisor and least common multiple

1◦ The greatest common divisor of natural numbers a1, a2, , a nis the largest natural number, b, which is a common divisor to a1, , a n

Suppose some positive numbers a1, a2, , a nare factored into products of primes so that

a1= p k111p k12

2 p k m1m, a2= p k121p k222 p k m2m, ., a n = p k1n1p k n2

2 p k m nm, where p1, p2, , p n are different prime numbers, the k ij are positive integers (i = 1,2,

, n; j =1,2, , m) Then the greatest common divisor b of a1, a2, , a nis calculated as

b = p σ11p σ2

2 p σ m m, σ j = min1≤i≤n k ij.

Example 2 The greatest common divisor of180 and 280 is 2 2 × 5 = 20 due to the following factorization:

180 = 2 2 × 3 2 × 5 = 2 2 × 3 2 × 5 1 × 7 0 ,

280 = 2 3 × 5 × 7 = 2 3 × 3 0 × 5 1 × 7 1

2◦ The least common multiple of n natural numbers a1, a2, , a

nis the smallest natural number, A, that is a multiple of all the a k

Suppose some natural numbers a1, , a nare factored into products of primes just as

in Item1◦ Then the least common multiple of all the a

kis calculated as

A = p ν1

1 p ν22 p ν m m, ν j = max1≤i≤n k ij.

Example 3 The least common multiple of180 and 280 is equal to 2 3 × 3 2 × 5 1 × 7 1 = 2520 due to their factorization given in Example 2.

1.1.2 Real, Rational, and Irrational Numbers

1.1.2-1 Real numbers

The real numbers are all the positive numbers, negative numbers, and zero Any real number can be represented by a decimal fraction (or simply decimal), finite or infinite The set of

all real numbers is denoted byR

All real numbers are categorized into two classes: the rational numbers and irrational

numbers

1.1.2-2 Rational numbers.

A rational number is a real number that can be written as a fraction (ratio) p/q with integer

p and q (q ≠ 0) It is only the rational numbers that can be written in the form of finite (terminating) or periodic (recurring) decimals (e.g.,1/8= 0.125and 1/6 =0.16666 .) Any integer is a rational number

The rational numbers form an infinite countable set The set of all rational numbers is

everywhere dense This means that, for any two distinct rational numbers a and b such that

a < b, there exists at least one more rational number c such that a < c < b, and hence there are infinitely many rational numbers between a and b (Between any two rational numbers,

there always exist irrational numbers.)

1.1.2-3 Irrational numbers

An irrational number is a real number that is not rational; no irrational number can

be written as a fraction p/q with integer p and q (q ≠ 0) To the irrational numbers there correspond nonperiodic (nonrepeating) decimals Examples of irrational numbers:

√

3=1.73205 , π =3.14159 .

The set of irrational numbers is everywhere dense, which means that between any two distinct irrational numbers, there are both rational and irrational numbers The set of irrational numbers is uncountable

1.2 Equalities and Inequalities Arithmetic Operations Absolute Value

1.2.1 Equalities and Inequalities

1.2.1-1 Basic properties of equalities

 Throughout Paragraphs 1.2.1-1 and 1.2.1-2, it is assumed that a, b, c, d are real numbers.

1 If a = b, then b = a.

2 If a = b, then a + c = b + c, where c is any real number; furthermore, if a + c = b + c, then

a = b.

3 If a = b, then ac = bc, where c is any real number; furthermore, if ac = bc and c≠ 0, then

a = b.

4 If a = b and b = c, then a = c.

5 If ab =0, then either a =0or b =0; furthermore, if ab≠ 0, then a≠ 0and b≠ 0

1.2.1-2 Basic properties of inequalities

1 If a < b, then b > a.

2 If a≤b and b≤a , then a = b.

3 If a≤b and b≤c , then a≤c

4 If a < b and b≤c (or a≤b and b < c), then a < c.

5 If a < b and c < d (or c = d), then a + c < b + d.

6 If a≤b and c >0, then ac≤bc

7 If a≤b and c <0, then ac≥bc

8 If0< a≤b (or a≤b<0), then1/a≥ 1/b

1.2.2 Addition and Multiplication of Numbers

1.2.2-1 Addition of real numbers

The sum of real numbers is a real number

Properties of addition:

a+0= a (property of zero),

a + b = b + a (addition is commutative),

a + (b + c) = (a + b) + c = a + b + c (addition is associative),

where a, b, c are arbitrary real numbers.

For any real number a, there exists its unique additive inverse, or its opposite, denoted

by –a, such that

a + (–a) = a – a =0

1.2.2-2 Multiplication of real numbers

The product of real numbers is a real number

Properties of multiplication:

a× 0=0 (property of zero),

ab = ba (multiplication is commutative),

a (bc) = (ab)c = abc (multiplication is associative),

a× 1=1 ×a = a (multiplication by unity),

a (b + c) = ab + ac (multiplication is distributive),

where a, b, c are arbitrary real numbers.

For any nonzero real number a, there exists its unique multiplicative inverse, or its reciprocal, denoted by a–1or1/a, such that

aa–1 =1 (a≠ 0)

1.2.3 Ratios and Proportions

1.2.3-1 Operations with fractions and properties of fractions

Ratios are written as fractions: a : b = a/b The number a is called the numerator and the number b (b≠ 0) is called the denominator of a fraction.

Properties of fractions and operations with fractions:

a

1 = a,

a

b = ab

bc = a : c

b : c (simplest properties of fractions);

a

b

c

b = a c

b , a

b

c

d = ad bc

bd (addition and subtraction of fractions);

a

b ×c= ac

b × c

d = ac

bc (multiplication by a number and by a fraction);

a

b : c = a

b : c

d = ad

bc (division by a number and by a fraction)

1.2.3-2 Proportions Simplest relations Derivative proportions.

A proportion is an equation with a ratio on each side A proportion is denoted by a/b = c/d

or a : b = c : d.

1◦ The following simplest relations follow from a/b = c/d:

ad = bc, a

c = b

d, a= bc

d, b= ad

c

2◦ The following derivative proportions follow from a/b = c/d:

ma + nb

pa + qb =

mc + nd

pc + qd ,

ma + nc

pa + qc =

mb + nd

pb + qd , where m, n, p, q are arbitrary real numbers.

Some special cases of the above formulas:

a b

b = c d

d , a – b

a + b =

c – d

c + d.

1.2.4 Percentage

1.2.4-1 Definition Main percentage problems

A percentage is a way of expressing a ratio or a fraction as a whole number, by using 100 as the denominator One percent is one per one hundred, or one hundredth of a whole number;

notation: 1%

Below are the statements of main percentage problems and their solutions

1◦ Find the number b that makes up p% of a number a Answer: b = ap

100.

2◦ Find the number a whose p% is equal to a number b Answer: a = 100b

p .

3◦ What percentage does a number b make up of a number a? Answer: p = 100b

a %.

1.2.4-2 Simple and compound percentage

1◦ Simple percentage Suppose a cash deposit is increased yearly by the same amount defined as a percentage, p%, of the initial deposit, a Then the amount accumulated after

tyears is calculated by the simple percentage formula

x = a

1+ pt

100



2◦ Compound percentage Suppose a cash deposit is increased yearly by an amount defined

as a percentage, p%, of the deposit in the previous year Then the amount accumulated after

tyears is calculated by the compound percentage formula

x = a

1+ p

100

t

,

where a is the initial deposit.

1.2.5 Absolute Value of a Number (Modulus of a Number)

1.2.5-1 Definition

The absolute value of a real number a, denoted by|a|, is defined by the formula

|a|=

a if a≥ 0,

–a if a <0

An important property: |a| ≥ 0

1.2.5-2 Some formulas and inequalities

1◦ The following relations hold true:

|a|=|–a|=√

a2, a≤ |a|,

|a|–|b|≤ |a + b| ≤ |a|+|b|,

|a|–|b|≤ |a – b| ≤ |a|+|b|,

|ab|=|a| |b|, |a/b|=|a|/|b|

2◦ From the inequalities|a| ≤Aand|b| ≤B it follows that|a + b| ≤A + B and|ab| ≤AB

1.3 Powers and Logarithms

1.3.1 Powers and Roots

1.3.1-1 Powers and roots: the main definitions

Given a positive real number a and a positive integer n, the nth power of a, written as a n,

is defined as the multiplication of a by itself repeated n times:

a n = a×a×a×· · ·×a

n multipliers

The number a is called the base and n is called the exponent.

Obvious properties: 0n=0,1n=1, a1 = a.

Raising to the zeroth power: a0=1, where a≠ 0 Sometimes00is taken as undefined,

but it is often sensibly defined as1

Raising to a negative power: a–n= 1

a n , where n is a positive integer.

If a is a positive real number and n is a positive integer, then the nth arithmetic root or radical of a, written as √ n

a , is the unique positive real number b such that b n = a In the case of n =2, the brief notation√

ais used to denote√2

a The following relations hold:

n

√

0=0, √ n

1=1, √ n

a n

= a.

Raising to a fractional power p = m/n, where m and n are natural numbers:

a p = a m/n = √ n

a m, a≥ 0

1.3.1-2 Operations with powers and roots.

The properties given below are valid for any rational and real exponents p and q (a >0,

b>0):

a–p = 1

a p, a p a q = a p+q,

a p

a q = a p–q, (ab) p = a p b p, a

b

q

= a q

b q, (a p)q = a pq.

In operations with roots (radicals) the following properties are used:

n

ab= n

a n

b, n a

b =

n

√ a

n

√

b, n

a m = n

a m

, n



m √

a= mn √

a

Remark It often pays to represent roots as powers with rational exponents and apply the properties of operations with powers.

1.3.2 Logarithms

1.3.2-1 Definition The main logarithmic identity

The logarithm of a positive number b to a given base a is the exponent of the power c to which the base a must be raised to produce b It is written as log a b = c.

Equivalent representations:

loga b = c ⇐⇒ a c = b, where a >0, a≠ 1, and b >0

Main logarithmic identity:

aloga b = b.

Simple properties:

loga1=0, loga a=1

1.3.2-2 Properties of logarithms The common and natural logarithms

Properties of logarithms:

loga (bc) = log a b+ loga c, logab

c



= loga b– loga c, loga (b k ) = k log a b, loga k b= 1

kloga b (k≠ 0), loga b= 1

logb a (b≠ 1), loga b= logc b

logc a (c≠ 1),

where a >0, a≠ 1, b >0, c >0, and k is any number.

The logarithm to the base10is called the common or decadic logarithm and written as

log10b = log b or sometimes log10b = lg b.

The logarithm to the base e (the base of natural logarithms) is called the natural logarithm and written as

loge b = ln b, where e = lim

n→∞ 1+ 1n n

=2.718281 .

a

Remark It often pays to represent roots as powers with rational exponents and apply the properties of operations with powers.

1.3.2...

loga1=0, loga a=1

1.3.2-2 Properties of logarithms The common and natural logarithms

Properties of logarithms:

loga (bc) = log a...

1.3.2-1 Definition The main logarithmic identity

The logarithm of a positive number b to a given base a is the exponent of the power c to which the base a must be raised to produce b It is