Summary College Algebra 1st chapter Schaums 4th ed

Fundamental Operations with Numbers

FOUR OPERATIONS
ARITHMETIC is the fundamental of mathematics that includes the operations of numbers such as:
 Addition
 Subtraction
 Multiplication
 Division

 Addition combines two or more values into a single term
When two numbers a and b are added, their sum is indicated by a + b
 Subtraction computes the difference between two value
When a number b is subtracted from a number a, the difference is indicated by a-b
 Multiplication also combines two values like addition and subtraction
The product of two numbers a and b is a number c such that a x b = c or a • b = c
 Division
When a number a is divided by a number b, the quotient obtained is written
𝒂
𝒂 ÷ 𝒃 𝒂/𝒃
𝒃
where a is called the dividend and b the divisor.
The expression a/b is also called a fraction, having numerator a and denominator b.
Division by zero is not defined.


SYSTEM OF REAL NUMBERS
The real number system consists of the collection of positive and negative rational and irrational numbers and zero.

 Natural numbers 1,2,3,4, . . . used in counting are also known as positive integers.
If two such numbers are added or multiplied, the result is always a natural number.
2 8
 Positive rational numbers or positive fractions are the quotients of two positive integers, such as or
3
.
5
The positive rational numbers include the set of natural numbers.
3
Thus the rational number is the natural number 3.
1
 Positive irrational numbers are numbers which are not rational, such as 2 , 𝜋

 Zero, written 0, arose in order to enlarge the number system so as to permit such operations as 6-6 or 10-10.
Zero has the property that any number multiplied by zero is zero.
Zero divided by any number ≠ 0(i.e., not equal to zero) is zero.
Zero is considered a rational number without sign.

 Negative integers (-3), negative rational numbers (-2/3) and negative irrational numbers(- 2)




Note:
Imaginary number −1

, PROPERTIES OF ADDITION AND MULTIPLICATION OF REAL NUMBERS
 COMMUTATIVE PROPERTY FOR ADDITION
The order of addition of two numbers does not affect the result.
a+b=b+a

 ASSOCIATIVE PROPERTY FOR ADDITION
The terms of a sum may be grouped in any manner without affecting the result
a + b +c = a + (b + c) = (a + b) + c

 COMMUTATIVE PROPERTY FOR MULTIPLICATION
The order of factors of a product does not affect the result
a•b=b•a

 ASSOCIATIVE PROPERTY FOR MULTIPLICATION
The factors of a product may be grouped in any manner without affecting the result
abc = (ab)c = a(bc)

 DISTRIBUTIVE PROPERTY FOR MULTIPLICATION OVER ADDITION
The product of a number a by the sum of two numbers (b + c) is equal to the sum of the products ab and ac
a(b + c) = ab + ac


RULES OF SIGNS
 To add two numbers with like signs, add their absolute values and prefix the common signs.
3+4=7 (-3) + (-4) = -7
 To add two numbers with unlike signs, find the difference between their absolute values and prefix the sign of the
numbers with greater absolute value
17 + (-8) = 9 (-6) + 4 = -2
 To subtract one number b from another a, change the operation to addition and replace b by its opposite, -b.
12 - 7 = 5 (-9) - (4) = -13 2 - (-8) =10
12 + (-7) = 5 (-9) + (-4) = -13 2 + 8 = 10

 To multiply or divide two numbers having like signs, multiply or divide their absolute values and prefix a plus sign
or no sign
(5) (3) = 15 −𝟔
= 𝟐
(-5)(-3) = 15 −𝟑
 To multiply or divide two numbers having unlike signs, multiply or divide their absolute values and prefix a minus
sign
(-3)(6) = -18 −𝟏𝟐
= −𝟑
(3)(-6) = -18 𝟒


EXPONENTS AND POWERS
When a number a is multiplied by itself n times, the product a•a•a•••a (n times) is indicated by the symbol: 𝒂𝒏
𝒂𝒏 : “nth power of a” , “a to the nth power” , “a to the nth”
2•2•2•2•2 = 𝟐𝟓 = 32 a•a•a•b•b = 𝒂𝟑 𝒃𝟐

, LAWS OF EXPONENTS
In 𝒂𝒏 , the number a is called base and the positive integer n is the exponent
If p and q are positive integers, then the following are laws of exponents
 𝒂𝒑 •𝒂𝒒 = 𝒂𝒑+𝒒
𝟐𝟑 𝟐𝟒 = 𝟐𝟑+𝟒 = 𝟐𝟕
𝒂𝒑 1
 = 𝒂𝒑−𝒒 =
𝒂𝒒 𝒂𝒒−𝒑
𝟑𝟓 𝟑𝟒 1 1
= 𝟑𝟓−𝟐 = 𝟑𝟑 = =
𝟑𝟐 𝟑𝟔 𝟑𝟔−𝟒 𝟑𝟐

 (𝒂𝒑 )q = 𝒂𝒑𝒒
(𝟒𝟐 )3= 𝟒𝟔
𝑎 𝒂𝒑
 (𝒂𝒃)𝒑 = 𝒂𝒑 𝒃𝒑 , ( )𝑝 = if b ≠ 0
𝑏 𝒃𝒒
(𝟒 • 𝟓)𝟐 = 𝟒𝟐 • 𝟓𝟐 𝟓 𝟓𝟑
( )𝟑 = 𝟑
𝟐 𝟐

OPERATIONS WITH FRACTIONS
 The value of a fraction remains the same if its numerator and denominator are both multiplied or divided by the same
number provided the number is not zero.
𝟑 𝟑•𝟐 𝟔 𝟏𝟓 𝟏𝟓 ÷ 𝟑 𝟓
= =
𝟒 𝟒•𝟐 𝟖
= =𝟔
𝟏𝟖 𝟏𝟖 ÷ 𝟑

 Changing the sign of the numerator or denominator of a fraction changes the sign of the fraction.
−𝟑 𝟑 𝟑
𝟓
= - 𝟓 = −𝟓

 Adding two fractions with a common denominator yields a fraction whose numerator is the sum of the numerators of
the given fractions and whose denominator is the common denominator.
𝟑 𝟒 𝟑+𝟒 𝟕
‘ + = 𝟓
𝟓 𝟓
=𝟓

 The sum or difference of two fractions having different denominators may be found by writing the fractions with a
common denominator.
𝟏 𝟐 𝟑 𝟖 𝟏𝟏
+ = +
𝟒 𝟑 𝟏𝟐 𝟏𝟐
= 𝟏𝟐

 The product of two fractions is a fraction whose numerator is the product of the numerators of the given fractions
and whose denominator is the product of the denominators of the fractions.
𝟐 𝟒 𝟐•𝟒 𝟖
• = = 𝟏𝟓
𝟑 𝟓 𝟑•𝟓

 The reciprocal of a fraction is a fraction whose numerator is the denominator of the given fraction and whose
denominator is the numerator of the given fraction.
Thus the reciprocal of
𝟑 𝟏 𝟓 𝟖 −𝟒 𝟑 −𝟑
3 (or ) reciprocal is reciprocal is reciprocal is or
𝟏 𝟑 𝟖 𝟓 𝟑 −𝟒 𝟒

 To divide two fractions, multiply the first by the reciprocal of the second.
𝒂 𝒄 𝒂 𝒅 𝒂𝒅
𝒃
÷ 𝒅 = 𝒃 • 𝒄 = 𝒃𝒄