Exponents
(1) Letnbe a positive integer and letabe a real number We definea n to be the real number given by a n =a| {z }ãaã ã ãa n factors
(2) Letnbe a negative integer n, that is, n = −kwhere kis a positive integer, and leta be a real number different from 0 We definea −k to be the real number given by a −k = 1 a k
(3) (i) Letabe a real number different from 0 We definea 0 =1.
(ii) We do not define 0 0 (thus the notation 0 0 is meaningless).
Terminology In the notationa n , the numbersnandaare called theexponentandbaserespectively.
Rules for Exponents Letaandbbe real numbers and letmandnbe integers (whena=0 orb=0, we have to add the condition:m,ndifferent from 0) Then we have
1 Simplify the following; give your answers without negative exponents.
Algebraic Identities and Algebraic Expressions
Identities Letaandbbe real numbers Then we have
RemarkThe above equalities are calledidentitiesbecause they are valid for all real numbersaandb.
Caution In general, (a+b) 2 ,a 2 +b 2 Note: (a + b) 2 = a 2 + b 2 if and only if a = 0 or b = 0.
0.2 Algebraic Identities and Algebraic Expressions 3
FAQ What is expected if we are asked to simplify an expression? For example, in (5), can we give 3x + 3 x 2 as the answer?
Answer There is no definite rule to tell which expression is simpler For (5), both 3(x + 1) x 2 and 3x + 3 x 2 are acceptable Use your own judgment
Solving Linear Equations
A linear equation in one real unknown \( x \) is expressed in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants, with \( a \neq 0 \) In this context, constants refer to fixed real numbers More broadly, any equation involving a single unknown \( x \) can be represented in a similar format.
Remark To be more precise, F should be a function from a subset of R into R See later chapters for the meanings of “function” and “R”.
Definition Asolutionto Equation (0.3.1) is a real numberx 0 such thatF(x 0 )=0.
Example The equation 2x+3=0 has exactly one solution, namely− 3
To solve an equation (in one unknown) means to find all solutions to the equation.
Definition We say that two equations areequivalentif the have the same solution(s).
Example The following two equations are equivalent:
To solve an equation, we use properties of real numbers to transform the given equation to equivalent ones until we obtain an equation whose solutions can be found easily.
Properties of real numbers Leta,bandcbe real numbers Then we have
(2) a=b=⇒ac=bc and ac=bc=⇒a=bifc,0
• =⇒is the symbol for “implies” The first part of Property (2) means that ifa=b, thenac=bc.
The symbol ⇐⇒ represents the logical equivalence of "if and only if" (iff), indicating that the statement a = b holds true if and only if a + c = b + c This means that the equality of a and b is directly linked to the equality of their respective sums with c.
Example Solve the following equations forx.
(2) a(b+x)=c−dx, wherea,b,canddare real numbers witha+d ,0.
(1) Using properties of real numbers, we get
5. FAQ Can we omit the last sentence?
Answer The steps above means that a real number xsatisfies 3x−5 = 2(7−x) if and only ifx = 19
5. It’s alright if you stop at the last line in the equation array because it tells that given equation has one and only one solution, namely 19
FAQ What is the difference between the word “solution” after the question and the word “solution” in the last sentence?
The term "solution" can refer to different concepts: one meaning is the answer to a problem, while the other pertains to the solution of a mathematical equation For instance, an equation like x² + 1 = 0 may have no solution, yet the methods used to determine this outcome represent a solution to the problem at hand.
FAQ Can we use other symbols for the unknown?
Answer In the given equation, if xis replaced by another symbol, for example, t, we get the equation
3t−5=2(7−t) in one unknownt Solution to this equation is also 19 In writing an equation, the symbol
6 Chapter 0 Revision for the unknown is not important However, if the unknown is expressed int, all the intermediate steps should usetas unknown:
(2) Using properties of real numbers, we get a(b+x) = c−dx ab+ax = c−dx ax+dx = c−ab
1 Solve the following equations forx.
Solving Quadratic Equations
A quadratic equation in one unknown is expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero To solve this equation, one can utilize methods such as factorization.
Factorization Method The method makes use of the following result on product of real numbers:
Fact Letaandbbe real numbers Then we have ab=0⇐⇒a=0 orb=0.
Solution Factorizing the left side, we obtain
FAQ Can we write “x=−5 andx=3”?
Answer The logic in solving the above equation is as follows x 2 +2x−15=0 ⇐⇒ (x+5)(x−3)=0
It means that a (real) number x satisfies the given equation if and only if x = −5 or x = 3 The statement
“x=−5 orx=3” cannot be replaced by “x=−5 andx=3”.
To say that there are two solutions, you may write “the solutions are−5and3” Sometimes, we also write
“the solutions are x 1 =−5and x 2 = 3” which means “there are two solutions−5and3and they are denoted by x 1 and x 2 respectively”.
In Chapter 1, you will learn the concept ofsets To specify a set, we may use “listing” or “description”.
Thesolution setto an equation is the set consisting of all the solutions to the equation For the above example, we may write
• the solution set is{−5,3} (listing);
• the solution set is{x:x=−5 orx=3} (description).
When we useand, we mean the listing method
Quadratic Formula Solutions to Equation (0.4.1) are given by x= −b±√ b 2 −4ac
(1) Ifb 2 −4ac>0, then (0.4.1) has two distinct solutions.
(2) Ifb 2 −4ac=0, then (0.4.1) has one solution.
(3) Ifb 2 −4ac