Preliminaries

The Sets of Numbers Used in Algebra

The sets of numbers used in algebra are, in general, subsets of R, the set of real numbers.

Natural Numbers N

The counting numbers, e.g., 1, 2, 3, 4, …

I ntegers Z

The counting numbers, together with their opposites and 0, e.g., 0, 1, 2, 3, $ $ -1, -2, -3, $ $

Rational Numbers Q

The set of all numbers that can be written as quotients $ a/b $ , $ b $ , a and b integers, e.g., 3/17, 10/3, -5.13, . . .

I rrational Numbers H

All real numbers that are not rational numbers, e.g., $ $ , $ $ , $ $ , $ -/3 $ , $ $

EXAMPLE 1.1 The number -5 is a member of the sets Z, Q, R. The number 156.73 is a member of the sets Q, R. The number $ 5$ is a member of the sets H, R.

Axioms for the Real Number System

There are two fundamental operations, addition and multiplication, that have the following properties $ (a, b, c $ arbitrary real numbers):

Closure Laws

The sum $ a + b $ and the product $ a b $ or ab are unique real numbers.

Commutative Laws

$ a + b = b + a $ : order does not matter in addition.

ab = ba: order does not matter in multiplication.

Associative Laws

$ a + (b + c) = (a + b) + c $ : grouping does not matter in repeated addition.

$ a(bc) = (ab)c $ : grouping does not matter in repeated multiplication.

Note (removing parentheses): Since $ a + (b + c) = (a + b) + c $ , $ a + b + c $ can be written to mean either quantity

Also, since $ a(bc) = (ab)c $ , abc can be written to mean either quantity.

Distributive Laws

$ a(b+c)=ab+ac; $ also $ (a+b)c=ac+bc: $ multiplication is distributive over addition.

I dentity Laws

There is a unique number 0 with the property that $ 0 + a = a + 0 = a $ .

There is a unique number 1 with the property that $ 1 a = a = a $ .

I nverse Laws

For any real number a, there is a real number -a such that $ a + (-a) = (-a) + a = 0 $ .

For any nonzero real number a, there is a real number $ a^{-1} $ such that $ aa^{-1} = a^{-1}a = 1 $ .

-a is called the additive inverse, or negative, of a.

$ a^{-1} $ is called the multiplicative inverse, or reciprocal, of a.

EXAMPLE 1.2 Associative and commutative laws: Simplify $ (3 + x) + 5 $ .

\[ \begin{aligned}(3+x)+5&=(x+3)+5\quad&Commutative law\\&=x+(3+5)\quad&Associative law\\&=x+8\end{aligned} \]

EXAMPLE 1.3 FOIL (First Outer Inner Last). Show that $ (a + b)(c + d) = ac + ad + bc + bd $

\[ \begin{aligned}(a+b)(c+d)&=a(c+d)+b(c+d)\quad&by the second form of the distributive law\\&=ac+ad+bc+bd\quad&by the first form of the distributive law\end{aligned} \]

Zero Factor Laws

  1. For every real number $ a, a = 0 $ .

  2. If ab = 0, then either a = 0 or b = 0.

Laws for Negatives

  1. $ -(-a)=a $

  2. $ (-a)(-b)=ab $

\[ -ab=(-a)b=a(-b)=-(-a)(-b) \]

  1. $ (-1)a = -a $

Subtraction and Division

Definition of Subtraction: $ a - b = a + (-b) $

Definition of Division: $ =ab=ab^{-1} $ . Thus, $ b^{-1}=1b^{-1}=1b= $ .

Note: Since 0 has no multiplicative inverse, $ a $ is not defined.

Laws for Quotients

\[ -\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}=-\frac{-a}{-b} \]

  1. $ = $

  2. $ = $ if and only if ad=bc.

  3. $ = $ , for k any nonzero real number. (Fundamental principle of fractions)

Ordering Properties

The positive real numbers, designated by $ R^{+} $ , are a subset of the real numbers with the following properties:

  1. If \(a\) and \(b\) are in \(\boldsymbol{R}^{+}\), then so are \(a + b\) and \(ab\).

  2. For every real number a, either a is in $ R^{+} $ , or a is zero, or -a is in $ R^{+} $ .

If a is in $ R^{+} $ , a is called positive; if -a is in $ R^{+} $ , a is called negative.

The number a is less than b, written a < b, if b - a is positive. Then b is greater than a, written b > a. If a is either less than or equal to b, this is written $ a b $ . Then b is greater than or equal to a, written $ b a $ .

EXAMPLE 1.4 3 < 5 because 5 - 3 = 2 is positive. -5 < 3 because 3 - (-5) = 8 is positive.

The following may be deduced from these definitions:

  1. a > 0 if and only if a is positive.

  2. If $ a $ , then $ a^{2} > 0 $ .

  3. If a < b, then $ a + c < b + c $ .

  4. If a < b, then $

    \[\begin{cases}ac < bc & if c > 0 \\ac > bc & if c < 0\end{cases}\]

    $

  5. For any real number a, either a > 0, or a = 0, or a < 0.

  6. If a < b and b < c, then a < c.

The Real Number Line

Real numbers may be represented by points on a line l such that to each real number a there corresponds exactly one point on l, and conversely.

EXAMPLE 1.5 Indicate the set $ {3, -5, 0, 2/3, , -1.5, -} $ on a real number line.

Absolute Value of a Number

The absolute value of a real number \(a\), written \(|a|\), is defined as follows:

\[ \left|a\right|=\left\{\begin{aligned}a&\quad if a\geq0\\ -a&\quad if a<0\end{aligned}\right. \]

Complex Numbers

Not all numbers are real numbers. The set C of numbers of the form $ a + bi $ , where a and b are real and $ i^{2} = -1 $ , is called the complex numbers. Since every real number x can be written as $ x + 0i $ , it follows that every real number is also a complex number.

EXAMPLE 1.6 $ 3 + = 3 + 2i $ , $ -5i $ , $ 2i $ , $ + i $ are examples of nonreal complex numbers.

Order of Operations

In expressions involving combinations of operations, the following order is observed:

  1. Perform operations within grouping symbols first. If grouping symbols are nested inside other grouping symbols, proceed from the innermost outward.

  2. Apply exponents before performing multiplications and divisions, unless grouping symbols indicate otherwise.

  3. Perform multiplications and divisions, in order from left to right, before performing additions and sub-tractions (also from left to right), unless operation symbols indicate otherwise.

EXAMPLE 1.7 Evaluate (a) $ -5 - 3^{2} $ , (b) $ 3 - 4[5 - 6(2 - 8)] $ , (c) $ [3 - 8 - (-1 - 2 )] (3^{2} - 5{2}){2} $

  1. $ -5 - 3^{2} = -5 - 9 = -14 $

\[ \begin{aligned}3-4[5-6(2-8)]&=3-4[5-6(-6)]\\&=3-4[5+36]\\&=3-4[41]=3-164=-161\end{aligned} \]

\[ \begin{aligned}(c)\ [3-8\cdot5-(-1-2\cdot3)]\cdot(3^{2}-5^{2})^{2}&=[3-8\cdot5-(-1-6)]\cdot(9-25)^{2}\\&=[3-(8\cdot5)-(-7)]\cdot(-16)^{2}\\&=[3-40+7]\cdot256\\&=-30\cdot256=-7,680\end{aligned} \]

SOLVED PROBLEMS

1.1. Prove the extended distributive law $ a(b + c + d) = ab + ac + ad $ .

\[ \begin{aligned}a(b+c+d)&=a[(b+c)+d]\quad&Associative law\\&=a(b+c)+ad\quad&Distributive law\\&=ab+ac+ad\quad&Distributive law\end{aligned} \]

1.2. Prove that multiplication is distributive over subtraction: $ a(b - c) = ab - ac $ .

\[ \begin{aligned}a(b-c)&=a[b+(-c)]\quad&Definition of subtraction\\&=ab+a(-c)\quad&Distributive law\\&=ab+(-ac)\quad&Laws for negatives\\&=ab-ac\quad&Definition of subtraction\end{aligned} \]

1.3. Show that $ -(a+b)=-a-b $ .

\[ \begin{aligned}-(a+b)&=(-1)(a+b)\\&=(-1)a+(-1)b\\&=(-a)+(-b)\\&=-a-b\end{aligned} \]

Laws for negatives

Distributive law

Laws for negatives

Definition of subtraction

1.4. Show that if $ = $ , then ad=bc.

Assume that \(\frac{a}{b}=\frac{c}{d}\). By the definition of division, \(\frac{a}{b}=\frac{c}{d}\) means \(ab^{-1}=cd^{-1}\). Hence,

\[ \begin{aligned}ad&=ad\cdot1\quad&Identity law\\&=adbb^{-1}\quad&Inverse law\\&=ab^{-1}db\quad&Associative and commutative laws\\&=cd^{-1}db\quad&By hypothesis\\&=c\cdot1\cdot b\quad&Inverse law\\&=bc\quad&Identity and commutative laws\end{aligned} \]

1.5. Prove that if a < b, then $ a + c < b + c $ .

Assume that \(a < b\). Then \(b - a\) is positive. But \(b - a = b - a + 0 = b - a + c + (-c)\) by the identity and inverse laws. Since \(b - a + c + (-c) = b - a + c - c = b + c - (a + c)\) by the definition of subtraction, the associative and commutative laws, and Problem 1.3, it follows that \(b + c - (a + c)\) is positive. Hence \(a + c < b + c\).

1.6. Identify as a member of the sets N, Z, Q, H, R, or C:

  1. -7

  2. 0.7

  3. $ $ ;

  4. $ $

  5. $ $

  6. -7 is a negative integer; hence it is also rational, real, and complex. -7 is in Z, Q, R, and C.

  7. 0.7 = 7/10; hence it is a rational number, hence real and complex. 0.7 is in Q, R, and C.

  8. $ $ ; is an irrational number; hence it is also real and complex. $ $ ; is in H, R, and C.

  9. $ $ is not defined. This is not a member of any of these sets.

  10. $ $ is not a real number, but it can be written as $ i $ ; hence, it is a complex number. $ $ is in C.

1.7. Identify as true or false:

  1. -7 < -8 (b) $ = 22/7 $ (c) $ x^{2} $ for all real x.

  2. Since $ (-8)-(-7)=-1 $ is negative, -8<-7, so the statement is false.

  3. Since $ $ is an irrational number and 22/7 is rational, the statement is false.

  4. This follows from property 2 for inequalities; the statement is true.

1.8. Rewrite the following without using the absolute value symbol, and simplify:

\[ \left|3-5\right| \]

\[ \left|3\right|-\left|5\right| \]

\[ \left|2-\pi\right| \]

  1. |x - 5| if x > 5

  2. |x + 6| if x < -6

  3. $ |3-5|=|-2|=2 $

  4. $ |3|-|5|=3-5=-2 $

  5. Since $ 2 < $ , $ 2 - $ is negative. Hence $ |2 - | = -(2 - ) = - 2 $ .

  6. Given that x > 5, x - 5 is positive. Hence |x - 5| = x - 5.

  7. Given that x < -6, x - (-6) = x + 6 is negative. Hence |x + 6| = -(x + 6) = -x - 6.

SUPPLEMENTARY PROBLEMS

1.9. Identify the law that justifies each of the following statements:

  1. $ (2x + 3) + 5 = 2x + (3 + 5) $

  2. $ 2x + (5 + 3x) = 2x + (3x + 5) $

  3. $ x^{2}(x + y) = x^{2} x + x^{2} y $

  4. $ 100[0.01(50 - x)] = 100(0.01) $

  5. If $ a + b = 0 $ , then b = -a.

  6. If $ (x - 5)(x + 3) = 0 $ , then either x - 5 = 0 or $ x + 3 = 0 $ .

Ans. (a) Associative law for addition

  1. Commutative law for addition

  2. Distributive law

  3. Associative law for multiplication

  4. Inverse law for addition

  5. Zero factor law

1.10. Are the following statements true or false?

  1. 3 is a real number.

  2. $ = 3.14 $

  3. $ |x-5|=x+5 $ (d) Every rational number is also a complex number.

Ans. (a) true; (b) false; (c) false; (d) true

1.11. Place the correct inequality sign between the following:

  1. 9?−8

  2. π?4

  3. $ $ ?0.33

  4. $ $ ?π

  5. −1.414?−√2

Ans. (a) >; (b) <; (c) >; (d) >; (e) >

1.12. Show that if ad = bc, then $ = $ . (Hint: Assume that ad = bc; then start with $ ab^{-1} $ and transform it into $ cd^{-1} $ in analogy with Problem 1.4.)

1.13. Show that $ = $ follows from the law that $ = $ if and only if ad=bc.

1.14. Rewrite the following without using the absolute value symbol, and simplify:

  1. $ |(-5)-[-(-9)]| $

  2. $ -|-1.4| $

  3. $ |6 - x| $ , if x > 6.

  4. $ -|-4 - x^{2}| $

Ans. (a) 14; (b) $ 1.4 - $ ; (c) x - 6; (d) $ -4 - x^{2} $

1.15. Evaluate (a) $ 2 - 4 ^{2} $ (b) $ 7 + 3[2(5 - 8) - 4] $ (c) $ {4 - 6[7 - (5 - 8)^{2}]}^{2} $

Ans. (a) -94; (b) -23; (c) 1936

1.16. Consider the set $ {-5, -, 0, , , , } $

  1. Which members of this set are members of N?

  2. Which members of this set are members of Z?

  3. Which members of this set are members of Q?

  4. Which members of this set are members of H?

Ans. (a) $ $ ; (b) -5, $ 0, $ ; (c) -5, $ - $ , 0, $ $ , $ $ ; (d) $ $ , $ $

1.17. A set is closed under an operation if the result of applying the operation to any members of the set is also a member of the set. Thus, the integers \(Z\) are closed under \(+,\), while the irrational numbers \(H\) are not, since, for example, \(\pi + (-\pi) = 0\) which is not irrational. Identify as true or false:

  1. \(Z\) is closed under multiplication.

  2. \(H\) is closed under multiplication.

  3. \(N\) is closed under subtraction.

  4. \(Q\) is closed under addition.

  5. \(Q\) is closed under multiplication.

Ans. (a) true; (b) false; (c) false; (d) true; (e) true