M0 (by Topics)mth Revise

25-03-2021

M0 (by Topics)mth Revise

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_{Sources: Abramson's 'College Algebra and Trigonometry' and Redden's 'Advanced Algebra'}

Real Numbers

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Algebra is often described as the generalization of arithmetic. The systematic use of variables, letters used to represent numbers, allows us to communicate and solve a wide variety of real-world problems. For this reason, we begin by reviewing real numbers and their operations.

A set is a collection of objects, typically grouped within braces ({ }), where each object is called an element. When studying mathematics, we focus on special sets of numbers.

[ begin{align*} mathbb { N } &= underbrace{{ 1,2,3,4,5 , dots }}_{color{Cerulean}{Natural: Numbers}}
& W &= underbrace{ { 0 , 1,2,3,4,5 , dots }}_{color{Cerulean}{Whole: Numbers}}
&mathbb{Z} &= underbrace{ {dots ,-3,-2,-1,0,1,2,3,dots}}_{color{Cerulean}{Integers}} end{align*}]

The three periods (…) are called an ellipsis and indicate that the numbers continue without bound. A subset, denoted (subseteq), is a set consisting of elements that belong to a given set. Notice that the sets of natural and whole numbers are both subsets of the set of integers and we can write (mathbb { N } subseteq mathbb{Z}) and (W subseteq mathbb{Z}).

A set with no elements is called the empty set and has its own special notation:

({:::}=varnothing: qquad color{Cerulean}{Empty: Set})

Rational numbers, denoted (mathbb{Q}), are defined as any number of the form (frac { a } { b }) where a and b are integers and b is nonzero. We can describe this set using set notation:

(mathbb { Q } = left{ frac { a } { b } | a , b in mathbb { Z } , b neq 0 right} qquad color{Cerulean}{Rational: Numbers})

The vertical line | inside the braces reads, “such that” and the symbol (in) indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all numbers (frac{a}{b}) such that a and b are elements of the set of integers and b is not equal to zero.” Decimals that terminate or repeat are rational. For example, (0.05=frac{5}{100}) and (0.overline{6}=0.6666…=frac{2}{3} ).

The set of integers is a subset of the set of rational numbers, (mathbb{Z}subseteqmathbb{Q}), because every integer can be expressed as a ratio of the integer and 1. In other words, any integer can be written over 1 and can be considered a rational number. For example, (7=frac{7}{1}).

Irrational numbers are defined as any numbers that cannot be written as a ratio of two integers. Nonterminating decimals that do not repeat are irrational. For example, (π=3.14159…) and (sqrt{2}=1.41421…).

Finally, the set of real numbers, denoted (mathbb{R}), is defined as the set of all rational numbers combined with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set of real numbers. In summary,

Notation Used to Define Subsets of Real Numbers

Sets of numbers can be described in several ways, including Interval Notation, Set-Builder Notation and Inequality Notation.

Interval Notation

In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded.

The smallest number in the interval, an endpoint, is written first.
The largest number in the interval, another endpoint, is written second, and is written after a comma.
Parentheses, (() or ()), are used to signify that an endpoint is not included in the interval.
Brackets, ([) or (]), are used to indicate that an endpoint is included in the interval.

Set Notation

Another way a set of numbers can be described is in set-builder notation. The braces ({}) are read as “the set of,” and the vertical bar (|) is read as “such that,” so we would read ( {x|10≤x<30}) as “the set of x-values such that 10 is less than or equal to x, and x is less than 30.”

Figure (PageIndex{2}) compares inequality notation, set-builder notation, and interval notation.

Given a line graph, describe the set of values using interval notation.

Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
At the left end of each interval, use [ with each end value included in the set (solid dot) or ( for each excluded end value (open dot).
At the right end of each interval, use ] with each end value included in the set (filled dot) or ) for each excluded end value (open dot).
Use the union symbol (cup) to combine all intervals into one set.
NOTE: The word 'or' (rather than (cup)) is used to combine sets described by inequalities or Set-builder notation.

Compound Inequalities

For sets with a finite number of elements, like for example the sets ({2,3,5}) and ({5,6}), the union of these sets is the set ({2,3,5,6}). The elements do not have to be listed in ascending order of numerical value, but if the original two sets have some elements in common, those elements should be listed only once in the union set. The intersection of these two set lists only those elements common to both sets, ({5})

Union of intervals

The union of two intervals is written differently, depending on the type of notation used.

When using inequality notation or set-builder notation (both which describe a condition that is either true or false), the word “or” is used.
When interval notation is used, two unconnected intervals are combined with the use of the union symbol,( cup ).

To find the union of two intervals, use the portion of the number line representing the total collection of numbers in the two number line graphs. For example,

(x<3) or (xgeq 6)

(color{Cerulean}{Interval notation:}) ((−∞,3)cup[6,∞))

(color{Cerulean}{Set notation:}) ({x| x<3 text{ or } xgeq 6})

Example (PageIndex{1}): Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure (PageIndex{4}) using inequality notation, set-builder notation, and interval notation.

Solution

To describe the values, (x), included in the intervals shown, we would say, “(x) is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

( begin{array} {cl}
1≤x≤3 text{ or }x>5 &text{Inequality}
{x|1≤x≤3 text{ or } x>5} qquad & text{Set-Builder Notation}
[1,3]cup(5,infty) & text{Interval Notation}
end{array})

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Try It (PageIndex{1})

Given Figure (PageIndex{5}), specify the graphed set in

words
set-builder notation
interval notation

Answers

a. Values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
b. ({x|x≤−2 text{ or } −1≤x<3})
c. (left(−∞,−2right]cupleft[−1,3right))

Intersection of intervals

The intersection of two intervals is also written differently, depending on the type of notation used.

The intersection of two intervals described using inequality notation or set-builder notation, can be written using the word “and”. However, the intersection can be more succinctly written as a compound inequality. For example, (−1leq x) and (x<3) can be written more concisely as (−1leq x<3) which reads “negative one is less than or equal to x and x is less than three.”
When interval notation is used, the logical “and” requires that both conditions must be true. Both inequalities will be satisfied by all the elements in the intersection, denoted (cap), of the solution sets of each.

To find the intersection of two intervals, take the portion of the number line that the two number line graphs have in common.

Example (PageIndex{2}):

Graph and give the interval notation and set notation equivalent to (x<3) and (xgeq −1).

Solution

Determine the intersection, or overlap, of the two solution sets to (x<3) and (xgeq −1). The solutions to each inequality are sketched above the number line as a means to determine the intersection, which is graphed on the number line below.

Here, (3) is not a solution because it solves only one of the inequalities. Alternatively, we may interpret (−1leq x<3) as all possible values for x between, or bounded by, (−1) and (3) where (−1) is included in the solution set.

Answer:

Interval notation: ([,−1, 3)); set notation: ({x|−1leq x<3})

Real Number Operations

Order of Operations

When we multiply a number by itself, we square it or raise it to a power of (2). For example, (4^2 =4times4=16). We can raise any number to any power. In general, the exponential notation (a^n) means that the number or variable (a) is used as a factor (n) times.

[a^n=acdot acdot acdots a qquad text{ n factors} nonumber ]

In this notation, (a^n) is read as the (n^{th}) power of (a), where (a) is called the base and (n) is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, (24+6 times dfrac{2}{3} − 4^2) is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

[24+6 times dfrac{2}{3} − 4^2 nonumber]

There are no grouping symbols, so move on to exponents. The number (4) is raised to a power of (2), so simplify (4^2) as (16).

[24+6 times dfrac{2}{3} − 4^2 nonumber ]

[24+6 times dfrac{2}{3} − 16 nonumber]

Next, perform multiplication or division, left to right.

[24+6 times dfrac{2}{3} − 16 nonumber]

[24+4-16 nonumber]

Lastly, perform addition or subtraction, left to right.

[24+4−16 nonumber]

[28−16 nonumber]

[12 nonumber]

Therefore,

[24+6 times dfrac{2}{3} − 4^2 =12 nonumber]

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

ORDER OF OPERATIONS

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

P(arentheses)
E(xponents)
M(ultiplication) and D(ivision)
A(ddition) and S(ubtraction)

HOW TO: Given a mathematical expression, simplify it using the order of operations.

Simplify any expressions within grouping symbols. Grouping symbols include parentheses ( ), brackets [ ], braces { }, fraction bars, radicals, and absolute value bars
Simplify any expressions containing exponents or radicals.
Perform any multiplication and division in order, from left to right.
Perform any addition and subtraction in order, from left to right.

Example (PageIndex{3}): Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

((3+2)^2+4 times(6+2))
(dfrac{5^2-4}{7}- sqrt{11-2})
(6−|5−8|+3times(4−1)) &&
(dfrac{14-3 times2}{2 times5-3^2})
(7times(5times3)−2times[(6−3)−4^2]+1)

Solution

[begin{align*} (3times2)^2-4times(6+2)&=(6)^2-4times(8) && qquad text{Simplify parentheses} &=36-4times8 && qquad text{Simplify exponent} &=36-32 && qquad text{Simplify multiplication} &=4 && qquad text{Simplify subtraction} end{align*}]
[begin{align*} dfrac{5^2-4}{7}- sqrt{11-2}&= dfrac{5^2-4}{7}-sqrt{9} && qquad text{Simplify grouping symbols (radical)} &=dfrac{5^2-4}{7}-3 && qquad text{Simplify radical} &=dfrac{25-4}{7}-3 && qquad text{Simplify exponent} &=dfrac{21}{7}-3 && qquad text{Simplify subtraction in numerator} &=3-3 && qquad text{Simplify division} &=0 && qquad text{Simplify subtraction} end{align*}]

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

[begin{align*} 6-mid 5-8mid +3times(4-1)&=6-|-3|+3times3 && qquad text{Simplify inside grouping symbols} &=6-3+3times3 && qquad text{Simplify absolute value} &=6-3+9 && qquad text{Simplify multiplication} &=3+9 && qquad text{Simplify subtraction} &=12 && qquad text{Simplify addition} end{align*}]
[begin{align*} dfrac{14-3 times2}{2 times5-3^2}&=dfrac{14-3 times2}{2 times5-9} && qquad text{Simplify exponent} &=dfrac{14-6}{10-9} && qquad text{Simplify products} &=dfrac{8}{1} && qquad text{Simplify differences} &=8 && qquad text{Simplify quotient} end{align*}]

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

[begin{align*} 7times(5times3)-2times[(6-3)-4^2]+1&=7times(15)-2times[(3)-4^2]+1 && qquad text{Simplify inside parentheses} &=7times(15)-2times(3-16)+1 && qquad text{Simplify exponent} &=7times(15)-2times(-13)+1 && qquad text{Subtract} &=105+26+1 && qquad text{Multiply} &=132 && qquad text{Add} end{align*}]

Try It (PageIndex{3})

Use the order of operations to evaluate each of the following expressions.

(sqrt{5^2-4^2}+7times(5-4)^2)
(1+dfrac{7times5-8times4}{9-6})
(|1.8-4.3|+0.4timessqrt{15+10})
(dfrac{1}{2}times[5times3^2-7^2]+dfrac{1}{3}times9^2)
([(3-8^2)-4]-(3-8))

Answer

a. (10) ( qquad ) b. (2) ( qquad ) c. (4.5) ( qquad ) d. (25) ( qquad ) e. (-60)

Real Number Properties

PROPERTIES OF REAL NUMBERS

Addition	Multiplication
Commutative Property	(a+b=b+a)	(atimes b=btimes a)
Associative Property	(a+(b+c)=(a+b)+c)	(a(bc)=(ab)c)
Distributive Property	(atimes (b+c)=atimes b+atimes c)
Identity Property	There exists a unique real number called the additive identity, 0, such that, for any real number (a)	There exists a unique real number called the multiplicative identity, 1, such that, for any real number (a) (atimes 1=a)
Inverse Property	Every real number (a) has an additive inverse, or opposite, denoted (–a), such that	Every nonzero real number (a) has a multiplicative inverse, or reciprocal, denoted (frac{1}{a}) , such that (atimes left(dfrac{1}{a}right)=1)

Constants and Variables

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as (x +5), (dfrac{4}{3}pi r^3), or (sqrt{2m^3 n^2}). In the expression (x +5), (5) is called a constant because it does not vary and (x) is called a variable because it does. (In naming the variable, ignore any exponents or radicals included with the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

Example (PageIndex{4}): Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

(x + 5)
(dfrac{4}{3}pi r^3)
(sqrt{2m^3 n^2})

Solution	a. (x + 5)	b. (dfrac{4}{3}pi r^3)	c. (sqrt{2m^3 n^2})
Constants	(5)	(dfrac{4}{3}), (pi)	(2)
Variables	(x)	(r)	(m),(n)

Try It (PageIndex{4})

List the constants and variables for each algebraic expression.

(2pi r(r+h))
(2(L + W))
(4y^3+y)

Answer

a. (2pi r(r+h))	b. (2(L + W))	c. (4y^3+y)
Constants	(2),(pi)	(2)	(4)
Variables	(r),(h)	(L), (W)	(y)

Evaluating Algebraic Expressions

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

[begin{align*} (-3)^5 &=(-3)times(-3)times(-3)times(-3)times(-3)Rightarrow x^5=xtimes xtimes xtimes xtimes x (2times7)^3&=(2times7)times(2times7)times(2times7)qquad ; ; quad Rightarrow (yz)^3=(yz)times(yz)times(yz) end{align*}]

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Example (PageIndex{5}): Evaluating an Algebraic Expression at Different Values

Evaluate the expression (2x−7) for each value for (x).

(x=0)
(x=1)
(x=12)
(x=−4)

Solution

Substitute (0) for (x). ( quad 2x-7 = 2(0)-7 = 0-7= -7 )
Substitute (1) for (x). ( quad2x-7 = 2(1)-7= 2-7= -5 )
Substitute (dfrac{1}{2}) for (x). ( quad 2x-7 = 2left (dfrac{1}{2} right )-7 = 1-7= -6 )
Substitute (-4) for (x). ( quad 2x-7 = 2(-4)-7= -8-7= -15 )

Try It (PageIndex{5})

Evaluate the expression (11−3y) for each value for (y).

(y=2)
(y=0)
(y=dfrac{2}{3})
(y=−5)

Answer

a. (5) ( qquad) b. (11)( qquad) c. (9)( qquad) d. (26)

Example (PageIndex{6}): Evaluating Algebraic Expressions

Evaluate each expression for the given values.

(x+5) for (x=-5)
(dfrac{t}{2t-1}) for (t=10)
(dfrac{4}{3}pi r^3) for (r=5)
(a+ab+b) for (a=11) , (b=-8)
(sqrt{2m^3 n^2}) for (m=2) , (n=3)

Solution

(-5) for (x). [begin{align*} x+5 &= (-5)+5 = 0 end{align*}]
Substitute (10) for (t). [begin{align*} dfrac{t}{2t-1} &= dfrac{(10)}{2(10)-1} &= dfrac{10}{20-1}= dfrac{10}{19} end{align*}]
Substitute (5) for (r). [begin{align*} dfrac{4}{3} pi r^3 &= dfrac{4}{3}pi (5)^3 &= dfrac{4}{3}pi (125)= dfrac{500}{3}pi end{align*}]
Substitute (11) for (a) and (-8) for (b). [begin{align*} a+ab+b &= (11)+(11)(-8)+(-8) &= 11-88-8 = -85 end{align*}]
Substitute (2) for (m) and (3) for (n). [begin{align*} sqrt{2m^3 n^2} &= sqrt{2(2)^3 (3)^2} &= sqrt{2(8)(9)} &= sqrt{144} = 12 end{align*}]

Try It (PageIndex{6})

Evaluate each expression for the given values.

(dfrac{y+3}{y-3}) for (y=5)
(7-2t) for (t=-2)
(dfrac{1}{3}pi r^2) for (r=11)
((p^2 q)^3) for (p=-2), (q=3)
(4(m-n)-5(n-m)) for (m=dfrac{2}{3}) (n=dfrac{1}{3})

Answer

a. (4) ( qquad) b. (11) ( qquad) c. (dfrac{121}{3}pi) ( qquad) d. (1728) ( qquad) e. (3)

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Example (PageIndex{7}): Simplifying Algebraic Expressions

Simplify each algebraic expression.

(3x-2y+x-3y-7)
(2r-5(3-r)+4)
(left(4t-dfrac{5}{4}sright)-left(dfrac{2}{3}t+2sright))
(2mn-5m+3mn+n)

Solution

[begin{align*} 3x-2y+x-3y-7&=3x+x-2y-3y-7 && qquad text{Commutative property of addition} &=4x-5y-7 && qquad text{Simplify} end{align*}]
[begin{align*} 2r-5(3-r)+4&=2r-15+5r+4 && qquad qquad qquad text {Distributive property} &=2r+5y-15+4 && qquad qquad qquad text{Commutative property of addition} &=7r-11 && qquad qquad qquad text{Simplify} end{align*}]
[begin{align*} left(4t-dfrac{5}{4}sright)-left(dfrac{2}{3}t+2sright)&=4t-dfrac{5}{4}s-dfrac{2}{3}t-2s && qquad text{Distributive property} &=4t-dfrac{2}{3}t-dfrac{5}{4}s-2s && qquad text{Commutative property of addition} &=dfrac{10}{3}t-dfrac{13}{4}s && qquad text{Simplify} end{align*}]
[begin{align*} 2mn-5m+3mn+n&=2mn+3mn-5m+n && qquad text{Commutative property of addition} &=5mn-5m+n && qquad text{Simplify} end{align*}]

Try It (PageIndex{7})

Simplify each algebraic expression.

(dfrac{2}{3}y−2left(dfrac{4}{3}y+zright))
(dfrac{5}{t}−2−dfrac{3}{t}+1)
(4p(q−1)+q(1−p))
(9r−(s+2r)+(6−s))