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MAT091 :: Lecture Note :: Week 01
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(due Thursday, 8/24/2017)
(due Tuesday, 8/29/2017)
(due Thursday, 8/31/2017)
It is the student's responsibility to read and understand the MAT091 syllabus.
I became an instructor at SCC during fall of 1997. During my time at SCC I have learned that students who attend class do better than those who don't.
Faculty were told the following:
"An instructor is paid to hold class from time A to time B. It is considered fraud if an instructor is consistently letting students out early or coming to class late."The Math Tutor Lab in CM441A is available for help learning arithmetic and algebra.
MAT108  Tutored Mathematics Course can be taken to help with respect to learning the MAT091 material.
Calculators are available for rent in the IT (Information Technology) building.
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Stand firm in your refusal to remain conscious during algebra.
In real life, I assure you, there is no such thing as algebra.
 Fran Lebowitz (01950) { American author; more... } [algebra]I don't know why I should have to learn Algebra...
I'm never likely to go there.
 Billy Connolly (01942?????) {Scottish comedian; more...} [math]As long as algebra is taught in school, there will be prayer in school.
 Cokie Roberts (01943?????) { American journalist and author; more... } [math]Algebra is the metaphysics of arithmetic.
 John Ray (0162701705) { English naturalist; more... } [math]To speak algebraically, Mr. M. is execrable, but Mr. G. is e(x+1)ecrable.
 Edgar Alan Poe (0180901849) { American poet and writer; more... } [math/algebra]My mother is a mathematician, so she knows how to induce good behavior.
"If I've told you n times, I've told you n+1 times...."
 Anonymous { She was only a mathematicians daughter, but she knew how to multiply. } [math]Men are liars. We'll lie about lying if we have to. I'm
an algebra liar. I figure two good lies make a positive.
 Tim Allen (01953) { American comedian; more... } [life]In a nutshell, Algebra is xsighting!
BAB:: Joke... Terror Alert  Algebra Movement
So... What is Algebra? Algebra is a branch of mathematics that provides a foundation for other branches of mathematics, science, engineering, computing, and so on. Via Wikipedia.org: "In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics." Algebra involves the evaluating of expressions and the solving of equations that contain one or more variables.
YouTube.com::Introduction to Algebra via KahnAcademy.org.
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Math.com tells us that exponents are a "shorthand way to show how many times a number, called the base, is multiplied times itself." In other words, just like multiplication is a form of "repeated addition," exponents are a form of "repeated multiplication."
5 + 5 + 5 = 3 * 5 (repeated addition) 5 * 5 * 5 = 5^{3} (repeated multiplication)Exponents imply the operation of "raising to a power." For example,
10^{5}
is read "10 raised to the 5th power," with10
being the base and5
the exponent (or power).By mathematical law (definition), any nonzero number raised to the power of zero is one.
5^{0} equals 1 9999^{0} equals 1 82^{0} equals 1 42^{0} equals 1 (42)^{0} equals 1 0^{0} equals ??? [Google says 1]Any number raised to the power of one is that number.
3^{1} equals 3 9999^{1} equals 9999 42^{1} equals 42Any number raised to the mth power (where
m > 1
) is that number multiplied by itself 'm' times.5^{4} = 5 * 5 * 5 * 5 3^{7} = 3 * 3 * 3 * 3 * 3 * 3 * 3 2^{3} = 2 * 2 * 2 5^{2} = (5 * 5) (5)^{2} = 5 * 5[special exponent values] A number squared is a number raised to the power of
2
and a number cubed is a number raised the power of3
.8 squared is 8^{2} which equals 8 * 8 4 cubed is 4^{3} which equals 4 * 4 * 4BAB:: Square Number Playing Results in Square Number Discovery [24 May 2007]Recall, the base10 (i.e. decimal) number system has the ones, tens, hundreds, thousands and so one. These positional values are based upon 10 being raised to the whole numbers 0, 1, 2, and so on.
10^{0}
equals1
(one)
10^{1}
equals10
(ten [deka])
10^{2}
equals100
(hundred [hecto])
10^{3}
equals1000
(thousand [kilo])
10^{4}
equals10,000
(10 thousand)
10^{5}
equals100,000
(100 thousand)
10^{6}
equals1,000,000
(million [mega] [1,000 thousand])
10^{7}
equals10,000,000
(10 million)
10^{8}
equals100,000,000
(100 million)
10^{9}
equals1,000,000,000
(billion [giga] [1,000 million])
...
10^{12}
equals1,000,000,000,000
(trillion [tera] [1,000 billion])
...
10^{15}
equals1,000,000,000,000,000
(quadrillion [peta] [1,000 trillion])More... On Notation
The caret
^
symbol is sometimes used to imply exponents. GDT calls this "calculator notation."2^4 = 2^{4} = 2 * 2 * 2 * 2 = 16 8^3 = 8^{3} = 8 * 8 * 8 = 512 1^2 = 1^{2} = 1 * 1 = 1More... One Expression That Contains Lots of Math
Let's take a peek at the following expression reading it from lefttoright.
a^n ⋅ 1 = a^n = a^(n + 0) = a^n ⋅ a^0More... Something From a Math Reading Group
GDT enjoyed seeing pictures drawn to scale that turned the earth into a pixel followed by reducing the sun into pixel. GDT also liked how powers of ten were used to demonstrate the base10 number system.
5555 = 5 * 10^3 + 5 * 10^2 + 5 * 10^1 + 5 * 10^0 209 = 2 * 10^2 + 0 * 10^1 + 9 * 10^0 790016 = ??? ddd,ddd,ddd = 3 * 10^8 + 5 * 10^5 + 2 * 10^3 + 0 * 10^2 + 10^0External Hyperlink(s)
 Wikipedia.org:: Exponent (Exponentiation) [opens new window]
 PurpleMath.com:: Exponents: Basic Rules [opens new window]
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A nonzero number (base) raised to a negative exponent is equal to one dividedby the number raised to the absolute value of the exponent.
1 a^{b} where 'b' is less than 0 equals  a^{b} 7^{2} equals 1/7^{2} equals 1/49 2^{5} equals 1/2^{5} equals 1/32 10^{2} equals 1/10^{2} equals 1/10010^1 1/10 tenth deci 10^2 1/100 hundreth centi 10^3 1/1000 thousandth milli 10^6 1/10^6 millionth micro 10^9 1/10^9 billionth nano 10^12 1/10^12 trillionth pico 10^15 1/10^15 quadrillionth femtoWhat's an absolute value?
The absolute value of a number is its distance from zero on the number line. For example,
5
is five ones away from zero; therefore, its absolute value is5
.Symbolically, two vertical bars with a number (or mathematical expression) between them represents an absolute value.
7 equals 7 7 equals 7 3  4 equals 1 equals 1 5 equals 5 3 * 2 = 3 * 2 = 6Function notation: Sometimes
abs(n)
is the absolute value function. The function outputs the absolute value of inputn
.External Hyperlinks
 PurpleMath.com:: Negative Exponents
 YouTube.com::Power of Ten  Are We Alone In The Universe
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This is a quickie terminology review that is intended to be augmented by lecture.
A mathematical expression is a combination of one or more terms that can be evaluated.
Expressions are evaluated resulting in a value. In other words, expressions "express" a value.
A term is either a number, variable (i.e. unknown number), function, or a combination of these objects.
In an expression terms are separated by addition and subtraction operators.
Expressions can contain grouping symbols (parenthesis, brackets, fraction bars) to alter order of operations when expressions are evaluated.
Numbers or variables that are multiplied together are called factors. A term can consist of multiple factors.
5 + 3 * 2 the expression has two terms: 5 and 3 * 2 the 3 * 2 term has two factors (3 and 2) + and * are arithmetic operators * has a higher precedence than +; therefore, it is evaluated first: 5 + 3 * 2 = 5 + 6 = 11 if + needs to be evaluated before *, then the expression is: (5 + 3) * 2 = 8 * 2 = 16An equation is a relationship between two expressions. Relational operators are used to relate expressions.
3 + 5 = 4 * 2 3 + 5 evaluates to 8 4 * 2 evaluates to 8 3 + 5 = 4 * 2 8 = 8Equations have a leftside and a rightside. The leftside is left of the relational operator. The rightside of an equation is to the right of the relational operator.
General syntax for a mathematical equation:
expression relationaloperator expression where relationaloperator is equal, not equal, greater than, greater than or equal, less than, less than or equal, approximate
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Evaluating expressions and solving equations can sometimes be made "easier" if they are "simplified." One step in simplifying expressions/equations is the "combining" of liketerms. Liketerms are terms having the same variables raised to the same exponents.
5x + 2x = 7x 2x  5x = 2x + (5x) = 3x x^{2} + 4x^{2} = 5x^{2} [x^{2} = 1x^{2}] 9x^{10} + 9y^{10} are not liketerms [different variables] 5x^{2} and 2x are not liketerms [different exponents]wyzant.com:: Simplifying Expressions Calculator
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An algebraic expression is a mathematical expresssion that contains at least one variable along with zero or more numbers (i.e. constants) and zero or more arithmetic operators.
The following are algebraic expressions.
n 3n 5n + 10 2x^2  2y^3 9a + 2b  3c 11(x + y^3 + 2w)Variables are unknown values that are represented by singlecharacter letters. For example,
3n
is3 times 'n'
, where 'n' is a variable. If a value is assigned to 'n', then the expression can be evaluated.3n + 8 if n = 3, then 3(3) + 8 = 100 if n = 5, then 3(5) + 8 = 7 if n = 0, then 3(0) + 8 = 8The replacement (or swapping) of variables with values can be described in a variety of ways: Values can be "plugged into" or "assigned to" variables; or variables can be "replaced with" or "substituted with" values; or we "let" variables be certain values.
5x + 1 assign 5 to x ........... 5(5) + 1 = 25 + 1 = 26 5x + 1 replace x with 5 ........ 5(5) + 1 = 25 + 1 = 26 5x + 1 plug 5 into x ........... 5(5) + 1 = 25 + 1 = 26 5x + 1 substitute x with 5 ..... 5(5) + 1 = 25 + 1 = 26 5x + 1 let x be 5 .............. 5(5) + 1 = 25 + 1 = 26Again, expressions are typically evaluated after variables have been assigned (i.e. given) values.
If a variable is multiplied by a constant, then the constant is called a coefficient.
3n ...... 'n' is a variable and 3 is a coefficient x ....... 'x' is a variable and 1 is the coefficient 1y ..... 'y' is a variable and 1 is the coefficientThe combination of a coefficent and variable is called a factor. Factors are numbers, variables, and expressions that are multiplied together to produce a product.
Factors separated by addition and subtraction operators are called terms.
3n + 1 .............. 1 factor, 2 terms 4a + 2b  10 ........ 2 factors (4a and 2b), 3 terms 7x  5 + 8y  11 .... 2 factors (7x and 8y), 4 terms 2x^3 + 10^2.......... 1 factor (2x^3), 2 terms 3(4)  2(1 + 3)...... 2 factors, 2 terms
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a. Simplify: 4  2(x + 3)4 + 2(x + 3) # rewrite a  b as a + b 4 + 2x + 6 # distribute 2 2x  2 # combine like terms (the constants) 2x + 2 # rewrite as an addition problem 2(x + 1) # factor out a 2 check work: 4  2(x + 3) 2(x + 1) let x = 0... 4  2(0 + 3) 2(0 + 1) 4  2(3) = 4  6 = 2 2(1) = 2 let x = 1... 4  2(1 + 3) 2(1 + 1) 4  2(4) = 4  8 = 4 2(2) = 4 let x = 1... 4  2(1 + 3) 2(1 + 1) 4  2(2) = 4  4 = 0 2(0) = 0 check work using the calculator: press Y= enter 42(x+3) into y1 enter 2(x+1) into y2 press GRAPH to see one line press 2nd GRAPH to see table values press Y= change y2 to 2(x+2) press GRAPH to see two lines press 2nd GRAPH and compare table valuesb. Simplify: (x^2 + x  5)  (2x^2  x + 3)# convert subtractions into additions (x^2 + x + 5) + (1)(2x^2 + x + 3) x^2 + x + 5 + 2x^2 + x + 3 # drop ()s and distribute 1 x^2 + 2x^2 + x + x + 5 + 3 # combine like terms (step 1) x^2 + 2x + 8 # combine like terms (step 2) check work: let x = 0 (x^2 + x  5)  (2x^2  x + 3) (0^2 + 0  5)  (2(0^2)  0 + 3) = 5  3 = 8 x^2 + 2x + 8 0^2 + 2(0) + 8 = 0 + 0 + 8 = 8 check work: let x = 1 (x^2 + x  5)  (2x^2  x + 3) (1^2 + 1  5)  (2(1^2)  1 + 3) ( 3 )  ( 4 ) = 7 x^2 + 2x + 8 1^2 + 2(1) + 8 1 + 2 + 8 = 7 check work: let x = 1 (x^2 + x  5)  (2x^2  x + 3) ((1)^2 + 1  5)  (2(1^2)  1 + 3) ( 1 + 1  5)  (2  1 + 3) ( 5 )  ( 6 ) = 11 x^2 + 2x + 8 (1)^2 + 2(1) + 8 = 1 + 2 + 8 = 11 check work using the calculator: press Y= enter (x^2 + x  5)  (2x^2  x + 3) into y1 enter x^2 + 2x + 8 into y2 on the y2 line... left arrow to the \ and press ENTER press GRAPH to see only one parabola press 2nd GRAPH to compare table values
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The horizon is the "apparent junction of earth and sky."
The following is a number line written horizontally.
<+++++++++++> 5 4 3 2 1 0 +1 +2 +3 +4 +5The vertical is "perpendicular to the plane of the horizon or to a primary axis." [Perpendicular is "being at right angles to a given line or plane."]
The following is a number line written vertically.
^  +5 +  +4 +  +3 +  +2 +  +1 +  0 +  1 +  2 +  3 +  4 +  5 +  vRight Angle
Perpendicular lines intersect and their point of intersection is called the origin. Perpendicular lines form four right angles at the origin. A right angle measures ninety degrees (
90°
).With respect the Cartesian coordinate system, the horizonal line is the
xaxis
and the vertical line is theyaxis
. < verticalaxis (yaxis)    o < horizonalaxis (xaxis) \  \  o is the origin Note: The Cartesian coordinate system was developed by French mathematician and philosopher Rene Descartes in approximately 1637.
RoadHacker:: Horizon/Vertical Slideshow [opens new window]
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The following are some inequalities graphed on the number line. An endpoint marked with 'x' (or ']') implies that value is included, while an endpoint marked with 'o' (or ')') is excluded.
x > 2 o=========> 5 4 3 2 1 0 1 2 3 4 5 x ≥ 4 x===========================> 5 4 3 2 1 0 1 2 3 4 5 x ≤ 3 <=========================x 5 4 3 2 1 0 1 2 3 4 5 x < 2 <==========o 5 4 3 2 1 0 1 2 3 4 5Linear inequalities are solved just like linear equations with one exception: the inequality sign is "flipped" whenever there is a multiply (or divide) by a negative. The term "flipped" is used to imply lessthan becomes greaterthan and vice versa; and that lessthanorequalto becomes greaterthanorequalto and vice versa.
Example 1: 8x + 1 < 2x  5 8x  2x + 1 < 5 6x + 1  1 < 5  1 6x < 6 x < 1 try x = 0... 8(0) + 1 = 9; 2(0)  5 = 5 9 is not less than 5 try x = 1... 8(1) + 1 = 7; 2(1)  5 = 7 7 is not less than 7 try x = 2... 8(2) + 1 = 15; 2(2)  5 = 9 15 is less than 9 Example 2: 2x < 5 2x / 2 > 5 / 2 # < flipped to > (divideby negative) x > 2.5 try x = 5... 2(5) < 5 10 < 5 is true try x = 5... 2(5) < 5 10 < 10 is falseThe following uses numbers to help understand when inequalities are flipped (switched).
10 > 5 is true multiply both sides by 1 10(1) = 10 and 5(1) = 5 10 > 5 is falseExternal Hyperlink(s)
 PurpleMath.com:: Graphing Linear Inequalities: y > mx + b, etc.
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