Download Chapter 1

Chapter 1
The Real Number System
Section 1.3 – Real Numbers and the Number Line
Homework Problems: 7-9, 12, 15, 16, 23-25, 28-29, 34-36, 46-48
Additional Problems
1) Indicate by letter which sets the following numbers to:
5
a) Real numbers
b) Irrational Numbers
1
7
c) Rational Numbers
2
d) Integers
e) Whole numbers
19
f) Natural numbers
-13
0
6.34
99
2) Write as a positive or negative number:
a) You have one hundred dollars.
b) You owe one hundred dollars.
3) Put 7, 3.4, and 2 on the number line below
4) Fill the blanks below with either < (less than) or > (greater than):
a) 12 ___ 14
b) 12 ___ 14
c) 3
3
4
___ 3
9
7
d) 77 ___ 80
Page 1 of 20
Chapter 1
The Real Number System
Section 1.4 & 1.5
Adding / Subtracting Real Numbers
Section 1.4 homework: 14-16, 23-25, 30-32, 63, 66
Section 1.5 homework: 1, 3, 11-13, 22-24, 31-33, 58, 63
Additional Problems
1) Add (+3) + (+4) =
2) Add (+3) + (-4) =
3) Add (-3) + (+4) =
4) Add (-3) + (-4) =
5) Subtract (+3) - (+4) =
6) Subtract (+3) - (-4) =
7) Subtract (-3) - (+4) =
8) Subtract (-3) - (-4) =
9) What is 3 + 4 + (-5) - (-2) =
Page 2 of 20
Chapter 1
The Real Number System
Section 1.6
Multiplying and Dividing Real Numbers
Homework: 9, 12-16, 23-27, 31, 33-35, 48-50
Additional Problems
1) Find the product (reduce fractions):
a) 8(7) =
b) 4(3) =
c) 12(0) =
d) 12.2(3.2) =
4 7
e) − 3 (2) =
2) Find the quotient (reduce fractions):
42
a) 7 =
b)
c)
−42
=
12
723.25
−1.1
=
3) Evaluate (reduce fractions):
4−14
a) 2(3)−1 =
b)
c)
3(11)−48
10
4(8)−103
10−3(7)
=
=
4) For a = 1, b = 2, c = 3 evaluate (reduce fractions)
2𝑎−𝑏
a) 3𝑐−𝑎𝑏 =
b)
c)
𝑏−3𝑐
𝑏𝑐−4𝑎
=
𝑎𝑐−𝑏(𝑏)
𝑐(𝑐)+𝑎𝑏
=
Page 3 of 20
Chapter 1
The Real Number System
Section 1.2
Variables, Expressions, and Equations
Homework: 1-3, 9, 11-13, 23, 47-50, 67-70
Additional Problems
1) Evaluate the following expressions when x = 3, y = 5, and z = 6:
a) 4x – 7
b) 7x – 3y
c) (4z – 2y – x)(z – 1)
3𝑎−2𝑏
2) Evaluate the expression 𝑎+𝑏+1 when:
a) a = 4 and b = 1
b) a = 2 and b = 6
3) For each of the following equations, is x = 6 a solution?
a) 7x = 43
b) 3x – 4 = 14
c) 48 – 3x = 42
Page 4 of 20
Chapter 1
The Real Number System
Section 1.7
Properties of Real Numbers
Homework: 1-8, 31, 33, 35, 37, 43, 46, 49
Additional Problems
1) Rewrite 1 + (2 + 3) using
a) The commutative property:
b) The associative property
2) For 5, what is the
a) The additive inverse?
b) The multiplicative inverse?
3) What is the identity for
a) Addition?
b) Multiplication?
4) Rewrite using the commutative property:
a) 3 + 5 =
b) 3 × 5 =
5) Rewrite using the associative property:
a) 3 + (5 + 7) =
b) (3 × 5) × 7 =
6) Use the distributive property to expand –4(5 – x) =
Page 5 of 20
Chapter 1
The Real Number System
Page 6 of 20
Section 1.1
Exponents, Order of Operations, and Inequality
Homework: 7, 11, 15, 25, 33, 40, 48, 51, 53, 59, 62, 77-80
Additional Problems
1) Evaluate:
a) 24 =
b) 53 =
c) 92 =
2) Evaluate:
a) (4 + 3)(5 − 2) =
b) 6 ÷ 2 + 1 =
c)
2+32
9−22
=
3) List which comparison symbols (from choice of =, <, >, ≠, ≤, ≥) that would form a true
statement if placed between the number below (in the blank):
a) 10 ___ 12
b) 11.12 ___ 11.021
c) 4 __ 4
Chapter 1
The Real Number System
Section 1.8
Simplifying Expressions
Homework: 2, 3, 8-10, 15-17, 27-30, 41-43
Additional Problems
1) Identify the following pair of term as like or unlike:
a) 4x, 7x
b) 4𝑥, −7𝑥 2 .
c) 4x, 7y
d) 4mar, 7ram
2) Simplify each expression (combine like terms)
a) 4r – 2r =
b) 4r – 2s – 5s +7r =
3) Give the numerical coefficient of each term:
a) 4r
b) –2s
c) –5s
d) 7r
4) Simplify each expression (expand and combine like terms)
a) 4(𝑎2 − 3𝑎) + 2𝑎 − 1 =
b) 3(𝑥 − 𝑦) − 2(𝑦 − 2𝑥) =
c) 4𝑎 + 𝑏 − 3(𝑎 + 𝑏) =
Page 7 of 20
Chapter 1
The Real Number System
Page 8 of 20
Vocabulary
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
absolute (value)
add
additive (inverse)
algebraic
associative
base
coefficient
combine
commutative
comparison
denominator
difference
distributive
division
divisor
equality
equations
evaluate
exponents
expression
factors
greater (than)
grouping
identify
identity
integers
irrational
less (than)
like (terms)
lowest
minuend
multiplication
(property)
33) multiplicative
(inverse)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
multiplying
natural (numbers)
negative
number (line)
numerator
numerical
operations
opposite
order
ordering
power
product
property
quotient
raised
rational
reciprocal
set
set-builder (notation)
signed
simplifying
solutions
subtract
subtrahend
sum
symbol
terms
undefined
unlike
variable
whole (numbers)
Chapter 1
The Real Number System
Section 1.3 – Real Numbers and the Number Line
Additional Answers
1) Indicate by letter which sets the following numbers to:
5
5
a, c, d, e, f a) Real numbers
1
b) Irrational Numbers
1 7
a,
c
7
c) Rational Numbers
2
2
d) Integers
a,
b
√19
e) Whole numbers
19
f) Natural numbers
-13 -13 a, c, d
0
0
a, c, d, e
6.34 6.34 a, c
99
99
a, c. d, e, f
2) Write as a positive or negative number:
c) You have one hundred dollars.
100
d) You owe one hundred dollars.
-100
3) Put 7, 3.4, and 2 on the number line below
4) Fill the blanks below with either < (less than) or > (greater than):
e) 12 _<_ 14
f) -12 _>_ -14
3
4
g) 3 7 _<_ 3 9
h) 77 _>_ -80
Vocabulary:
absolute (value)
additive (inverse)
greater (than)
integers
irrational
less (than)
natural (numbers)
negative
number (line)
opposite
ordering
rational
set
set-builder (notation)
whole (numbers)
Page 9 of 20
Chapter 1
The Real Number System
Section 1.4 & 1.5
Adding / Subtracting Real Numbers
Additional Answers
1)
2)
3)
4)
5)
6)
7)
8)
9)
Add (+3) + (+4) = 7
Add (+3) + (4) = 1
Add (3) + (+4) = 1
Add (-3) + (-4) = 
Subtract (+3)  (+4) = 
Subtract (+3) - (-4) = 7
Subtract (-3) - (+4) = 
Subtract (-3) - (-4) = 1
What is 3 + 4 + (-5) - (-2) = 7  3 = 4
Vocabulary:
add
difference
minuend
signed
subtract
subtrahend
sum
Page 10 of 20
Chapter 1
The Real Number System
Section 1.6
Multiplying and Dividing Real Numbers
Additional Answers
1) Find the product (reduce fractions):
f) 8(7) = 56
g) 4(3) = 12
h) 12(0) = 0
i) 12.2(3.2) = 39.04
4 7
28
𝟏𝟒
j) − 3 (2) = − 6 = − 𝟑
2) Find the quotient (reduce fractions):
42
d) 7 = 𝟔
e)
f)
−42
𝟕
= −𝟐
12
723.25
−1.1
= −𝟔. 𝟓𝟕𝟓 = −
𝟐𝟔𝟑
𝟒𝟎
, Note: Either answer is OK.
3) Evaluate (reduce fractions):
4−14
−10
d) 2(3)−1 = 5 = −𝟐
e)
f)
3(11)−48
10
4(8)−103
10−3(7)
=
−15
10
−71
𝟑
= − 𝟐 = −𝟏. 𝟓, Note: Either answer is OK.
𝟕𝟏
= −11 = 𝟏𝟏
3) For a = 1, b = 2, c = 3 evaluate (reduce fractions)
d)
e)
f)
2𝑎−𝑏
2(1)−(2)
2−2
0
= 3(3)−(1)(2) = 9−2 = 7 = 𝟎
3𝑐−𝑎𝑏
𝑏−3𝑐
(2)−3(3)
2−9
−7
𝟕
= (2)(3)−4(1) = 6−4 = 2 = − 𝟐
𝑏𝑐−4𝑎
(1)(3)−(2)(2)
𝑎𝑐−𝑏(𝑏)
3−4
−𝟏
𝑐(𝑐)+𝑎𝑏
=
3(3)+(1)(2)
= 9+2 =
Vocabulary:
denominator
division
divisor
lowest
multiplication (property)
multiplying
numerator
product
quotient
reciprocal
terms
undefined
𝟏𝟏
= −𝟑. 𝟓, Note: Either answer is OK.
𝟏
= − 𝟏𝟏, Note: Either answer is OK.
Page 11 of 20
Chapter 1
The Real Number System
Page 12 of 20
Section 1.2
Variables, Expressions, and Equations
Additional Problems
1) Evaluate the following expressions when x = 3, y = 5, and z = 6:
d) 4x – 7
4(3) – 7 = 12 – 7 = 5
e) 7x – 3y
7(3) – 3(5) = 21 – 15 = 6
f) (4z – 2y – x)(z – 1)
[4(6) – 2(5) – (3)][(6) – 1] = (24 – 10 – 3)(5) = 11(5) = 55
3𝑎−2𝑏
2) Evaluate the expression 𝑎+𝑏+1 when:
c) a = 4 and b = 1
d) a = 2 and b = 6
3(4)−2(1)
(4)+(1)+1
3(2)−2(6)
(2)+(6)+1
=
=
12−2
6
6−12
9
=
=
𝟏𝟎
𝟔
−𝟔
𝟗
𝟓
=𝟑
=−
−𝟐
𝟑
Note: Do not have to reduce unless instructions ask you to reduce answers.
3) For each of the following equations, is x = 6 a solution?
d) 7x = 43
7(6) = 43
42 = 43
e) 3x – 4 = 14
3(6) – 4 = 14 18 – 4 = 14 14 = 14
f) 48 – 3x = 42 48 – 3(6) = 42 48 – 18 = 42 30 = 42
Vocabulary:
algebraic
equality
equations
evaluate
expression
identify
solutions
symbol
variable
No
Yes
No
Chapter 1
The Real Number System
Section 1.7
Properties of Real Numbers
Additional Answers
1) Rewrite 1 + (2 + 3) using
c) The commutative property:
d) The associative property
2) For 5, what is the
c) The additive inverse?
d) The multiplicative inverse?
3) What is the identity for
c) Addition?
d) Multiplication?
= (2 + 3) + 1 or = 1 + (3 + 2)
= (1 + 2) + 3
5
𝟏
𝟓
0
1
4) Rewrite using the commutative property:
c) 3 + 5 = 5 + 3
d) 3 × 5 = 5 × 3
5) Rewrite using the associative property:
c) 3 + (5 + 7) = (3 + 5) + 7
d) (3 × 5) × 7 = 3 × (5 × 7)
6) Use the distributive property to expand –4(5 – x) = –20 + 4x
Vocabulary:
associative
commutative
distributive
identity
multiplicative (inverse)
property
Page 13 of 20
Chapter 1
The Real Number System
Page 14 of 20
Section 1.1
Exponents, Order of Operations, and Inequality
Additional Problems
1) Evaluate:
d) 24 = 16
e) 53 = 125
f) 92 = 81
2) Evaluate:
d) (4 + 3)(5 − 2) = (7)(3) = 𝟐𝟏
e) 6 ÷ 2 + 1 = 3 + 1 = 𝟒
f)
2+32
9−22
=
2+9
9−4
=
𝟏𝟏
𝟓
3) List which comparison symbols (from choice of =, <, >, ≠, ≤, ≥) that would form a true
statement if placed between the number below (in the blank):
d) 10 ___ 12
<, ≠, ≤
e) 11.12 ___ 11.021
>, ≠, ≥
f) 4 __ 4
=, ≤, ≥
Vocabulary:
base
comparison
exponents
grouping
operations
order
power
raised
Chapter 1
The Real Number System
Section 1.8
Simplifying Expressions
Homework: 7, 11, 15, 25, 33, 40, 48, 51, 53, 59, 62, 77-80
Additional Problems
1) Identify the following pair of term as like or unlike:
e) 4x, 7x
like
2
f) 4𝑥, −7𝑥
unlike
g) 4x, 7y
unlike
h) 4mar, 7ram like (put in alphabetical order and moth are arm)
2) Simplify each expression (combine like terms)
c) 4r – 2r = 2r
d) 4r – 2s – 5s +7r = 11r – 7s
3) Give the numerical coefficient of each term:
e) 4r
4
f) –2s
–2
g) –5s
–5
h) 7r
7
4) Simplify each expression (expand and combine like terms)
d) 4(𝑎2 − 3𝑎) + 2𝑎 − 1 = 4𝑎2 − 12𝑎 + 2𝑎 − 1 = 𝟒𝒂𝟐 − 𝟏𝟎𝒂 − 𝟏
e) 3(𝑥 − 𝑦) − 2(𝑦 − 2𝑥) = 3𝑥 − 3𝑦 − 2𝑦 + 4𝑥 = 𝟕𝒙 − 𝟓𝒚
f) 4𝑎 + 𝑏 − 3(𝑎 + 𝑏) = 4𝑎 + 𝑏 − 3𝑎 − 3𝑏 = 𝒂 − 𝟐𝒃
Vocabulary:
coefficient
combine
factors
like (terms)
numerical
simplifying
terms
unlike
Page 15 of 20
Chapter 1
The Real Number System
Page 16 of 20
Glossary
Absolute value: (See number)
Addends: (See Addition).
Addition: is combining several quantities together. Note: For real numbers, addition can be
considered as combining lengths together. From where you are, move the length (size of the
number) you want to go in the direction of the number. At the beginning you start from zero.
Addends: are the quantities being added together.
Sum: the result of addition.
Associative law of addition: says that the order that the results of combining the addends
together is independent of the order you add the addends together. (Just make sure you use
each addend once and only once).
Commutative property of addition: says that adding the first number to the second
produces the same result as adding the second number to the first.
Zero Property of addition: states that adding zero to any number results in the same
number. For the reason, zero is called the additive identity.
Additive Identity: Zero. Adding zero does not change the result.
Algebraic Expression: is a finite collection of numbers, variables, groupings combined together
with the operations of addition, subtraction, multiplication, division, and exponentiation. Note:
This can be extended to include algebraic functions (which can be expressed as an algebraic
expression in just the variables of the function). Normally the exponents are rational, because
otherwise one tends to get outside the algebraic realm.
Associative Property: is when, for the same operation, grouping (what you do first) does not
make any difference. (See Addition and Multiplication).
Base: (See Exponential Notation)
Coefficient: (See Terms)
Combine like terms: (See terms)
Commutative property: is when for a numerical operation the numbers can be reversed and the
result is the same. I.E., A operation B has the same value as B operation A. (See Addition and
multiplication).
Comparison Relations: Give an indication which of a pair of numbers is larger and or indicate
they could be the same value. We say a number is larger than another number if there is a
positive number that can be added to the second number to give the first number.
Equal (=): Says the two values on either side of the equal relation have the same value.
Greater than (>): Says the value on the left of the greater than relation is larger than the
value on the right.
Greater than or equal to (≥): Says the value on the left of the greater than or equal to
relation is either larger than or equal to the value on the right.
Less than (<): Says the value on the right of the greater than relation is larger than the
value on the left.
Less than or equal to (≤): Says the value on the right of the greater than or equal to
relation is either larger than or equal to the value on the left.
Not equal to (≠): Says the two values on either side of the not equal relation have
different values. There is no indication which is larger.
Denominator: (See Fraction)
Difference: (See Subtraction)
Distributive law of multiplication: (See Multiplication)
Dividend: (See Division)
Chapter 1
The Real Number System
Page 17 of 20
Division: is the inverse operation of multiplication. To divide by a number, multiply by its
reciprocal.
Dividend: is the number being divided. In the expression 30 divided by 10 (or 10 into 30),
30 is the dividend.
Divisor: is the number doing the dividing. In the expression 30 divided by 10 (or 10 into 30),
10 is the divisor.
Reciprocal: To find the reciprocal of a value, write the number as a fraction and then
interchange the numerator and denominator. Note: For mixed numbers, you need to rewrite
1
the number as an improper fraction first. The reciprocal of n is 𝑛.
Quotient: is the (principal) result of division. In the expression 30 divided by 10 is 3, 3 is the
quotient. This class we will not be using remainders.
Note: Division by zero is undefined.
Note: If you divide zero by any number, except zero, the quotient is zero.
Divisor: (See Division)
Element: (See Set)
Equal (=): (See Comparison Relations)
Equation: Is a mathematical statement that states two expressions have the same value.
Solutions of an equation: is the set of points (values for the variables) that make the
equation a true statement. Note: In a manner of speaking, an equation is something we
would like to be true. Finding (hopefully all) the solutions of the equation is called
solving the equation.
Evaluate:
1) Simplify an expression to determine its value.
2) Substitute given values for variables in an mathematical expression, and the determine
2+𝑥
the value for that expression. Example: To evaluate 4+𝑥, when x = 6. First replace x in
2+(6)
8
4
that expression by (6) to get 4+(6). Next evaluate that expresion to get 10 = 5.
Exponent: (See Exponential Notation and Order of Operations)
Exponential Notation: Indicates that a number is to be multiplied by itself several times.
Base: is the number that is to be multiplied by itself several times.
Exponent: Indicates how many times a number is to be multiplied by itself.
Example: The notation 53 indicates that 5, the base, is to be multiplied by itself 3 (the
exponent) times; i.e. 53  5  5  5  25  5  125 .
Power: is a synonym for exponent. Normally the word power is preceded by an ordinal
number. 54 is read as “five to the fourth power”. For the exponent of 2, the number can be
read as “squared” instead of “to the second power”. For the exponent of 3, the number can be
read as “cubed” instead of “to the third power”.
Factors: (See multiplication).
Fraction: In this course, this is the usual way of doing division. It is normally seen as
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
.
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
Denominator: is the bottom half of the fraction, it is the divisor of the division operation.
Fraction bar: Is the line that separates the numerator from the denominator. It is also a
grouping symbol (see Order of Operations).
Numerator: is the top half of the fraction, it is the dividend of the division operation.
Greater than (>): (See Comparison Relations)
Greater than or equal to (≥): (See Comparison Relations)
Groupings: (See Order of Operations)
Chapter 1
The Real Number System
Page 18 of 20
Identity: (See Additive Identity and Multiplicative Identity).
Also refers to an expression that is always true as x + x = 2x.
Integers: Any number that can be rewritten (is equal to a number) as having no fractional part or
digits past the decimal point.
Natural number: is a positive integer – i.e. integers starting with one. They are sometimes
referred to as the counting numbers.
Whole number: is a nonnegative integer – i.e. integers starting with zero.
Irrational: (See Rational Number)
Less than (<): (See Comparison Relations)
Less than or equal to (≤): (See Comparison Relations)
Like terms: (See Terms)
Minuend: (See Subtraction)
Multiplicand: (See Multiplication)
Multiplicative Identity: is one. Anything multiplied (or divided by 1) does not change its value.
Multiplication: In the expression 2 times 12 is 24:
Multiplicand: 2 is the multiplicand, the number being multiplied.
Multiplier: 12 is the multiplier; multiply the multiplicand by the multiplier.
Product: 24 is the product, the result of multiplication.
Factors: are things multiplied together to produce a product. Both the multiplicand and the
multiplier are considered a factor of the product.
Distributive law of multiplication: says that adding (or subtracting) two numbers together
then multiply the result by a third number produces the same result as multiplying the first
two numbers by the third number first then adding (or subtracting) the two products together.
Example: 6  (2  3)  6  5  30 and (6  2)  (6  3)  12  18  30 .
Zero property of multiplication: refers to the fact that the product is zero if any of the
factors of the product are zero.
Multiplier: (See Multiplication)
Natural number: (See Integers)
Negative: (See Number)
Not equal to (≠): (See Comparison Relations)
Number: A number has a size and direction.
Absolute value: indicates the size of the number. When the number represents length,
the absolute value of a number indicates how far the number is from the value zero.
Sign: indicates the direction of the number. A positive sign (+) means it is in the
direction of numbers getting larger – on the number line that is normally to the right. A
negative sign (–) means it is in the direction of numbers getting smaller – on the number
line that is normally to the left. Zero has no direction associated with it. However, one
may write zero as either 0, +0, or –0. Note: For a non-zero number, if the sign is omitted,
the sign is assumed to be positive.
Number line: is used to represent numbers as points on a straight line. The distance
between two numbers is proportional to the difference between the two numbers. On the
standard number line, the line is horizontal with larger numbers on the right. For a
vertical number line, the larger numbers are higher and the smaller numbers are lower.
Note: The number line can at best represent just part of the real numbers. Zero does not
have to be on any number line you draw. You are also free to choose a desirable scale.
Numerator: (See Fraction)
Numerical Coefficient: (See Terms)
Chapter 1
The Real Number System
Page 19 of 20
Order of operations: dictates the order an arithmetic expression is correctly evaluated. In brief
the order is:
Groupings as parenthesis ( ), brackets [ ], or braces {}. Essentially this means to work from
the inside to the outside. Other examples of (implied) groupings are denominators of
fractions, numerators of fractions, expressions under the radical / square root sign,
3 5
expressions in the exponent (as 2 , you evaluate the exponent 3+5 first). In fact anything
used as a subscript, superscript, or index, or parameter needs to be evaluated before it is used.
Exponents, evaluating a base raised to a power is done next;
Multiplication and Division is done next. Either of these two may be done first. (However,
for ease of calculation, we suggest doing multiplications before divisions. Note you can
rearrange the order of multiplying and dividing so the divisions come last);
Addition and Subtraction are done last.
Note: Because the addition and subtractions are done last, you can evaluate everything between
the outermost addition and subtraction signs separately from each other and you will still get the
correct evaluation. Note there are other rules, like the distribution law of multiplication, which
allows one to restate the expression before evaluating it.
Positive: (See number)
Power: (See Exponential Notation)
Product: (See Multiplication)
Quotient: (See Division)
Rational Number: is any number that can be written as the quotient of two integers. If a number
cannot be written as the quotient of two integers, it is called irrational.
Reciprocal: (See Division)
Set: is a collection of objects.
Element: is what each objects in the set is called.
Set-list notation: The name of each element in the set is listed inside braces as
{1, 3, 5, 7}.
Set-Builder notation: describes the members in the set instead of listing the members.
Example: {n | n is an even whole number}. The prior example is read as “The set n where
n is an even whole number”.
Sign: (See number)
Solutions of an equation: (See Equation)
Subtraction: adds the opposite to a quantity. Note: For real numbers, subtraction can be
considered as combining lengths together. From where you are, move the length (size of the
number) you want to go in the opposite direction of the number. At the beginning you start from
zero.
Difference: the result of subtraction. In the expression 5 – 3 = 2, 2 is the difference.
Sometimes the difference is called the remainder.
Minuend: In the expression 5 – 3 = 2, 5 is the minuend.
Subtrahend: In the expression 5 – 3 = 2, 3 is the subtrahend.
Subtrahend: (See Subtraction)
Sum: (See Addition)
Symbol: (See variable)
Chapter 1
The Real Number System
Page 20 of 20
Terms: Are things (numbers, variables) that are algebraically added or subtracted together. Note:
there is some ambiguity in this definition. 5xy is an algebraic expression with just one term; it is
not added or subtracted to anything, but could be. That term contains three factors: 5, x, and y.
4x + 5y – 2xy contain three terms: 4x, 5y, and 2xy. The easiest way to count the number of terms
is to count the outermost addition (+) and subtraction (–) operators and add 1 to the sum.
Outermost here means not inside a grouping symbol. The confusion starts when you find out that
for 2(x + 2y + 3z) = 2x + 4y + 6z , the expression on the left contains just one term and the
expression on the right contains 3 terms. But the right side of the equation is just the expansion
of the left hand side! By the way the factor (x + 2y + 3z) contains 3 terms inside parenthesis.
Numerical Coefficient: In a term, there is normally just one outermost numerical factor.
That number is called the numerical coefficient. For the term 5xy, the numerical
coefficient is 5. If there is no visible number, then the numerical coefficient is considered
to be 1 since ab = 1ab. We will avoid weird expressions as 2x5y in this course. (Not sure
if the numerical coefficient should be 2, 5, or 10). However, for the term 2x(5y) the
numerical coefficient is 2 since 5 is not “outermost”. The numerical coefficient is often
just called the coefficient.
Like terms: Two terms in an algebraic expression are considered like terms if their nonnumerical factors can be rearranged to be the same. Remember an exponent indicates that
a factor has been repeated that many times. Since multiplication is commutative, it is
easiest to list the factors in alphabetical order (assuming alphabetical variables) and check
if they are the same. Example 2mar would become 2amr, – 3ram,would become
– 3amr, and 7arm would become 7amr; so all three of those terms are like terms.
However, the term 5𝑎2 𝑚𝑟 woulb become 5aamr is not like the other three terms because
it has an extra factor of a.
Combine like terms: Like terms can be added or subtracted together by just adding
subtracting the numerical coefficients together and then placing the common factors after
the sum. Example: 3x + 4y + 5 – 6y + 2x = 5x – 2y + 5 since for the like terms 3x + 2x we
have 3 + 2 = 5 and for the like terms + 4y – 6y we have + 4 – 6 = –2.
Unlike terms: If two different terms are not like terms, they are called unlike terms.
Undefined: (See Division)
Unlike terms: (See terms).
Variable: is a symbol, usually a letter of (any) alphabet that represents (or can be replaced by) a
number.
Whole number: (See Integers)
Zero property of multiplication: (See Multiplication).