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02. Number Systems
02-1 Number Systems
(1) Number Systems:
Real numbers
(R)
Rational
numbers
(Q)
Integers
(Z)
Fractions
(a) N  Z  Q  R, I  R
(b) Q  I  R, Q  I  
(c) Q C  I, I C  Q
Whole Numbers (W)
Terminating decimals: ¼, -½, 1.8, -0.7,…
Repeating decimals: 0.555, 3.1232323
Irrational number (I): Infinite nonrecurring decimals – π, e, √2
(2) Venn Diagram of Number Systems:
[Note]
Natural numbers (N): 1, 2, 3, 
0
Negative integers: -1, -2, -3, 
Infinite decimals
Real Numbers
 Real Numbers: The real numbers R are "all the numbers" on the number line. They include the rational and
irrational numbers together.
(a) On the SAT the word number always means “real number.”
(b) With real numbers all operations can be performed, except for the root of an even index and negative
radicand, and division by zero.
 Rational Numbers: Rational Numbers can be written as a fraction (i.e. as a ratio of two integers).
P
Q{
| p  Z, q  Z, and q  0}
q
 Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as fractions, terminating
decimals, or repeating decimals.
 Integers: All the whole numbers together with their opposites – not fractions or decimals
Z  {…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, …}
(a) Zero is neither positive nor negative.
(b) Consecutive integers: integers that follow in sequence; ex.) 22, 23, 24, 25
(c) Even integers: numbers that can be divisible by 2; …, 4, 2, 0, 2, 4, …, 2n
(d) Odd integers: numbers that are not even; …, 5, 3, 1, 1, 3, 5, …, 2n  1
(e) Consecutive numbers differ by 1, and consecutive even numbers and consecutive odd numbers differ
by 2.
02-2 Number Lines
Number Lines: A number line is used to geometrically represent the relationships between
numbers: integers, fractions, and/or decimals.
(a) The numbers to the right of 0 are positive numbers, and those to the
left of 0 are negative numbers.  If a number is to the left of a
number on the number line, it is less than the other number. If it is to
the right then it is greater than that number.
(b) Every real number can be represented on the number line.

The real number line is composed of infinitely many points.

Each point on the real number line corresponds to a unique number.
(c) The distance between two points on the number line is the absolute value of the difference of their coordinates.
I. Number and Operations
02-3 Absolute Values
(1) Definition of Absolute Number: The Absolute value of a
number is its distance from 0 on the number line.
For example, 4 and 4 have the same
absolute value (4)  |4|  |4| 4
 Symbol: | | is used to indicate the absolute value.
 The absolute value of x is the number that has a distance of b from the origin  |x|  b  x  b or  b
 As the number is further away from the origin, its absolute value is greater.
(2) Absolute Value
For any real value b defines the absolute value |b| to be the magnitude of b
|b| 




b
b
if b  0
if b  0
The absolute value of zero is zero: |0|  0
The absolute values of positive numbers are equal to themselves: |b|  b (b  0)
The absolute values of negative numbers are equal to the negative of the numbers: |b|  b (b  0)
The absolute value of a number is equal to the absolute value of the negative of the number: |b|  |b|
(3) Properties of Absolute value
(a) |a|  |a|
(b) |a|  |b|  a   b
(c) |ab|  |a||b|
(d)
(e) |a |  |b|  |a  b|  |a|  |b|
a
a

b
b
[Example]
(2)
 |3|  3,
|0|  0,
|2|  (2)  2
(3)
(e) |3  (2)|  |1|  1, but |3|  |2|  3  2  5
02-4 Rules of Basic Arithmetic
(1) Basic Operations of Arithmetic
Operation
(a) Addition
(b) Subtraction
(c) Multiplication
(d) Division
Symbol
Result


Sum



Difference


, ( ), , no sign
Product


, :, 
Quotient

(2) Distributive Property
Property
Addition is a communicative operator: a  b  b  a
Addition is an associative operator: (a  b)  c  a  (b  c)
Subtraction is not a communicative operator: a  b  (b  a)
Subtraction is not an associate operator: (a  b)  c  a  (b  c)
Multiplication is a communicative operator: a  b  b  a
Multiplication is an associative operator: (a  b)  c  a  (b  c)
Division is not a communicative operator: a  b  b  a
Subtraction is not an associate operator: (a  b)  c  a  (b  c)
For any real numbers a, b, and c (c  0):
(a) Multiplication over Addition/Subtraction: a  (b  c)  a  b  a  c
a b a b
(b) Division over Addition/Subtraction:
 
c
c c
(3) Order of Operations: PEMDAS (
)  Exponent   or    or 
(a) Clear parentheses: ( )  { }, braces  [ ], brackets
(b) Multiplication and Division: from left to right
(c) Addition and subtraction: from left to right
[Example]
(2)
 If a  3(x  7) and b  3x  7, what is the value of a  b?
a  b  3(x  7)  (3x  7)  3x  21  3x  7  14
(3)
 Simplify the following expression
(i) 3  [5  {2  (8  4)}]  3  [5  {2  4}]  3  [5  2]  3  7  4
(ii) 7  6(4  1)  (15  3)  2[20  (2  8)  5]  7  6  3  5  2[20  16  5]  7  18  5  2  9  30  18  12
I. Number and Operations
02-5 Signed Numbers
(1) Signed Numbers: A number preceded by either a plus or a minus sign is called a signed
number.
(a) Positive numbers: Positive numbers are greater than 0. They can be written with a positive sign (), but they are
usually written without it. They can represent gains, increases, above, greater than….
(b) Negative numbers: Negative numbers are less than 0. They are always written with a negative sign (). They can
represent losses, decreases, below, less than…..
(2) Rules of Signed Numbers
Rules
(1) To add two numbers with like signs, add their absolute values and
attach the common sign.
(2) To add two numbers with unlike signs, find the difference between
their absolute values and attach the sign of the number with larger
absolute value.
(3) To subtract one number from another number, change the operation
to addition and replace by its opposite.
(4) To multiply (or divide) two numbers having like signs, multiply (or
divide) their absolute values and attach a plus sign.
(5) To multiply (or divide) two numbers having unlike signs, multiply (or
divide) their absolute values and prefix a minus sign.
(6) The product and the quotient of more than two signed numbers is
negative if an odd number of negative signs is involved and positive if
there is an even number of negative signs.
347
(3)  (4)  7
Examples
17  (8)  9, (6)  4  2
(18)  15  3
12  7  12  (7)  5, 2  (8)  2  8  10
(9)  4  (9)  (4)  13
5  3  15, (5)  (3)  15, (6)  (3)  2
(3)  6  18, 3  (6)  18
(12)  4  3
(3)  6  2  36, (1)  (2)  (3)  6
I. Number and Operations