Download Real Numbers: Natural Numbers: N= {1,2,3,· · ·} Integers: Z= {0,−1,1

Real Numbers:
Natural Numbers: N= {1, 2, 3, · · ·}
Integers: Z= {0, −1, 1, −2, 2, −3, 3, · · ·}
Note that every natural number is an integer. There are
integers (negative numbers) that are not natural numbers.
2 −1
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Rational Numbers: Q={ ,
, 0, 3, − , · · ·}
5 3
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A rational number is a ratio of two integers. Note that
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every integer is a rational number (e.g. 8 = ), but
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there are rational numbers that are not integers (frac2
tions whose denominator is not 1. E.g. ).
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When expressed as a decimal, every rational number either terminate after some point, or continue indefinitely
in a repeated pattern.
E.g.
3
= 0.6
5
79
= 2.393939 · · · = 2.39
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Irrational numbers are numbers that
√ cannot be expressed as ratios of two integers. e.g. 2, π. When expressed as decimals, irrational numbers don’t terminate
and don’t repeat. E.g. π = 3.14159 · · ·
Note that rational numbers are never irrational and vice
versa.
Putting together rational and irrational numbers give us
the set of Real Numbers, R.
I.e., real numbers are the collection of all the rational
and irrational numbers.
Please note that the terms natural, rational, irrational, when used in the above case, are not adjectives,
but nouns. The set of Real Numbers is a specific collection of numbers that have particular properties in mathematics. The word real is not used to describe the nature
of the numbers and does not imply that the real numbers
are any more or less real (used as a description) than any
other numbers.
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An inequality involves one of four inequality symbols:
< less than
> greater than
≤ less than or equal to
≥ greater than or equal to
3 < 5 is a true statement
5 < 5 is a false statement
5 ≤ 5 is a true statement
We can graph an inequality statement like x < 10 on a
number line. We use open parentheses ( ) for < or >,
and use bracket [ ] for ≤ or ≥.
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The absolute value of a number is the distance the
number is from zero. Algebraically,


|x| = 
x if x ≥ 0
−x if x < 0
To add two negative numbers, add the absolute value of
the two numbers, and turn the result into negative.
E.g. −4 + −3 = −(4 + 3) = −7
To add a positive number with a negative number. First
determine which number has a larger absolute value. If
the negative number has a larger absolute value, the result is negative, otherwise the result is positive. Subtract
the smaller absolute value from the larger absolute value
and assign a sign as given above.
E.g. −5 + 3 = −(5 − 3) = −2
To subtract a negative number is to add the positive:
E.g. 3 − −4 = 3 + 4 = 7
To add a negative is to subtract the positive:
4 + −6 = 4 − 6 = −2
The product of two negative numbers is positive. The
product of a negative number with a positive number is
negative.
Note: |6 − 3| =
6 6+3
The absolute value does not change any operation inside
it, the absolute value only changes a negative number
to its opposite when the final result inside the absolute
value, after all the operations are done, is a negative
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number.
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Exponents:
Any number to a positive integer power means that number times itself that many times.
xn = x
| · x ·{zx · · · x}
n times
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E.g. x = x · x · x
E.g. 34 = 3 · 3 · 3 · 3 = 81
E.g. 25 = 2 · 2 · 2 · 2 · 2 = 32.
In evaluating an expression, we need to always follow the
Order of Operations:
Parenthesis, acting as a grouping symbol, always has the
highest priority.
Exponents, or raising to a power, is the next highest
priority.
Multiplication and division should be done after exponents, they have the same priority. If an expression has
both multiplication and division written, it should be
evaluated from left to right.
Addition and subtraction has the same, lowest priority.
E.g.
4 + 32 − (2 + 3) + 12 − 2 · 5
= 4 + 32 − 5 + 12 − 2 · 5
= 4 + 9 − 5 + 12 − 2 · 5
= 4 + 9 − 5 + 12 − 10
= 13 − 5 + 12 − 10
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= 8 + 12 − 10
= 20 − 10
= 10
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Simplifying Algebraic Expressions:
A variable is a letter used to represent a number. A
variable is used in one of two purposes:
1. It acts as a pronoun for an unknown quantity, for example, we may not know the age of John, but we know
that John is 5 years older than Mary, we can let x represent the age of John, and y represent the age of Mary,
we can then write x = y + 5.
2. A variable also can be used as a generic expression to
represent a collection of numbers (or other objects) that
share a particular characteristic when we need to write
formulas about the numbers:
E.g. For any real number x, y, we have: x + y = y + x.
A constant is a number whose value is fixed. The number 3, for example, is a constant. A constant can also
be represented by a letter (or any other symbol) as long
as it is understood that the number represented by the
letter has a fixed value.
An expression is a collection of variables and constants
with mathematical operations:
E.g. 3x + 5
E.g. x + 4y −
√
x
Each expression can have a value if each variable of the
expression is assigned a value. For example, in the expression:
5x + 3y − 4z, if x = 4, y = 1, and z = 2, then the
expression has the value:
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5(4) + 3(1) − 4(2) = 20 + 3 − 8 = 15.
To find the value of the expression given the values of
the variables is to evaluate the expression.
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If we have the equal sign (=) between two expressions,
we have an equation. An equation can be true or false.
E.g. 3 + 2 = 9
E.g. 4 − 2 = 2
E.g. x + 3 = 5
The first equation is always false, the second equation is
always true (called an identity), and the third equation
is sometimes true (if x = 2).
To find the value(s) of the variable that would make the
equation a true equation is to solve the equation.
Please understand the difference between an expression
and an equation. An expression does not need to be
solved. An expression may be evaluated if all the values
of the variables are known or assigned. An equation, on
the other hand, can be solved. An equation is either true
or false.
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A term in an expression is separated by addition or subtraction from other terms.
E.g. 4x − 3xy + 2z has three terms, they are 4x, −3xy,
and 2z. Note that we say the first term is 4x, not 4 and
x, and we say that the second term is −3xy, because the
sign in front of a term is part of the term. The constant
(the number) in a term is called the coefficient of the
term. In the example, the coefficient of the first term
is 4, the coefficient of the second term is −3, and the
coefficient of the third term is 2.
If a term does not have a number explicitely written, the
coefficient is 1.
E.g. x + 3y. In this expression with two terms, the
coefficient of the first term is 1.
E.g. 4x2 −y +12xz 2 . In this expression with three terms,
the coefficient of the second term is −1.
Two terms are like terms if they involve the same variables to the same power:
E.g. 4x2 y, 6x2 y are like terms, they both have variables
x and y, and the power of x are both 2, and the power
of y are both 1.
E.g. 3x, 4xy are unlike terms, since the first term does
not have y.
E.g. 4x2 , −8x are unlike terms, because the first term
has x to the 2 power, and second term has x to the first
power.
If the terms are alike, we can combine like terms by
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adding their coefficients.
E.g. 3x + 5x = 8x
E.g. 4x2 + 3x − 2x2 = 2x2 + 3x
E.g. x + 4x = 5x.
Notice in the third example that there is the invisible
coefficient of 1 in the first term, and that’s why the simplified expression is 5x.
When simplifying an expression, we are trying to
write an expression to an equivalent expression that is
easier to read or evaluate.
Two expressions are equivalent if they would have the
same value for the same values of the variables. In the
above 3 examples, the expression on the right is equivalent to the expression on the left.
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Properties of the Real Numbers:
For any real numbers x, y, z,
Commutative property of Addition:
x+y =y+x
Commutative property of Multiplication:
xy = yx
Associative property of Addition:
(x + y) + z = x + (y + z)
Associative property of Multiplication:
(xy)z = x(yz)
Distributive property
x(y + z) = xy + xz
(y + z)x = yx + zx = xy + xz
x(y − z) = xy − xz
(y − z)x = yx − zx = xy − xz
Notice that when we use the distributive property with a
negative number outside of the parentheses, the negative
sign will also need to be distributed.
E.g. 4x − 3(x − 5) = 4x − 3x + 15 = x + 15.
Notice that the negative sign in front of 3 is also distributed to the x (making it −3x) and to the −5, making
it +15.
We can simplify an expression using the above properties
and collect like terms:
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E.g. [10(x + 3) − 5x] + [2(x − 1) + 3]
= 10x + 30 − 5x] + [2x − 2 + 3] (distributive property)
= [5x + 30] + [2x + 1] (collect like terms)
= 5x + 30 + 2x + 1 (remove parentheses)
= 7x + 31 (collect like terms)
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Rules of exponents:
xm xn = xm+n
E.g. 23 · 22 = 25
xm
= xm−n
n
x
At this stage we only consider the case when m > n.
Later we will see that the formula makes sense even if
m ≤ n.
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E.g. 2 = 53 = 125
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m n
(x ) = xmn
E.g. (42 )3 = 46 = 4096
(xy)n = xn y n
E.g. (3 · 5)2 = 32 · 52 = 9 · 25 = 225
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