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MAT 101 – Lecture 1 Notes
Definitions from the Text, sections 1.1 –
1.4
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1.1 – Real Numbers
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Natural Numbers (N) – aka counting numbers
= {1,2,3,…}
Whole Numbers = {0,1,2,3,…}
Integers (Z) = {…,-3,-2,-1,0,1,2,3,…}
Rational Numbers (Q) – Let a and b
represent integers, with b ≠ 0. Then the set
Q = { a/b | a,b are integers and b ≠ 0}
Why can’t b=0?
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Division by 0 is undefined!
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1.1 – Real Numbers
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The number line is a diagram that helps us
visualize numbers in relationship to other
numbers.
Each number, represented as a point on the
number line, is called a coordinate.
Define a unit equal to the distance between
any two consecutive integers.
Define the origin at the coordinate 0.
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1.1 – Real Numbers
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On the number line, the larger of 2 numbers
is ALWAYS to the right of the smaller one.
Notice that every rational number can be
represented on the number line. What else
can be represented on it?
Examples 1 and 2
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1.1 – Real Numbers
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Real Numbers (R) – the set of numbers that
corresponds to all points represented by the
number line.
Real Numbers include the sets of rational and
irrational numbers.
Irrational Numbers cannot be written as a ratio
of integers.
2 and
 Examples:
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Pi is the ratio of the circumference to the diameter of
any circle,  3.14
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1.1 – Real Numbers
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Example 3: True or False (Create Figure
1.10 in the text for a visual aid)
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Every rational number is an integer
Every counting number is an integer
Every irrational number is a real number
Every whole number is a counting number
Answers: F (counterexample: 2/5), T, T, F
(counterexample: 0)
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1.1 – Real Numbers
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Interval notation is used to represent intervals of
real numbers.
Intervals can be finite (bounded) or infinite
(unbounded).
Finite intervals have endpoints that can be
represented graphically with coordinates on the
number line (set of real numbers).
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4 types of intervals (open and closed endpoints)
Infinite intervals make use of – ∞ and/or ∞ to
represent at least one endpoint.
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5 types of intervals
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1.1 – Real Numbers
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Examples 4 and 5
The absolute value of a number is the
distance (number of units) from 0 on the
number line.
|a|=|-a| >= 0 for any real number, a
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|0|=0, since distance from 0 to itself is 0.
|a|>0 for any real number, a ≠ 0, since distance
must be positive from a to 0.
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1.1 – Real Numbers
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Two numbers located on opposite sides of 0
on the number line that have the same
absolute value are called opposites.
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What is the opposite of -5?
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0 is its own opposite, by this definition
Write this expression as –(-5)
For any real number a, -(-a)=a
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Note: Square roots of negative numbers are not
real numbers, so be careful where you place the
negative signs with radicals!
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1.1 – Real Numbers
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Absolute value in symbolic notation:
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| a | {
a if a ≥ 0
-a if a < 0
Example 6 (7 is more of the same)
Questions – Section 1.1?
Break Time!
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1.2 – Fractions
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A fraction fits into which set of numbers from
section 1?
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Rational numbers (a/b)
Though integers can be written as fractions
(divide by 1), we consider fractions to include
only the rational numbers that are not
integers.
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2/3 is a fraction
2/1 = 2 is an integer
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1.2 – Fractions
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Every fraction can be written in infinitely
many equivalent forms.
Converting a fraction into an equivalent
fraction with a larger denominator is called
building up the fraction.
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This is done by multiplying the numerator and
denominator of the fraction by the same nonzero
number. The fraction changes appearance, but
not value!
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1.2 – Fractions
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Converting a fraction to an equivalent fraction
with a smaller denominator is called reducing
the fraction.
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When we reduce fractions, we are factoring the
numerator and denominator and dividing out the
common factor(s).
When a fraction cannot be reduced any further, it
is written in lowest terms.
Again, fractions change appearance, not value!
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1.2 – Fractions
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Examples 1 and 2
Multiplication of fractions is as simple as
multiplying straight across (numerators and
denominators) and then reducing the result.
Example 3
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1.2 – Fractions
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Unit Conversion can be achieved by
multiplying a conversion factor expressed as
a fraction.
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This method is called cancellation of units,
because we can cancel units, similar to the way
we have been canceling common factors when
reducing fractions.
Example 4
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1.2 – Fractions
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For m ÷ n = p, n is called the divisor and the
result, p, is called the quotient of m and n.
The reciprocal, or multiplicative inverse, of a
fraction a/b where a,b ≠ 0 is b/a.
Dividing Fractions is equivalent to multiplying by
the reciprocal of the divisor:
For b,c,d ≠ 0,
a/b ÷ c/d = a/b d/c
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A reciprocal is found 
by flipping the fraction a/b into b/a
Then 1/3 ÷ 2 = 1/3 *1/2 = 1/6
Example 5
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1.2 – Fractions
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Adding and Subtracting Fractions require us to find
a check the denominator first prior to doing the
addition or subtraction in the numerator.
Adding and subtracting two fractions with the same
denominator is as simple as adding or subtracting
across the numerator and leaving the denominator
the same.
An improper fraction (a/b, where b ≠0) is a fraction in
which the numerator is larger than the denominator
(a > b).
An improper fraction can be written instead as a
mixed number – a natural number along with a
fraction.
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9/8 = 8/8 + 1/8 = 1 ⅛
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1.2 – Fractions
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Adding and subtracting fractions with different
denominators requires us to find a common
denominator first.
A least common denominator is the least
common multiple in the denominators of two
or more fractions.
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1/6 and 2/3 have a least common denominator of
6.
2/3,4/9, and 5/6 have a least common
denominator of 18
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1.2 – Fractions
2/3, 4/9, and 5/6 have a least common
denominator of 18. How do we figure this
out systematically?
Strategy for Finding the LCD:
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1)
2)
3)
Factor each denominator completely
Determine the maximum number of times each
distinct factor occurs in any denominator
The LCD is the product of all of the distinct
factors, where each factor is used the maximum
number of times (identified in Step 2).
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1.2 – Fractions
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Prime number – any number 2 or larger that
cannot be factored into anything other than
itself and 1.
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Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
A number has been factored completely once
it is written as a product of only prime
numbers.
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Note that 1 is not prime by definition, though it
cannot be factored into anything else.
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1.2 – Fractions
Once we have found the LCD, we can add
or subtract fractions with different
denominators by:
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1)
2)
3)
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Building up each denominator to the LCD (by
multiplying numerator and denominator by the
LCD factors which are missing in the
denominator)
Adding or subtracting numerators
Reducing the quotient to lowest terms
Examples 6 and 7
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1.2 – Fractions
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Fractions, Decimals, and Percentages
In the decimal system, a fraction with a denominator
of 10, 100, 1000, and so on, is commonly written as
a decimal number.
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3/10 = 0.3, 25/100 = 0.25, 5/1000 = 0.005
Fractions with a denominator of 100 are often
written as percentages.
To convert between fractions, decimals, and
percentages, we can make use of our knowledge of
building up and reducing fractions
Examples 8 and 9
Break
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1.3 – Addition and Subtraction of Real Numbers
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Sum of Two Numbers with Like Signs
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Add their absolute values and keep the sign the
same as in the given numbers.
Think of this in terms of distance on the number
line: If we are a distance of 5 units away from the
origin in the negative direction, and we want to
continue in that direction for a distance of 12 units,
we have gone a total of 17 units in the negative
direction.
Example: (-5) + (-12) = - (|-5| + |-12|) =
- (5+12) = -17
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1.3 – Addition and Subtraction of Real Numbers
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Addition of Numbers with Unlike Signs
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When adding two numbers with Unlike Signs, subtract their
absolute values. The sign of the number with the larger
absolute value will remain in the answer.
Again, in terms of distance, if we are 3 units to the left of
the origin and we want to travel 2 units to the right of the
origin, we are still 1 unit away from the origin in the
negative direction. (-3 + 2 = -1)
In the case where we are adding opposites, the result will
be 0. a and –a are called additive inverses for this reason.
Additive Inverse Property: (-a) + a = a + (-a) = 0
Examples 1, 2 and 3
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1.3 – Addition and Subtraction of Real Numbers
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Subtraction of Signed Numbers
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For any real numbers a and b,
a - b = a + (-b)
So, subtraction of a number is equivalent to
adding its additive inverse!
It can often be helpful to interpret negative
real numbers (especially when dealing with $)
as debts and positive real numbers as
assets.
Examples 4 and 5
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1.4 – Multiplication and Division of Real Numbers
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For m · n = p, p is called the product of the
numbers m and n, and m and n are known as
the factors.
Multiplication Notation: We can denote the
product of variables m and n as mn OR m · n.
We can denote the product of numbers with
raised dots or parentheses:
3 · 5 OR 3(5)
When multiplying a number and a variable,
no symbol is used between them:
6x represents the product of 6 and x.
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1.4 – Multiplication and Division of Real Numbers
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The product of two nonzero real numbers is
positive if the numbers have the same signs
(both positive or both negative), and negative
if the numbers have different signs.
Example 1
Division can be defined in terms of
multiplication as follows:
If a, b, and c are any real numbers with b ≠ 0,
then a ÷ b = c provided that c · b = a.
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1.4 – Multiplication and Division of Real Numbers
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We can solve division problems by using
multiplicative inverses, just as we were able
to solve subtraction problems by using
additive inverses!
Recall that the multiplicative inverse, or
reciprocal, of a/b where a,b ≠ 0 is b/a
Example: Solve 10 ÷ 2.
10 ÷ 2 = 10 · ½ = 10/2 = 5/1 (reduced)
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1.4 – Multiplication and Division of Real Numbers
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The quotient of two nonzero real numbers is positive
if the numbers have the same signs (both positive or
both negative), and negative if the numbers have
different signs. -- just like the product!
Division by Zero in terms of multiplication:
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If we write 10 ÷ 0 = c, then we need to find c such that c · 0
= 10. This is impossible!
Similarly, if we write 0 ÷ 0 = c, then we need to find c such
that c · 0 = 0. Any c will do in this case!
Since either task does not result in a single result, we call
any quotient with 0 in the denominator undefined.
Examples 2 and 3
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