1 MAT 101 – Lecture 1 Notes Definitions from the Text, sections 1.1 – 1.4.

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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

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? Division by 0 is undefined!

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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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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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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. Examples: and Pi is the ratio of the circumference to the diameter of

any circle,

2 14.3


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Example 3: True or False (Create Figure 1.10 in the text for a visual aid) 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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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). 4 types of intervals (open and closed endpoints)

Infinite intervals make use of – ∞ and/or ∞ to represent at least one endpoint. 5 types of intervals


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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 |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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Two numbers located on opposite sides of 0 on the number line that have the same absolute value are called opposites. 0 is its own opposite, by this definition

What is the opposite of -5? Write this expression as –(-5)

For any real number a, -(-a)=a 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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Absolute value in symbolic notation: a if a ≥ 0

-a if a < 0 Example 6 (7 is more of the same) Questions – Section 1.1? Break Time!

{|| a


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A fraction fits into which set of numbers from section 1? 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. 2/3 is a fraction 2/1 = 2 is an integer

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. This is done by multiplying the numerator and

denominator of the fraction by the same nonzero number. The fraction changes appearance, but not value!

1.2 – Fractions

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Converting a fraction to an equivalent fraction with a smaller denominator is called reducing the fraction. 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!

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

1.2 – Fractions

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Unit Conversion can be achieved by multiplying a conversion factor expressed as a fraction. 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

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 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

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. 9/8 = 8/8 + 1/8 = 1 ⅛

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. 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

1.2 – Fractions

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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:1) Factor each denominator completely

2) Determine the maximum number of times each distinct factor occurs in any denominator

3) 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).

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. 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. Note that 1 is not prime by definition, though it

cannot be factored into anything else.

1.2 – Fractions

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Once we have found the LCD, we can add or subtract fractions with different denominators by:

1) Building up each denominator to the LCD (by multiplying numerator and denominator by the LCD factors which are missing in the denominator)

2) Adding or subtracting numerators

3) Reducing the quotient to lowest terms Examples 6 and 7

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. 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

1.2 – Fractions

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1.3 – Addition and Subtraction of Real Numbers

Sum of Two Numbers with Like Signs 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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Addition of Numbers with Unlike Signs 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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Subtraction of Signed Numbers 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

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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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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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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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: 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


1 MAT 101 – Lecture 1 Notes Definitions from the Text, sections 1.1 – 1.4.

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