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

2. Natural Numbers (N) – aka counting numbers = {1,2,3,…}
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!

3. Define the origin at the coordinate 0.
1.1 – Real Numbers
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.

4. 1.1 – Real Numbers
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

5. Real Numbers include the sets of rational and irrational numbers.
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,

6. Answers: F (counterexample: 2/5), T, T, F (counterexample: 0)
1.1 – Real Numbers
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)

7. 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

8. |a|=|-a| >= 0 for any real number, a
1.1 – Real Numbers
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.

9. What is the opposite of -5? For any real number a, -(-a)=a
1.1 – Real Numbers
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!

10. Absolute value in symbolic notation:
1.1 – Real Numbers
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!

11. A fraction fits into which set of numbers from section 1?
1.2 – Fractions
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

12. Every fraction can be written in infinitely many equivalent forms.
1.2 – Fractions
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!

13. 1.2 – Fractions
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!

14. 1.2 – Fractions
Examples 1 and 2
Multiplication of fractions is as simple as multiplying straight across (numerators and denominators) and then reducing the result.
Example 3

15. 1.2 – Fractions
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

16. 1.2 – Fractions
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

17. 1.2 – Fractions
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

18. 1.2 – Fractions
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

19. Strategy for Finding the LCD:
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:
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).

20. 1.2 – Fractions
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.

21. 1.2 – Fractions
Once we have found the LCD, we can add or subtract fractions with different denominators by:
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

22. 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

23. 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

24. 1.3 – Addition and Subtraction of Real Numbers
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. ( = -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

25. 1.3 – Addition and Subtraction of Real Numbers
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

26. 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.

27. Division can be defined in terms of multiplication as follows:
1.4 – Multiplication and Division of Real Numbers
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.

28. 1.4 – Multiplication and Division of Real Numbers
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)

29. Division by Zero in terms of multiplication:
1.4 – Multiplication and Division of Real Numbers
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