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MAT 101 – Lecture 1 Notes

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

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MAT 101 – Lecture 1 Notes

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

  2. 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. 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. 1.1 – Real 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. 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. 1.1 – Real Numbers • 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. 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. 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. 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. 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. 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. 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. 1.2 – Fractions • 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. (-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

  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. 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. 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

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