Math 195 Lecture 1 - The Real Numbers, Fractions, LCD, and Interval Notation

The instructor begins by emphasizing the importance of paying attention in class, even to familiar topics, to refine observational skills that will be crucial later. The lecture delves into the foundation of real numbers, starting with natural numbers (positive integers), denoted in boldface as ℕ. It then covers integers (whole numbers including negative and zero), denoted as ℤ, and rational numbers, which are fractions of integers where the denominator is non-zero, denoted as ℚ. The real numbers, which fill in the gaps between rational numbers, are introduced with the notation ℝ, and the instructor mentions the complexity of defining them precisely. The concept of irrational numbers, which cannot be expressed as fractions but exist on the number line, is also briefly introduced.

This paragraph discusses the properties of real numbers, such as commutative and associative properties, which dictate the order of operations does not affect the outcome for addition and multiplication. The distributive property is also highlighted, which allows for the distribution of multiplication over addition. These properties are fundamental axioms, or statements accepted without proof, that all derived mathematical principles must adhere to. The instructor uses the analogy of area calculation to explain the intuitive nature of these properties, emphasizing their importance in understanding real numbers and their operations.

The instructor introduces key mathematical terminology related to real numbers, such as additive and multiplicative identities, which are numbers that do not change the value of another number when added or multiplied, respectively. The concept of negatives and the additive inverse is explained, along with the idea of moving units on the number line to represent addition and subtraction. The instructor also clarifies that zero is neither positive nor negative but is considered non-negative. The properties of negative numbers, such as their equidistant position from zero on the number line, are also discussed.

The paragraph focuses on the multiplication of real numbers, introducing the concept of multiplicative identity and the reciprocal of a number. It explains that the reciprocal of a non-zero number is the number that, when multiplied by the original, yields the multiplicative identity. The instructor also covers the basic operations with fractions, including how to add, multiply, and divide them. The importance of never dividing by zero is emphasized, and the process of finding a common denominator for adding fractions is discussed. The concept of the least common denominator (LCD) is introduced as a way to simplify the process of adding fractions.

The instructor discusses methods for simplifying complex fractions, which are fractions with other fractions in their numerator and/or denominator. Two primary methods are presented: one involves finding a common denominator and combining the fractions by definition, while the other uses the least common denominator (LCD) to simplify the complex fraction more efficiently. The benefits of using the LCD are highlighted, as it can reduce the number of steps required to simplify an expression, thus minimizing the potential for errors.

This paragraph emphasizes the importance of factoring in mathematics, especially for simplifying expressions and solving equations. The instructor demonstrates the trial and error method for factoring quadratic expressions and contrasts it with the AC method, which will be covered later. The discussion highlights the value of efficiency in mathematical problem-solving, encouraging students to practice different methods to find the most efficient one for their needs. The instructor also stresses the importance of practicing to improve mental manipulation of numbers and expressions, especially in the absence of calculators.

The instructor introduces the concept of sets and the two main notations used to represent them: the roster method, which lists the elements of a set, and the set builder notation, which describes the properties that elements of a set must satisfy. The use of ellipses to denote continuation of a pattern and the symbols for membership and non-membership in a set are explained. Additionally, the paragraph covers the union and intersection of sets, explaining how to combine and find common elements between two sets, respectively.

This paragraph delves into interval notation, which is used to express the range of values that a variable can take, such as being greater than, less than, or between two numbers. The instructor explains the different types of intervals and their corresponding notations, including those that include or exclude the endpoints. The use of number lines to visually represent these intervals is also discussed, providing a pictorial understanding of the concepts introduced.

In the concluding paragraph, the instructor reminds students that they are starting from the very beginning, ensuring that everyone has a solid foundation in the material. The instructor also previews upcoming topics, such as the handling of intervals and their intersections and unions, which will be covered in future lectures. The paragraph ends with a note that there is no homework assigned yet, but it will be given once the current topic is completed.