Podcast
Questions and Answers
What is a real number?
What is a real number?
- A number that can be expressed only as a whole number.
- A number that only has a finite decimal representation.
- A number that can be represented as a point on the real line. (correct)
- A number that can be expressed as a fraction.
What is the completeness property of real numbers primarily concerned with?
What is the completeness property of real numbers primarily concerned with?
- The representation of numbers as decimals.
- The lack of gaps in the real number line. (correct)
- The ability to divide real numbers by zero.
- The ability to combine real numbers using various operations.
Which of the following describes a rational number?
Which of the following describes a rational number?
- A number that is always whole and positive.
- A number that cannot be expressed as a fraction.
- A number with a finite decimal expansion only.
- A number that can be expressed as a fraction m/n where n ≠0. (correct)
When multiplying an inequality by a negative number, what happens to the direction of the inequality?
When multiplying an inequality by a negative number, what happens to the direction of the inequality?
Which of the following is NOT included in the natural numbers?
Which of the following is NOT included in the natural numbers?
What is the result of dividing by zero in the context of real numbers?
What is the result of dividing by zero in the context of real numbers?
Which of these can be expressed in the form of m/n where n ≠0?
Which of these can be expressed in the form of m/n where n ≠0?
What does the symbol R represent in mathematics?
What does the symbol R represent in mathematics?
If a < b and c > 0, what conclusion can be drawn?
If a < b and c > 0, what conclusion can be drawn?
Which of the following statements regarding infinite decimal expansions is true?
Which of the following statements regarding infinite decimal expansions is true?
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Study Notes
Real Numbers
- Real numbers can be expressed in decimal form and include values such as -3/4, 1/3, and √2.
- Infinite decimal expansions, such as 0.999..., represent the same real number as 1.000... .
- Real numbers can be represented as points on a number line, known as the real line, denoted by the symbol R.
- Properties of real numbers are categorized into algebraic properties, order properties, and completeness.
Algebraic Properties
- Real numbers can be added, subtracted, multiplied, and divided (except by 0).
- Division by zero is undefined.
Order Properties
- If a, b, and c are real numbers, certain rules for inequalities apply:
- a < b implies a + c < b + c.
- a < b implies a - c < b - c.
- a < b and c > 0 implies ac < bc.
- a < b and c < 0 implies bc < ac, with the special case that a < b implies -b < -a.
- If a > 0, then 1/a > 0.
- For both positive or negative a and b, a < b implies 1/a < 1/b.
- Multiplying by a positive number preserves an inequality, while multiplying by a negative number reverses it.
Completeness Property
- The completeness property ensures there are no "holes" in the real numbers, which is crucial for the concept of limits.
- This property is foundational for many calculus theorems.
Special Subsets of Real Numbers
- Natural numbers include 1, 2, 3, 4, and so on.
- Integers encompass 0, positive, and negative whole numbers (±1, ±2, ±3, ...).
- Rational numbers are numbers expressible as a fraction m/n, where m and n are integers and n ≠0. Examples include 1/3, -9, and 4/200.
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