Number Systems

Here are the study notes for the text in detailed bullet points, focusing on key facts with context:

Introduction to Number Systems

• We have learned about the number line and how to represent various types of numbers on it.
• The number line is infinite in both directions, and we can collect different types of numbers and store them in a bag.

Natural Numbers, Whole Numbers, and Integers

• Natural numbers (N) are positive integers, starting from 1 (1, 2, 3, ...).
• Whole numbers (W) include zero and all natural numbers (0, 1, 2, 3, ...).
• Integers (Z) include all whole numbers, both positive and negative (...,-3, -2, -1, 0, 1, 2, 3, ...).

Rational Numbers

• A rational number (Q) is a number that can be written in the form p/q, where p and q are integers, and q ≠ 0.
• Rational numbers include all natural numbers, whole numbers, and integers.
• Rational numbers can be represented on the number line.
• There are infinitely many rational numbers between any two given rational numbers.

Irrational Numbers

• An irrational number is a number that cannot be written in the form p/q, where p and q are integers, and q ≠ 0.
• Irrational numbers cannot be represented as a finite decimal or fraction.
• Examples of irrational numbers include √2, √3, and π.
• There are infinitely many irrational numbers.

Real Numbers

• Real numbers (R) are either rational or irrational numbers.
• Every real number can be represented by a unique point on the number line.
• Every point on the number line represents a unique real number.

Decimal Expansions of Real Numbers

• The decimal expansion of a rational number is either terminating or non-terminating recurring.
• A number with a non-terminating non-recurring decimal expansion is irrational.
• Examples of decimal expansions include 0.5 (terminating), 0.142857 (non-terminating recurring), and π (non-terminating non-recurring).

Key Results

• The decimal expansion of a rational number is either terminating or non-terminating recurring.
• A number with a non-terminating non-recurring decimal expansion is irrational.
• Every number with a non-terminating non-recurring decimal expansion is irrational.

Examples and Exercises

• Examples of locating irrational numbers on the number line, such as √2 and √3.
• Exercises to practice identifying and working with rational and irrational numbers, and their decimal expansions.Here are the study notes:

Rational and Irrational Numbers

• A rational number is a number that can be written in the form p/q where p and q are integers and q ≠ 0.
• An irrational number is a number that cannot be written in the form p/q where p and q are integers and q ≠ 0.

Decimal Expansions

• The decimal expansion of a rational number is either terminating or non-terminating recurring.
• A number whose decimal expansion is terminating or non-terminating recurring is rational.
• The decimal expansion of an irrational number is non-terminating non-recurring.
• A number whose decimal expansion is non-terminating non-recurring is irrational.

Operations on Real Numbers

• When you add, subtract, multiply, or divide two rational numbers, the result is always a rational number.
• When you add, subtract, multiply, or divide a rational number and an irrational number, the result may be rational or irrational.
• The sum, difference, product, or quotient of two irrational numbers may be rational or irrational.

Square Roots and Radical Signs

• A square root of a positive real number a is a number b such that b² = a and b > 0.
• The symbol √ is called the radical sign.
• a can be extended to a^(1/n) where a is a real number and n is a positive integer.

Rationalizing Denominators

• To rationalize the denominator of an expression, you multiply it by a suitable form of 1 to make the denominator a rational number.
• This is useful when working with expressions that involve square roots.

Laws of Exponents

• The laws of exponents can be extended to rational exponents.
• Let a > 0 be a real number and p and q be rational numbers. Then:
  • a^p × a^q = a^(p+q)
  • (a^p)^q = a^(pq)
  • a^p / a^q = a^(p-q)
  • a^p × b^p = (ab)^p

I hope these notes are helpful! Let me know if you have any questions.