1.1 Properties of Real Numbers

Real numbers have some cool tricks up their sleeve. They follow rules that make math work smoothly, like always giving you another real number when you add, subtract, multiply, or divide them (except dividing by zero, which is a no-go).

These properties are the building blocks of algebra. They let us simplify expressions, solve equations, and prove important stuff about numbers. Understanding them is key to tackling more complex math problems and seeing how numbers really work.

Properties of Real Numbers

Closure, Identities, and Inverses

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• The set of real numbers is closed under addition, subtraction, multiplication, and division (except division by zero) performing these operations on real numbers always results in another real number
• The additive identity is 0, and the multiplicative identity is 1 for real numbers $a + 0 = a$ and $a \times 1 = a$ for any real number $a$
• Real numbers have additive inverses (negatives) and multiplicative inverses (reciprocals), except for 0, which does not have a multiplicative inverse
  • For any real number $a$, there exists a unique real number $-a$ such that $a + (-a) = 0$ (additive inverse)
  • For any non-zero real number $a$, there exists a unique real number $\frac{1}{a}$ such that $a \times \frac{1}{a} = 1$ (multiplicative inverse)

Commutativity, Associativity, and Distributivity

• The commutative property holds for addition and multiplication of real numbers, stating that the order of the operands does not affect the result: $a + b = b + a$ and $ab = ba$
  • Example: $3 + 5 = 5 + 3 = 8$ and $2 \times 4 = 4 \times 2 = 8$
• The associative property applies to addition and multiplication of real numbers, allowing for regrouping of operands without changing the result: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$
  • Example: $(2 + 3) + 4 = 2 + (3 + 4) = 9$ and $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$
• The distributive property relates multiplication and addition of real numbers, allowing for the distribution of multiplication over addition: $a(b + c) = ab + ac$
  • Example: $2(3 + 4) = 2 \times 3 + 2 \times 4 = 14$

Simplifying Expressions with Properties

Combining Like Terms and Factoring

• Simplifying algebraic expressions involves using the properties of real numbers to combine like terms, distribute, factor, or perform other operations to obtain an equivalent, more concise expression
• The commutative and associative properties allow for the rearrangement of terms in an expression, enabling simplification and the combination of like terms
  • Example: $3x + 2y + 5x = 8x + 2y$ (combining like terms)
• The distributive property is used to expand or factor expressions, such as $a(b + c) = ab + ac$ or $ab + ac = a(b + c)$
  • Example: $2(3x + 4) = 6x + 8$ (distributing) and $6x + 9 = 3(2x + 3)$ (factoring)

Solving Equations

• When solving equations, the properties of real numbers justify the use of inverse operations to isolate the variable, maintaining the equality of both sides of the equation
• The additive and multiplicative identities can be used to simplify expressions or solve equations by adding 0 or multiplying by 1 without changing the value of the expression or the solution to the equation
  • Example: $3x + 0 = 12$ simplifies to $3x = 12$
• The additive and multiplicative inverses can be used to cancel out terms or factors in an expression or equation, simplifying the problem
  • Example: $3x + 5 = 11$ can be solved by subtracting 5 from both sides, yielding $3x = 6$, and then dividing both sides by 3 to get $x = 2$

Rational vs Irrational Numbers

Definitions and Examples

• Real numbers can be classified as either rational or irrational numbers, forming a disjoint partition of the real number set
• Rational numbers are numbers that can be expressed as the ratio of two integers, with the denominator not equal to zero, in the form $\frac{p}{[q](https://www.fiveableKeyTerm:q)}$, where $p$ and $q$ are integers and $q \neq 0$
  • Examples of rational numbers include integers (1, -3), fractions ($\frac{2}{5}$), and terminating or repeating decimals (0.75, 0.3333...)
• Irrational numbers are numbers that cannot be expressed as the ratio of two integers and have non-repeating, non-terminating decimal expansions
  • Examples of irrational numbers include non-perfect square roots ($\sqrt{2}$, $\sqrt{3}$), pi ($\pi$), and Euler's number ($e$)

Density of Rational Numbers

• The set of rational numbers is dense in the set of real numbers, meaning that between any two real numbers, there exists a rational number
  • For any two distinct real numbers $a$ and $b$, there exists a rational number $r$ such that $a < r < b$
  • This property implies that there are infinitely many rational numbers between any two real numbers, even if they are very close together

Proving Irrationality

Proof by Contradiction

• A number is considered irrational if it cannot be expressed as the ratio of two integers, with the denominator not equal to zero
• To prove a number is irrational, one common method is to use a proof by contradiction, assuming the number is rational and deriving a logical contradiction
• For square roots of non-perfect squares, such as $\sqrt{2}$ or $\sqrt{3}$, a proof by contradiction can be used to demonstrate their irrationality
  • The proof typically assumes the number is rational, expresses it as a fraction in lowest terms, and then derives a contradiction by showing that the fraction cannot be in lowest terms
  • The contradiction arises from the fact that if the number is rational, the fraction's numerator and denominator must both be divisible by the square root of the non-perfect square, which is impossible for a fraction in lowest terms

Other Methods

• Other methods for proving irrationality include using the fundamental theorem of arithmetic or the properties of rational numbers to derive contradictions
• The irrationality of specific numbers, such as $\pi$ or $e$, can be proven using more advanced techniques, such as calculus or infinite series expansions
  • For example, the irrationality of $\pi$ can be proven using the fact that its decimal expansion is non-repeating and non-terminating, which is a property of irrational numbers
  • The irrationality of $e$ can be proven using its definition as the limit of $(1 + \frac{1}{n})^n$ as $n$ approaches infinity and showing that it cannot be expressed as a rational number