1.1 Properties of Real Numbers and Algebraic Operations

Real numbers are the building blocks of algebra. They include whole numbers, fractions, and irrational numbers like pi. Understanding their properties and how to work with them is crucial for solving equations and simplifying expressions.

Commutative, associative, and distributive properties help us rearrange and simplify expressions. Inverse operations and identities allow us to undo operations and maintain equality. These concepts form the foundation for more advanced algebraic techniques.

Properties of Real Numbers

Commutative, Associative, and Distributive Properties

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• The commutative property states that the order of the operands does not affect the result for addition and multiplication: $a + b = b + a$ and $a \times b = b \times a$
• The associative property states that the grouping of the operands does not affect the result for addition and multiplication: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$
• The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products: $a(b + c) = ab + ac$
• These properties can be used to simplify expressions by rearranging terms (commutative property), grouping like terms (associative property), and distributing factors (distributive property)
• These properties can also be used to solve equations by isolating the variable on one side of the equation

Applying Properties to Simplify Expressions and Solve Equations

• Example of using the commutative property to simplify an expression: $3 + x + 2 = x + 5$
• Example of using the associative property to simplify an expression: $(2 + 3) + x = 2 + (3 + x)$
• Example of using the distributive property to simplify an expression: $2(3x + 4) = 6x + 8$
• Example of using properties to solve an equation: If $2x + 3 = 11$, then $2x = 8$ (by subtracting 3 from both sides) and $x = 4$ (by dividing both sides by 2)

Inverse Operations for Real Numbers

Additive Inverses

• The additive inverse of a number is the opposite of that number
• When a number is added to its additive inverse, the result is zero (the additive identity)
• For any real number $a$, the additive inverse is $-a$
• Example: The additive inverse of 5 is -5, and $5 + (-5) = 0$

Multiplicative Inverses (Reciprocals)

• The multiplicative inverse (or reciprocal) of a number is the number that, when multiplied by the original number, results in 1 (the multiplicative identity)
• For any non-zero real number $a$, the multiplicative inverse is $\frac{1}{a}$
• Example: The multiplicative inverse of 4 is $\frac{1}{4}$, and $4 \times (\frac{1}{4}) = 1$
• The multiplicative inverse of a fraction $\frac{a}{b}$ is $\frac{b}{a}$, as long as $a \neq 0$ and $b \neq 0$

Identities in Real Number Operations

Additive Identity

• The additive identity is the number that, when added to any real number, results in the original number
• The additive identity is 0
• For any real number $a$, $a + 0 = a$
• Example: $7 + 0 = 7$

Multiplicative Identity

• The multiplicative identity is the number that, when multiplied by any real number, results in the original number
• The multiplicative identity is 1
• For any real number $a$, $a \times 1 = a$
• Example: $3 \times 1 = 3$

Arithmetic Operations on Real Numbers

Types of Real Numbers

• Real numbers include rational numbers (numbers that can be expressed as the ratio of two integers) and irrational numbers (numbers that cannot be expressed as the ratio of two integers)
• Rational numbers include integers (positive and negative whole numbers and zero), fractions, and terminating or repeating decimals
• Examples of rational numbers: -3, $\frac{2}{5}$, 0.75
• Irrational numbers include non-terminating, non-repeating decimals and numbers that cannot be expressed as decimals, such as square roots of non-perfect squares and pi ($\pi$)
• Examples of irrational numbers: $\sqrt{2}$, $\pi$

Performing Operations on Real Numbers

• Addition, subtraction, multiplication, and division can be performed on any two real numbers, with the exception of division by zero, which is undefined
• When adding or subtracting rational and irrational numbers, the result is always an irrational number
• Example: $2 + \sqrt{3}$ is irrational
• When multiplying rational and irrational numbers, the result is rational if the irrational factor has an even exponent; otherwise, the result is irrational
• Example: $2\sqrt{3}$ is irrational, but $(\sqrt{3})^2 = 3$ is rational
• When dividing two real numbers, the result is rational if both numbers are rational or if the irrational factors cancel out; otherwise, the result is irrational
• Example: $\frac{3}{\sqrt{2}}$ is irrational, but $\frac{\sqrt{8}}{\sqrt{2}} = 2$ is rational