1.1 Properties of Real Numbers and Algebraic Operations

Properties of Real Numbers

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.

The commutative, associative, and distributive properties let you rearrange and simplify expressions. Inverse operations and identities let you undo operations and maintain equality. These concepts form the foundation for everything else in this course.

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Properties of Real Numbers

Commutative, Associative, and Distributive Properties

The commutative property says that order doesn't matter for addition and multiplication:

$a + b = b + a$ $a \times b = b \times a$

Note that subtraction and division are not commutative. $5 - 3 \neq 3 - 5$, and $6 \div 2 \neq 2 \div 6$.

The associative property says that grouping doesn't matter for addition and multiplication:

$(a + b) + c = a + (b + c)$ $(a \times b) \times c = a \times (b \times c)$

Again, this does not apply to subtraction or division. $(10 - 5) - 2 \neq 10 - (5 - 2)$.

The distributive property connects multiplication and addition. Multiplying a sum by a factor is the same as multiplying each term separately, then adding:

$a(b + c) = ab + ac$

This also works with subtraction: $a(b - c) = ab - ac$.

Together, these properties let you:

• Rearrange terms to group like terms (commutative)
• Regroup terms for easier computation (associative)
• Expand or factor expressions (distributive)

Applying Properties to Simplify Expressions and Solve Equations

Here's how each property looks in practice:

• Commutative: $3 + x + 2 = x + 3 + 2 = x + 5$
• Associative: $(2 + 3) + x = 2 + (3 + x) = 5 + x$
• Distributive: $2(3x + 4) = 6x + 8$

When solving equations, you'll often use several properties together. For example, to solve $2x + 3 = 11$:

1. Subtract 3 from both sides: $2x = 8$
2. Divide both sides by 2: $x = 4$

Each step relies on performing the same operation on both sides to maintain equality, which connects directly to inverse operations below.

Inverse Operations for Real Numbers

Additive Inverses

The additive inverse of a number is its opposite. When you add a number to its additive inverse, you get zero (the additive identity).

For any real number $a$, its additive inverse is $-a$.

• The additive inverse of 5 is $-5$, because $5 + (-5) = 0$
• The additive inverse of $-3$ is $3$, because $-3 + 3 = 0$

This is exactly what you're doing when you "subtract from both sides" of an equation.

Multiplicative Inverses (Reciprocals)

The multiplicative inverse (or reciprocal) of a number is the value that, when multiplied by the original, gives 1 (the multiplicative identity).

For any nonzero real number $a$, its multiplicative inverse is $\frac{1}{a}$.

• The multiplicative inverse of 4 is $\frac{1}{4}$, because $4 \times \frac{1}{4} = 1$
• The multiplicative inverse of $\frac{2}{3}$ is $\frac{3}{2}$, because $\frac{2}{3} \times \frac{3}{2} = 1$

More generally, the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$, as long as both $a \neq 0$ and $b \neq 0$.

Zero has no multiplicative inverse because no number multiplied by 0 gives 1. This is closely related to why division by zero is undefined.

Identities in Real Number Operations

Additive Identity

The additive identity is 0. Adding 0 to any number leaves it unchanged:

$a + 0 = a$

This might seem obvious, but it matters when you're justifying algebraic steps. For instance, "adding zero in a useful form" (like $+3 - 3$) is a technique you'll use when completing the square later in the course.

Multiplicative Identity

The multiplicative identity is 1. Multiplying any number by 1 leaves it unchanged:

$a \times 1 = a$

Similarly, multiplying by "1 in a useful form" (like $\frac{3}{3}$) is how you find common denominators or rationalize denominators without changing the value of an expression.

Arithmetic Operations on Real Numbers

Types of Real Numbers

The real number system has two main categories:

• Rational numbers can be expressed as the ratio of two integers $\frac{p}{q}$ where $q \neq 0$. Their decimal forms either terminate or repeat.
  • Examples: $-3$, $\frac{2}{5}$, $0.75$, $0.\overline{3}$
• Irrational numbers cannot be expressed as such a ratio. Their decimal forms go on forever without repeating.
  • Examples: $\sqrt{2}$, $\pi$, $\sqrt[3]{5}$

Together, rational and irrational numbers make up the entire set of real numbers. Every point on the number line corresponds to exactly one real number.

Performing Operations on Real Numbers

You can add, subtract, multiply, and divide any two real numbers, with one exception: division by zero is undefined.

The trickier part is knowing whether your result is rational or irrational. Here are the key rules:

• Rational $+$ or $-$ irrational = irrational. For example, $2 + \sqrt{3}$ is irrational. You can't add a rational number to an irrational one and get a "clean" result.
• Nonzero rational $\times$ irrational = irrational. For example, $2\sqrt{3}$ is irrational. However, multiplying by 0 gives 0, which is rational.
• Irrational $\times$ irrational can go either way. $\sqrt{3} \times \sqrt{3} = 3$ is rational, but $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ is irrational. The result depends on whether the irrational parts "cancel out."
• Division follows similar logic. $\frac{3}{\sqrt{2}}$ is irrational, but $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{4} = 2$ is rational because the irrational factors simplify.