1.5 Properties of Real Numbers

Real numbers form the foundation of algebra. They include all the numbers you'll work with in this course: whole numbers, integers, fractions, decimals, and irrational numbers like $\pi$ and $\sqrt{2}$. The properties below govern how these numbers behave under addition and multiplication, and they're the rules that justify every step you take when rearranging or simplifying an expression.

Properties of Real Numbers

Rearranging algebraic terms

The commutative and associative properties let you reorder and regroup terms. This flexibility is what makes it possible to rearrange expressions into simpler or more useful forms.

• Commutative property of addition
  • $a + b = b + a$
  • Order doesn't matter when adding: $3 + 5 = 5 + 3$
  • This is why you can rearrange terms like $x + 7$ into $7 + x$ without changing the value.
• Commutative property of multiplication
  • $a \times b = b \times a$
  • Order doesn't matter when multiplying: $2 \times 7 = 7 \times 2$
  • So $5 \cdot x$ and $x \cdot 5$ mean the same thing.
• Associative property of addition
  • $(a + b) + c = a + (b + c)$
  • Grouping doesn't matter when adding: $(2 + 3) + 4 = 2 + (3 + 4)$
  • You can regroup to make mental math easier. For instance, $17 + 8 + 2$ is simpler as $17 + (8 + 2) = 17 + 10 = 27$.
• Associative property of multiplication
  • $(a \times b) \times c = a \times (b \times c)$
  • Grouping doesn't matter when multiplying: $(2 \times 3) \times 4 = 2 \times (3 \times 4)$
  • Useful for rearranging to find friendlier products, like computing $25 \times 7 \times 4$ as $25 \times 4 \times 7 = 100 \times 7 = 700$.

A common point of confusion: subtraction and division are not commutative or associative. $5 - 3 \neq 3 - 5$, and $(12 \div 4) \div 2 \neq 12 \div (4 \div 2)$. These properties apply only to addition and multiplication.

Properties for equation solving

These properties define the special numbers (0 and 1) and the idea of "undoing" an operation. They come up constantly when you isolate a variable.

• Additive identity property
  • $a + 0 = a$
  • Zero is the identity element for addition because adding it leaves any number unchanged: $5 + 0 = 5$.
• Multiplicative identity property
  • $a \times 1 = a$
  • One is the identity element for multiplication because multiplying by it leaves any number unchanged: $7 \times 1 = 7$.
• Additive inverse property
  • $a + (-a) = 0$
  • Every number has an opposite that brings you back to zero: $3 + (-3) = 0$.
  • This is the property you use when you "subtract from both sides" of an equation. You're really adding the additive inverse.
• Multiplicative inverse property
  • $a \times \frac{1}{a} = 1$, where $a \neq 0$
  • Every nonzero number has a reciprocal that brings you back to one: $4 \times \frac{1}{4} = 1$.
  • This is the property behind "dividing both sides" of an equation. You're really multiplying by the multiplicative inverse.
• Zero property of multiplication
  • $a \times 0 = 0$
  • Any number times zero is zero: $6 \times 0 = 0$.
  • This property is especially important later when solving equations like $(x - 2)(x + 5) = 0$, because it tells you at least one of those factors must be zero.

Distributive property applications

The distributive property is the bridge between addition and multiplication. It's the single most-used property when simplifying algebraic expressions.

• The property itself
  • $a(b + c) = ab + ac$
  • Multiplying a number by a sum is the same as multiplying by each part and then adding: $2(3 + 4) = 2 \times 3 + 2 \times 4 = 6 + 8 = 14$.
  • It also works with subtraction: $a(b - c) = ab - ac$.
• Expanding expressions (distributing)
  1. Multiply the outside factor by each term inside the parentheses.
  2. Simplify each product.
  3. Combine like terms if possible.
  • Example: $3(2x + 5) = 3 \times 2x + 3 \times 5 = 6x + 15$
• Factoring expressions (the reverse of distributing)
  1. Identify the greatest common factor (GCF) of all terms.
  2. Write the GCF outside parentheses.
  3. Divide each original term by the GCF and write the results inside the parentheses.
  • Example: $6x + 15 = 3(2x + 5)$, since 3 is the GCF of 6x and 15.

Expanding and factoring are opposite operations. Distributing removes parentheses; factoring puts them back. Being comfortable going in both directions is key for equation solving throughout this course.

Fundamental concepts in algebra

• Number sets build on each other. Natural numbers ($1, 2, 3, \ldots$) sit inside the whole numbers ($0, 1, 2, \ldots$), which sit inside the integers ($\ldots, -2, -1, 0, 1, 2, \ldots$), which sit inside the rational numbers (any number expressible as $\frac{p}{q}$ where $q \neq 0$), which sit inside the real numbers (all points on the number line, including irrationals like $\sqrt{2}$).
• Equality ($=$) means two expressions represent the same value. Inequalities ($<, >, \leq, \geq$) describe how values compare when they're not equal.
• Axioms are the accepted starting rules of algebra. The properties listed above are axioms of the real number system. The closure property says that when you add or multiply any two real numbers, the result is always another real number. The set of real numbers is "closed" under addition and multiplication.