Algebra I - Property of Arithmetic Operations

• Mathematics •

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The **Properties of Arithmetic Operations** section in the PDF breaks down the rules of real numbers into three main categories: ### 1. Properties of Real Numbers Under Addition and Multiplication Assuming $a$, $b$, and $c$ are real numbers, real numbers satisfy 11 core properties: 5 for addition, 5 matching ones for multiplication, and 1 property that connects addition and multiplication. * **Commutative Property:** The order in which you add or multiply two numbers does not change the result. * *Addition:* $a + b = b + a$ * *Multiplication:* $a \cdot b = b \cdot a$ * **Associative Property:** How you group three numbers when adding or multiplying does not change the result. * *Addition:* $(a + b) + c = a + (b + c)$ * *Multiplication:* $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ * **Identities Exist:** There are special real numbers that leave other numbers unchanged when added or multiplied. * *Addition:* $0$ is a real number and acts as the **additive identity**, meaning $a + 0 = 0 + a = a$. * *Multiplication:* $1$ is a real number and acts as the **multiplicative identity**, meaning $a \cdot 1 = 1 \cdot a = a$. * **Inverses Exist:** Every real number has an opposite that brings it back to the identity element. * *Addition:* $-a$ is a real number, and $a + (-a) = (-a) + a = 0$. Also, $-(-a) = a$. * *Multiplication:* If $a \neq 0$, then $\frac{1}{a}$ is a real number, and $a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1$. Also, $\frac{1}{1/a} = a$. * **Closure:** Combining two real numbers through addition or multiplication always produces another real number. * *Addition:* $a + b$ is a real number. * *Multiplication:* $ab$ is a real number. * **Distributive Property (of addition over multiplication):** This single property connects addition and multiplication. Multiplying a number by a sum is the same as multiplying the number by each part of the sum and then adding the products: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$. **Derivative Properties Involving Zero:** * **Multiplication by zero:** Any real number multiplied by zero equals zero ($a \cdot 0 = 0 \cdot a = 0$). * **Zero product property:** If the product of two real numbers is zero ($ab = 0$), then either $a = 0$ or $b = 0$, or both. --- ### 2. Inequality Symbols When reading inequality signs, the sharp end always points toward the smaller number, and the open end points toward the larger number. * **$<$ (less than):** Example: $1 < 2$ and $4 < 56$. * **$>$ (greater than):** Example: $1 > 0$ and $56 > 4$. * **$\neq$ (not equal to):** Example: $0 \neq 3$ and $-1 \neq 1$. * **$\le$ (less than or equal to):** Example: $1 \le 1$ and $1 \le 2$. * **$\ge$ (greater than or equal to):** Example: $1 \ge 1$ and $3 \ge -29$. --- ### 3. Properties of Equality and Inequality Assuming $a$, $b$, and $c$ are real numbers: * **Trichotomy:** For any two real numbers $a$ and $b$, exactly one of the following statements is true: $a < b$, $a = b$, or $a > b$. * **Reflexive Property:** * *Equality:* Any number is equal to itself ($a = a$). * *Inequality:* Does not apply to strict inequalities ($<$ and $>$). * **Symmetric Property:** * *Equality:* If $a = b$, then $b = a$. * *Inequality:* Does not apply to inequalities. * **Transitive Property:** * *Equality:* If $a = b$ and $b = c$, then $a = c$. * *Inequality:* If $a < b$ and $b < c$, then $a < c$. * **Addition and Subtraction:** * *Equality:* Adding or subtracting the same number from both sides keeps them equal. If $a = b$, then $a + c = b + c$ and $a - c = b - c$. * *Inequality:* Adding or subtracting the same number from both sides keeps the inequality pointing the same way. If $a < b$, then $a + c < b + c$ and $a - c < b - c$. * **Multiplication and Division:** * *Equality:* If $a = b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$ (provided $c \neq 0$). * *Inequality (with a positive number):* Multiplying or dividing by a positive number keeps the inequality the same. If $a < b$ and $c > 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$. * *Inequality (with a negative number):* Multiplying or dividing by a negative number requires you to **switch the inequality sign**. If $a < b$ and $c < 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.