Chapter 1.3: Operations on Real Numbers

Vocabulary flashcards covering fundamental real number operations, properties of multiplication and division, exponents, square roots, order of operations, and algebraic expressions from Chapter 1.3.

Adding two numbers with the same sign

Add their absolute values and use the common sign as the sign of the sum.

Adding two numbers with different signs

Take the difference of their absolute values (smaller subtracted from larger) and use the sign of the larger absolute value as the sign of the sum.

Subtracting Real Numbers

If $a$ and $b$ are real numbers, then $a - b = a + (-b)$.

Multiplying or dividing two real numbers with the same sign

The result is a positive number.

Multiplying or dividing two real numbers with different signs

The result is a negative number.

Multiplication property of zero

If $b$ is a real number, $0 \cdot b = b \cdot 0 = 0$.

Reciprocals

Two numbers whose product is $1$.

Quotient of any real number and zero

The quotient of any real number and $0$ is undefined.

Quotient of zero and any real number

The quotient of $0$ and any real number $a \neq 0$ equals $0$.

Exponent ($a^n$)

If $a$ is a real number and $n$ is a natural number, then the $n\text{th}$ power of $a$, written as $a^n$, is the product of $n$ factors, each of which is $a$.

Base

In an exponential expression such as $3^4$, the base is the factor being repeatedly multiplied (in this expression, $3$).

Exponent (in exponential notation)

In an exponential expression such as $3^4$, the exponent is the natural number notation that indicates the number of repeating factors (in this expression, $4$).

Square Root

A number $b$ is a square root of a number $a$ if $b^2 = a$.

Principal Square Root

The positive square root of a number, noted as $\sqrt{a}$.

Negative Square Root

The negative square root of a number, noted as $-\sqrt{a}$.

Order of Operations

Rules for simplifying expressions: simplify within grouping symbols first (innermost set first) and fraction bars separately, then: 1. Evaluate exponential expressions, roots, or absolute values left to right; 2. Multiply or divide left to right; 3. Add or subtract left to right.

Value of an Expression

The result obtained when numbers are substituted for the variables in an algebraic expression and the operations are performed.

Evaluating an Expression

The entire process of substituting numbers for the variables in an algebraic expression and performing the operations.