Problem 49 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$\frac{-21}{3}$$

Step-by-Step Solution

Answer

-7

Step 1: Identifying the Operation and Numbers Involved

The operation to be done is division. The numbers to be divided here are -21 and 3.

Step 2: Performing the Division

Now proceed to do the division. In this case, \(-21\) divided by 3 equals \(-7\).

Key Concepts

Division

Division is one of the basic arithmetic operations, which involves splitting a number into equal parts. This operation helps in distributing an amount or quantity evenly across a specified number of parts. In the exercise you came across, you needed to divide \[ -21 \] by \[ 3 \], which means you were tasked to find out how many times \[ 3 \] fits into \[ -21 \].

To perform division:

• Identify the dividend (the number to be divided). Here, it is \[ -21 \].
• Identify the divisor (the number that divides the dividend). Here, it is \[ 3 \].
• Calculate the quotient by determining how many times the divisor can fit into the dividend. In this example, \[ 3 \] can fit exactly \[ -7 \] times into \[ -21 \].

Remember, division can sometimes lead to decimals or fractions when the divisor doesn’t evenly divide the dividend. However, for this exercise, the division is clean with an integer result of \[ -7 \].

Negative Numbers

Understanding negative numbers is crucial when dealing with division operations like the one in the exercise. Negative numbers have values less than zero and are used to represent deficits, temperatures below zero, and similar scenarios.

When dividing negative numbers, the sign rules are:

• If both numbers (dividend and divisor) have the same sign, the quotient is positive.
• If the numbers have different signs, the quotient is negative.

In this case, since \[ -21 \] is negative and \[ 3 \] is positive, the quotient is negative. This happens because the numbers have different signs, resulting in \[ -7 \] as the solution.

Negative numbers can sometimes be tricky, but keeping these sign rules in mind can significantly ease your calculations.

Integer Mathematics

In the world of math, integer mathematics deals with whole numbers and their operations. Integers include positive numbers, negative numbers, and zero. It is important to note that integers do not include fractions or decimals. In the exercise, you dealt with the integer division of \[ -21 \] by \[ 3 \], resulting in another integer, \[ -7 \].

Key notes about integer mathematics:

• Operations like addition, subtraction, multiplication, and division often remain within integer boundaries unless you divide and get a non-integer result.
• Integer division is direct when both the dividend and divisor are integers that divide cleanly, like in this case.
• When division doesn't result in an integer, it usually involves a decimal or implies the presence of a remainder in integer mathematics.

With integer division, the focus is on determining whole-number results and understanding how negative operations affect sign and magnitude. Remember, integer arithmetic is a bedrock for working with more complex numbers and mathematical concepts.