Problem 58 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises 47-76, perform the indicated division or state that the expression is undefined. $$ -80 \div 8 $$

Step-by-Step Solution

Answer

The result of the division is -10.

Step 1: Identify the division operation

The given expression \( -80 \div 8 \) represents a division operation where -80 is being divided by 8.

Step 2: Perform the division operation

Divide -80 by 8. Remember that division of a negative number by a positive number will result in a negative answer. So, \( -80 \div 8 = -10 \).

Key Concepts

Negative Numbers

When we talk about negative numbers, we refer to numbers that are less than zero. These numbers are represented with a minus sign (-).
Negative numbers are often used in real life scenarios such as temperatures below freezing, depths below sea level, or financial debts. Why is it important to understand negative numbers in division? Because knowing how they behave helps in correctly solving problems involving them.

• Any number with a minus sign is negative.
• Negative numbers are used to represent opposite values, such as loss or removal.

In mathematical operations, especially in division, multiplying or dividing a negative by a positive number results in a negative number, which may sometimes seem unintuitive at first.

Positive Numbers

Positive numbers are the numbers greater than zero. They are represented without any sign or specifically with a plus sign (+). For example, 1, 2, 3, and so forth are all positive numbers.
Positive numbers are used commonly and represent values such as gains or accumulations. Understanding positive numbers is the first step in learning basic arithmetic.

• Positive numbers are used in most daily counting.
• In division, positive numbers play a critical role in determining the sign of the result.

When dividing a positive number by another positive number, the resulting value is positive. This rule is consistent across mathematics and ensures uniform results.

Division Operation

Division is a basic arithmetic operation used to determine how many times one number is contained within another. It can also be thought of as distributing a number into equal parts. In division, the number being divided is called the dividend, the number by which we divide is called the divisor, and the result is called the quotient.

• The division operation is denoted by symbols like / or \(\div\).
• The division of numbers can yield either a whole number or a fraction.

When dividing numbers with different signs, such as a negative dividend and a positive divisor, the quotient will be negative, as seen with \(-80 \div 8\), resulting in \(-10\). However, when both numbers are negative, the result becomes positive.

Undefined Expressions

In mathematics, an undefined expression occurs when you try to divide by zero. This is because division by zero does not result in a finite number. Mathematically, it is considered undefined, and there's no meaningful way to interpret the result.
Division by zero is to be avoided because it doesn’t provide valid mathematical outcomes. Here are a few points to consider:

• Any number divided by zero is undefined.
• This concept is crucial to ensure mathematical reasoning remains consistent.

For example, in the expression \( x \div 0 \), no matter what \(x\) is, the result is undefined. Always check your divisors!Understanding when expressions are undefined is a key part of successfully navigating algebra and overall mathematical reasoning.