Problem 59
Question
In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$120 \div(-10)$$
Step-by-Step Solution
Verified Answer
Therefore, the result of the expression \(120 \div(-10)\) is -12.
1Step 1: Division Among Positive and Negative Numbers
Since -10 is a non-zero number, the operation is defined. The rule while dividing integers is if the signs of both the integers are same, the quotient is positive and if the signs of the integers are different, the quotient is negative.
2Step 2: Performing Division
So, 120 divided by -10 equals to -12. Because 120 is a positive number and -10 is a negative number, the quotient is negative.
Key Concepts
Division of Positive and Negative NumbersUndefined Mathematical ExpressionsInteger Arithmetic
Division of Positive and Negative Numbers
Understanding how to divide integers, especially when dealing with positive and negative numbers, forms the foundation of integer arithmetic. Simple rules make this process quite straightforward.
When dividing two integers with different signs—one positive and one negative—the result is always a negative number. This is because the integers are considered to be on opposite sides of the number line, and dividing them is like measuring the distance between two points in opposite directions.
To apply this in practice, consider the following example: \(120 \text{ divided by } -10\). Since 120 is positive and -10 is negative, according to the rule, the quotient will be negative. Thus, \(120 \text{ divided by } -10 = -12\). This tells us that if \(-10\) were added together 12 times, we would reach \(-120\), which is the negation of \(120\).
When dividing two integers with different signs—one positive and one negative—the result is always a negative number. This is because the integers are considered to be on opposite sides of the number line, and dividing them is like measuring the distance between two points in opposite directions.
To apply this in practice, consider the following example: \(120 \text{ divided by } -10\). Since 120 is positive and -10 is negative, according to the rule, the quotient will be negative. Thus, \(120 \text{ divided by } -10 = -12\). This tells us that if \(-10\) were added together 12 times, we would reach \(-120\), which is the negation of \(120\).
Undefined Mathematical Expressions
In the grand scheme of mathematical expressions, there are situations where a certain operation cannot be performed due to restrictions inherent to the mathematics itself. One classic example of an undefined expression is division by zero.
Why is division by zero undefined? Consider what division means: it is essentially the operation of determining how many times a number (the divisor) can be subtracted from another number (the dividend). But if the divisor is zero, you cannot subtract zero from the dividend an infinite number of times; the operation is senseless and has no meaningful answer in the context of arithmetic.
In the exercise at hand, however, the divisor is \(-10\), which is not zero. Therefore, the division is defined. It's crucial for students to remember this rule: any number divided by zero is undefined, but zero divided by any non-zero number is simply zero.
Why is division by zero undefined? Consider what division means: it is essentially the operation of determining how many times a number (the divisor) can be subtracted from another number (the dividend). But if the divisor is zero, you cannot subtract zero from the dividend an infinite number of times; the operation is senseless and has no meaningful answer in the context of arithmetic.
In the exercise at hand, however, the divisor is \(-10\), which is not zero. Therefore, the division is defined. It's crucial for students to remember this rule: any number divided by zero is undefined, but zero divided by any non-zero number is simply zero.
Integer Arithmetic
Integer arithmetic encompasses the basic operations—addition, subtraction, multiplication, and division—performed with whole numbers which include positive numbers, negative numbers, and zero. Each of these operations follows specific rules that maintain consistency within the number system.
In the context of division, we've seen how the quotient behaves when we divide a positive number by a negative number. Similarly, when both numbers are positive or both are negative, the result will always be positive. The signs of the numbers dictate the sign of the result in both multiplication and division.
For instance, when dividing two negative integers, say \(-120 \text{ divided by } -10\), the result is positive \(12\) because the two negative signs cancel each other out. Integer arithmetic is all about these simple consistent rules, helping you predict the outcome of different operations easily.
In the context of division, we've seen how the quotient behaves when we divide a positive number by a negative number. Similarly, when both numbers are positive or both are negative, the result will always be positive. The signs of the numbers dictate the sign of the result in both multiplication and division.
For instance, when dividing two negative integers, say \(-120 \text{ divided by } -10\), the result is positive \(12\) because the two negative signs cancel each other out. Integer arithmetic is all about these simple consistent rules, helping you predict the outcome of different operations easily.
Other exercises in this chapter
Problem 58
Determine whether the given number is a solution of the equation. $$3(w+2)=4(w-3) ; 10$$
View solution Problem 58
Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$3 \div \frac{12}{7}$$
View solution Problem 59
Simplify each algebraic expression. $$8(4 y+3)+(-35 y)$$
View solution Problem 59
Use the order of operations to simplify each expression. $$4|10-(8-20)|$$
View solution