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Chapter 3: Problem 12

\((-1)(-1)(-1)(-1) \div(-1)=\) ?

Short Answer

Expert verified
The result is \(-1\).

Step by step solution

01

Simplify Multiplication of Negatives

Multiply the four negative ones together. We start by noticing that the multiplication of two negative numbers is positive: \((-1) \times (-1) = 1\).Applying the same multiplication to the other pair:\((-1) \times (-1) = 1\).Now, multiply the results:\(1 \times 1 = 1\).
02

Divide by Negative One

Take the result from Step 1, which is \(1\), and divide by \(-1\):\[1 \div (-1) = -1\]Dividing a positive number by a negative number gives a negative result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication of Negatives
When multiplying negative numbers, there are some rules that can help you understand the result.
  • If you multiply two negative numbers together, the result is positive: \[(-a) \times (-b) = ab\]
  • This might seem counterintuitive at first but think of it as a double-negative situation. Just like negating a negative statement in English turns it positive, the same applies to numbers.
  • In the original problem, we multiplied four \(-1\)s together by pairing them: \((-1) \times (-1)\) and another \((-1) \times (-1)\), giving us two positive ones: \(1 \times 1 = 1\).
Breaking the problem down into parts makes it much more manageable and helps avoid mistakes.
Division of Integers
Dividing integers follows similar rules as multiplication, with an added layer of complexity when dealing with negative numbers.
  • When you divide a positive number by a negative number, the quotient is negative: \[a \div (-b) = -c\]
  • Conversely, if both the dividend and divisor are negative, the quotient is positive: \[(-a) \div (-b) = c\]
In the problem given, we had to divide the result of the multiplication, \(1\), by \(-1\). Since the multiplication result was positive, dividing by a negative switched it to negative, resulting in \(-1\).
Always remember that the sign of the quotient depends on the signs of the numbers you are dividing.
Negative Numbers
Negative numbers are those less than zero, and they have unique properties when used in arithmetic operations.
  • They are often represented by a minus sign \(-\) before the number, such as \(-3\).
  • Adding a negative is like subtracting a positive: \[a + (-b) = a - b\]
  • Subtracting a negative is like adding a positive: \[a - (-b) = a + b\]
  • Understanding this helps simplify complicated expressions, especially when negative numbers interact in multiplication and division.
Grasping the concept of negative numbers and mastering these basic operations give you the foundation needed to tackle more complex math problems.

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Most popular questions from this chapter

Convert the following to percents: \(\begin{array}{lll}\text { (a) } .75 & \text { (b) } .7532 & \text { (c) } 1.5 & \text { (d) } \frac{2}{3} & \text { (e) } 1-.77\end{array}\)

\(20 \%\) of \(100=\) ?

\(3 \cdot 4=?\)
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