Problem 28 - Algebra for College Students

Question

Perform the following operations with real numbers. $$0 \div(-11)$$

Step-by-Step Solution

Answer

0

Step 1: Identify the Operation

Look at the expression provided, which is a division operation: \(0 \div (-11)\). The goal is to divide zero by negative eleven.

Step 2: Recall the Property of Division

Recall the property that dividing zero by any non-zero number results in zero. Mathematically, this can be expressed as \(0 \div a = 0\) where \(a\) is any non-zero real number.

Step 3: Apply the Property of Division

Apply the property by substituting \(a = -11\) into the expression. Since \(-11\) is a non-zero real number, the result of the division \(0 \div (-11)\) is \(0\).

Step 4: Conclusion

Based on the previously stated property, the solution to \(0 \div (-11)\) is zero.

Key Concepts

Division

Division is one of the four basic operations in mathematics. It is essentially the process of finding out how many times one number, called the divisor, is contained within another number, called the dividend. Think of division as the opposite of multiplication.

For example, if we have 15 divided by 3, we are essentially asking: "How many 3s fit perfectly into 15?" The answer is 5.

When dividing, it’s important to remember that the divisor cannot be zero. Dividing by zero is undefined in mathematics because it doesn't produce a meaningful or valid result. However, if the dividend is zero, like in our exercise, dividing zero by any non-zero number will simply result in zero. This is a special property of division that highlights the uniqueness of zero in arithmetic operations.

Property of Zero

Zero holds a unique place in mathematics with several interesting properties. One key property of zero is its behavior in division. When zero is divided by any non-zero number, the result is always zero.

This property is rooted in the idea that zero, being the total absence of value, can be split into any number of parts without altering its value. Mathematically, it is expressed as:

• \( 0 \div a = 0 \)
• where \( a \) is any non-zero real number.

This rule helps to simplify expressions and solve equations where zero appears as a term. In our exercise, dividing zero by negative eleven is straightforward thanks to this property: the result remains zero.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-) in front of them, like -1, -2, or -11 as in our example. They are used to represent quantities that go below zero, such as temperatures below freezing, or debts.

In division, the sign of the dividend and the divisor affects the sign of the result. When dividing -11 into zero, the negative sign of -11 does not influence the result because the dividend is zero. This reinforces the property of division we discussed earlier: zero divided by any non-zero number, positive or negative, remains zero.

Understanding how negative numbers interact with division is crucial, especially when dealing with more complex mathematical or real-world problems.