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Chapter 2: Problem 98

Determine whether each statement is true or false. The sum of zero and a negative number is always a negative number.

Short Answer

Expert verified
True: the sum remains negative.

Step by step solution

01

Understanding the Problem Statement

We are asked to assess the statement: 'The sum of zero and a negative number is always a negative number.' We need to determine if this statement is generally true or false by examining the properties of addition involving zero and negative numbers.
02

Summing Zero and a Negative Number

Consider a negative number, represented as \( -a \), where \( a > 0 \). If we add zero to this negative number, the expression is \( 0 + (-a) \). According to the identity property of addition, adding zero to any number results in that number itself. Therefore, \( 0 + (-a) = -a \), which is negative.
03

Conclusion on the Sum

Since adding zero to a negative number, \( -a \), results in the negative number \( -a \) itself, the outcome remains negative. Therefore, the original statement that the sum of zero and any negative number is always a negative number is true.

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Key Concepts

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

Identity Property of Addition
The identity property of addition is a fundamental principle in mathematics. This property states that adding zero to any number leaves that number unchanged. It is like a mirror reflection in arithmetic鈥攚hatever you have will stay exactly the same if zero is involved in addition.
  • If you have a positive number and add zero, the result is still that positive number.
  • If you have a negative number and add zero to it, you'll still wind up with that negative number.
In formal terms, for any number \( a \), the equation is \( a + 0 = a \). This supports mathematical operations by ensuring that when operations involve zero, the numbers' values are not altered. This concept is particularly helpful in understanding that the sum of zero and any number鈥攑ositive or negative鈥攚ill always be the number itself.
Negative Numbers
Negative numbers are those less than zero and are represented with a minus sign. Imagine a thermometer with negative temperatures during winter鈥攖he numbers below zero are negative! These numbers can represent debts, temperatures, or any decrease.
Adding zero to a negative does not change its value. Take \( -5 \) as an example. When you add zero (\( -5 + 0 \)), the result remains \( -5 \). This property is important in mathematical operations, ensuring stability when zeroes come into play.
Understanding how to work with negative numbers includes remembering these key points:
  • Negative numbers are on the left of zero on a number line.
  • Adding two negative numbers will give another negative number.
  • Adding zero to a negative number doesn't change that negative number.
Mathematical Reasoning
Mathematical reasoning is the logical process of solving problems based on mathematical principles. It's like a detective solving a mystery bypiecing together clues!
When faced with a statement or problem, using mathematical reasoning involves:
  • Understanding the problem thoroughly by breaking it down into smaller, more manageable parts.
  • Applying known properties鈥攍ike the identity property of addition鈥攖o guide your thinking.
  • Checking the validity of statements by considering established facts in mathematics.
In the case of determining whether the sum of zero and a negative number is negative, we need to recall the identity property. By applying it, we ascertain that the statement is indeed true, as zero added to a negative number results in that number itself, preserving its negativity.
This logical sequence demonstrates how mathematical reasoning involves using rules and patterns to validate concepts and solve problems effectively.

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Most popular questions from this chapter

Write the given numbers in order from least to greatest. $$ |-1|,-|-6|,-(-6),-|1| $$

Insert \(<,>,\) or \(=\) between each pair of numbers to make a true statement. $$ |-24| \quad-(-24) $$

Insert \(<,>,\) or \(=\) between each pair of numbers to make a true statement. $$ 0 \quad-9 $$

Find the opposite of each integer. $$ 5 $$

Choose all numbers for \(x\) from each given list that make each statement true. \(|x|>8\) a. -9 b. -5 c. 8 d. -12
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