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Chapter 1: Problem 88

If the sum of two integers is \(0,\) what can we conclude about the two integers? Why?

Short Answer

Expert verified
The two integers must be opposites of each other.

Step by step solution

01

Understand the Given Information

The problem states that the sum of two integers is zero. Let's denote these two integers as \(a\) and \(b\). Therefore, we have the equation: \(a + b = 0\).
02

Isolate One of the Integers

To gain more insight, let's isolate one of the integers. We can rewrite the equation as \(a = -b\). This means that integer \(a\) is the negative of integer \(b\).
03

Interpret the Equation

Since \(a = -b\), it implies that each integer has the same magnitude but opposite signs. For example, if \(b\) is \(5\), then \(a\) must be \(-5\). Conversely, if \(b\) is \(-3\), then \(a\) must be \(3\).
04

Conclusion

The two integers must be additive inverses of each other. This means for any integer \(a\), its additive inverse is \(-a\). Hence, if two integers sum to zero, they must be opposites (one positive and one negative).

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Key Concepts

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

Introduction to Integers
Integers are numbers that include both positive and negative whole numbers, as well as zero. These numbers are integral to understanding basic mathematical concepts.

In the number line, integers are represented by both positive numbers to the right of zero and negative numbers to the left of zero.

Some examples include:
  • Positive integers: 1, 2, 3, 4, ...
  • Negative integers: -1, -2, -3, -4, ...
  • Zero is considered an integer but is neither positive nor negative.
Understanding integers is fundamental for solving problems that involve adding, subtracting, and finding relationships between numbers.
Additive Inverses
An additive inverse of an integer is another integer that, when added to the original, results in a sum of zero. This concept is central to problems involving the sum of integers equating to zero.

For any integer 'a', its additive inverse is '-a'. In other words, a + (-a) = 0.

Some examples include:
  • The additive inverse of 7 is -7; therefore, 7 + (-7) = 0.
  • The additive inverse of -3 is 3; hence, -3 + 3 = 0.
Knowing how to identify additive inverses helps in solving equations like the one discussed where the sum is zero, ensuring that the numbers must be opposites.
Opposite Signs
When two numbers have opposite signs, one is positive, and the other is negative. This relationship is key to understanding why their sum can be zero.

If two integers, denoted as 'a' and 'b', have opposite signs and equal magnitudes, their sum will always be zero鈥攂ecause each positive unit is canceled out by a corresponding negative unit.

For example, consider these pairs:
  • (5 and -5): 5 + (-5) = 0
  • (-8 and 8): -8 + 8 = 0
As seen here, both pairs have opposite signs and equal absolute values, demonstrating that their sum will be zero in each case. This understanding is crucial for solving many integer-related problems.

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