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Chapter 5: Problem 137

If you are given two integers, explain how to determine which one is smaller.

Short Answer

Expert verified
Directly compare the two integers. The one with the lower value is the smaller integer, unless they are equal, in which case neither is smaller.

Step by step solution

01

Understand the problem

Getting a grasp on the underlying problem is the first step. We are given two integers and our task is to discern which one of them is smaller. That means comparing the two numbers to each other.
02

Compare the integers

To determine which of the two integers is smaller, we simply compare them. This can be done directly because integers have inherent order due to their definition. If the first integer is less than the second, it is the smaller integer. Otherwise, the second integer is smaller.
03

Consider equal integers

If the two given integers are equal, then neither integer is smaller than the other. In this case, we note that the integers are equal.

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Key Concepts

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

Comparing Integers
When we talk about comparing integers, we are essentially looking at which number is bigger, or smaller, or if they are the same. This might sound simple, but understanding how to do it systematically is very important. Comparing integers involves checking their value against each other. You will see that you can do this by looking at the number line.
  • If one integer lies to the left of another on the number line, it is smaller.
  • If it lies to the right, it is larger.
  • If both numbers are at the same point, they are equal.
Knowing the order of numbers helps simplify this task because it allows you to see which is bigger or smaller immediately by their position relative to each other.
Number Order
Number order is fundamental to understanding how numbers relate to each other. This concept states that each number has a specific place in the sequence of integers. The idea is simple:

- Numbers increase in value as they move from the left to the right on the number line.
- All integers have an inherent order. For example, -3 is always less than 1 because it comes earlier on the number line.

By understanding number order, you can quickly tell which number is smaller and which is larger, facilitating the process of comparing integers effectively.
Equal Integers
Sometimes when comparing integers, they are found to be equal. This happens when both numbers are the same. It's like having two cookies with the same number of chocolate chips.

When integers are equal, they sit at exactly the same point on the number line. This means they have the same value.

To identify equal integers, simply compare them:
  • If both have the same numerical value, they are equal.
  • No integer is greater or lesser; they are exactly the same.
Recognizing equal integers is important, especially in mathematical problems where distinction between values determines the solution.
Mathematical Reasoning
Mathematical reasoning is the logical thought process that helps you solve math problems. It's like a roadmap that guides you through solving questions step-by-step.

When comparing integers, mathematical reasoning helps you decide which integer is smaller. You think through the problem:
  • Understand what the problem is asking (which number is smaller).
  • Observe and compare both integers.
  • Apply logical steps to determine if one is lesser, greater, or if they are equal.
By developing strong mathematical reasoning skills, you not only solve the problem at hand, but you also enhance your ability to tackle various math challenges in the future.

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Most popular questions from this chapter

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=40,000, r=0.1\).

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=3\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{4}, r=2\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=-2, r=-3\).
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