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Chapter 0: Problem 143

If you are given two real numbers, explain how to determine which is the lesser.

Short Answer

Expert verified
To determine which of two given real numbers is lesser, compare the numbers on a number line. The number which lies to the left will be the lesser number. We can verify this by subtracting the smaller number from the larger one and confirming the result is positive.

Step by step solution

01

Understand Numerical Comparison

Firstly, it is necessary to understand that in numerical comparison, a number is considered less than the other if it is further to the left on the number line.
02

Apply Concept on Real Numbers

If we get two real numbers, we keep them on number line in their respective positions. The number which lies on the left will be the lesser number.
03

Verification

To verify, we can subtract the smaller supposed number from the larger one, if the result is a positive number, we can confirm that the smaller supposed number is indeed lesser.

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Key Concepts

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

Number Line
The number line is an excellent visual tool to understand numerical comparison between two real numbers. It is a straight line where each point corresponds to a real number. The value of numbers increases as we move right, and decreases as we move left.
This makes it easy to see which numbers are smaller and which are larger.
  • For instance, on a number line, -3 will be to the left of 2 because it is smaller.
  • Every whole number, fraction, or decimal can find its place on the line.
By placing numbers on this line, one can quickly ascertain which one is lesser by observing their positions. Just remember: left is less!
This intuitive method helps clarify the relationship between numbers and can aid in mastering concepts such as ordering and magnitude.
Real Numbers
Real numbers encompass all the numbers on the number line, including integers, fractions, and irrational numbers.
They are defined to give meaningful results for various mathematical operations such as addition, subtraction, and comparison.
  • Real numbers include rational numbers like 3/4 and whole numbers, such as 5.
  • Irrational numbers like \(\sqrt{2}\) or \(\pi\) are also considered real numbers because they have specific places on the number line.
Being comfortable with real numbers allows you to use them effectively in numerical comparisons, solve equations, and execute all kinds of arithmetic operations.
Remember, these encompass every point along the continuous line. They are central to many branches of mathematics and are foundational for higher mathematics.
Subtraction in Verification
Subtraction is a simple mathematical procedure that helps verify which of two numbers is smaller when comparing them. Let's assume we suspect that number A is less than number B. By subtracting A from B, we can confirm this.
  • If \(B - A = C\) and \(C\) is positive, A is indeed less than B.
  • Positive results affirm the initial assumption about the smaller number.
This method of verification offers a numerical check to back up the visual representation provided by the number line. It's a straightforward process by which you can confirm your estimates using concrete calculations.
This aligns with basic arithmetic principles, reinforcing concepts such as borrowing and understanding positive and negative results in real-life numerical situations.

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Most popular questions from this chapter

Solve each equation. $$7-7 x=(3 x+2)(x-1)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You grouped the polynomial's terms using different groupings than I did, yet we both obtained the same factorization.

Explain why \(|x|<-4\) has no solution.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
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