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Chapter 2: Problem 114

Answer true or false. Zero is always less than a positive number.

Short Answer

Expert verified
True, zero is always less than a positive number.

Step by step solution

01

Understanding Zero

Zero is a unique number and it serves as a neutral element in addition. It is neither positive nor negative.
02

Understanding Positive Numbers

Positive numbers are all numbers greater than zero. Examples include 1, 2, 3, and so on. Mathematically, any number satisfying the inequality \( x > 0 \) is a positive number.
03

Comparing Zero to Positive Numbers

When comparing zero to a positive number, zero is always less than any positive number because by definition, positive numbers are greater than zero.
04

Conclusion

Since zero is always less than any positive number, the statement "Zero is always less than a positive number" is true.

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Key Concepts

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

Zero in Mathematics
Zero is quite a fascinating number that holds a unique place in mathematics. It acts as the neutral, or identity element, particularly in addition. This means that any number plus zero will result in the original number being unchanged. For example, adding zero to five gives you five:
  • 5 + 0 = 5
  • Zero has no impact on addition, making it neutral.
Beyond addition, zero is neither positive nor negative. It sits right in the middle of the number line as a dividing point, distinguishing negative numbers from positive numbers. Being aware of this neutral property is critical when comparing zero with other numbers.
Inequalities
Inequalities refer to mathematical expressions that show the relationship between two values in terms of their sizes. They clarify whether numbers are greater than, less than, or equal to each other. Some common inequality symbols include:
  • \( > \): greater than
  • \( < \): less than
  • \( \geq \): greater than or equal to
  • \( \leq \): less than or equal to
Understanding inequalities is essential when determining the value of zero compared to other numbers. For instance, when assessing the claim "Zero is always less than a positive number":- Positive numbers are defined as numbers greater than zero.- So, any positive number \( x \) satisfies \( x > 0 \).This inequality, \( 0 < x \) confirms that zero is indeed always less than a positive number.
Number Line Comparison
The number line is a visual tool that helps in understanding numerical relationships, including the position of zero and positive numbers. Picture a horizontal line with equally spaced marks at regular intervals. Zero is often located at the center.
  • On the number line, positive numbers extend to the right of zero.
  • Negative numbers stretch to the left of zero.
Looking at a number line makes it clear that positive numbers are further right than zero, reinforcing the notion that they are larger. This simple visual representation confirms why zero is always less than any positive number. Remember, the farther right a number is, the greater it is! This straightforward concept is crucial for anyone grappling with math and serves as a basis for understanding broader mathematical ideas.

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Most popular questions from this chapter

Insert \(<\) or \(>\) between each pair of integers to make a true statement. $$ -8 \quad 0 $$

Multiply. See Examples 5 through 7. $$ -2(3)(-7) $$

Subtract. $$ 3-8 $$

Find each quotient. See Examples 10 through 14. $$ \frac{0}{-21} $$

The temperature on one January day in Minneapolis was \(10^{\circ}\) below \(0^{\circ}\) Celsius. Represent this quantity by an integer and tell whether this temperature is cooler or warmer than \(5^{\circ}\) below \(0^{\circ}\) Celsius.
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