State whether the following statement is true or false.

The smallest integer is zero.

(a) True

(b) False

The statement "The smallest integer is zero" is false. The correct answer is (b) False. To understand why, let's delve into the concept of integers and their properties.

Understanding Integers

Integers are a set of numbers that include all whole numbers, both positive and negative, as well as zero. This set can be represented as follows:

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero: 0

Identifying the Smallest Integer

When we talk about the smallest integer, we need to consider the entire set of integers. The integers extend infinitely in both the positive and negative directions. Therefore, while zero is indeed an integer, it is not the smallest one. In fact, there are infinitely many negative integers that are smaller than zero, such as -1, -2, -3, and so on.

Visualizing the Number Line

To visualize this, think of a number line:

• On the right side, you have positive integers increasing towards infinity.
• On the left side, negative integers decrease towards negative infinity.
• Zero sits in the middle, separating positive and negative integers.

This arrangement clearly shows that there is no "smallest" integer because for any integer you choose, you can always find a smaller one by subtracting one. For example, if you take -100, the next smaller integer would be -101.

Conclusion

In summary, the statement claiming that zero is the smallest integer is incorrect. The set of integers includes infinitely many negative numbers, making it impossible to identify a smallest integer. Thus, the answer is (b) False.