In basic arithmetic, what is the product of a nonzero number and its reciprocal?

Introduction / Context:
This question tests understanding of the concept of a reciprocal, also called the multiplicative inverse, and how it behaves under multiplication. Knowing what happens when a number is multiplied by its reciprocal is fundamental to algebra, fractions and many higher level topics. The focus is on nonzero numbers, since zero does not have a reciprocal.

Given Data / Assumptions:

• We are considering a nonzero real number, which we can call a.
• The reciprocal or multiplicative inverse of a is defined as 1 divided by a.
• The question asks for the product a * (1 / a).
• Zero is excluded, because division by zero is undefined and zero has no reciprocal.
• Standard rules of multiplication and division of real numbers apply.

Concept / Approach:
The reciprocal of a nonzero number a is the number that, when multiplied by a, gives 1. This is the defining property of the multiplicative inverse. In symbolic form, if a is not zero, then a * (1 / a) = 1. So the correct choice must reflect this fundamental identity. The approach here is to apply this definition directly rather than trying to plug in specific numerical examples only.

Step-by-Step Solution:
Step 1: Let a be an arbitrary nonzero real number.Step 2: By definition, the reciprocal or multiplicative inverse of a is 1 / a.Step 3: Consider the product of a and its reciprocal: a * (1 / a).Step 4: Use the rule of multiplication and division: a * (1 / a) can be viewed as a divided by a.Step 5: For any nonzero number a, a / a = 1, because a is exactly one times itself.Step 6: Therefore the product of a nonzero number and its reciprocal is always equal to 1.

Verification / Alternative check:
We can confirm this by trying some concrete examples. If the number is 2, its reciprocal is 1 / 2, and 2 * (1 / 2) = 1. If the number is 5, its reciprocal is 1 / 5, and 5 * (1 / 5) = 1. If the number is -3, the reciprocal is 1 / -3, and -3 * (1 / -3) = 1 as well. These examples from both positive and negative numbers show that the result is consistently 1 whenever the number is nonzero.

Why Other Options Are Wrong:
Option a claims the product is 0, which would be true only if one of the factors were zero, but that contradicts the condition that the number is nonzero. Option c suggests that the product is the negative of the number, which does not match the definition of a reciprocal and fails in any simple test, since 2 * (1 / 2) is not equal to -2. Option d suggests that the product equals the number itself, which is also not correct; for example, 3 * (1 / 3) equals 1, not 3. Only 1 is consistent with the definition for every nonzero number.

Common Pitfalls:
Some learners confuse additive inverse (which gives a sum of 0) with multiplicative inverse (which gives a product of 1). Another error is to forget that the reciprocal is always 1 divided by the number, not the number itself or its negative. Clear separation of these concepts is very important for success in algebra.

Final Answer:
The product of a nonzero number and its reciprocal is always 1.