For a non zero number, what is the product of the number and its multiplicative inverse (reciprocal)?

Introduction / Context:
This conceptual question checks your understanding of the idea of a multiplicative inverse, also known as a reciprocal. The multiplicative inverse of a non zero number is defined so that when you multiply the number by its inverse, the result is the multiplicative identity. Understanding this idea is important in algebra, fractions, and many areas of higher mathematics, because it underlies division and solving equations.

Given Data / Assumptions:

• We are working with a non zero real number, call it a.
• The multiplicative inverse of a is another number that we can denote as 1/a.
• The question asks for the product a * (1/a).
• Zero is excluded because its multiplicative inverse does not exist.

Concept / Approach:
By definition, the multiplicative inverse of a non zero number a is the number that, when multiplied by a, gives the multiplicative identity, which is 1 in standard arithmetic. For any non zero real number a, its reciprocal is 1/a. This definition is built into how we understand division: dividing by a is the same as multiplying by 1/a. Therefore, when you multiply a by 1/a, the a in the numerator and denominator cancel, leaving 1. This is independent of whether a is positive or negative, as long as it is not zero.

Step-by-Step Solution:
Let a be a non zero number. Its multiplicative inverse is 1/a. Compute the product: a * (1/a). Since a is non zero, the factor a in the numerator cancels with a in the denominator. So a * (1/a) = 1. Therefore, the product of a non zero number and its multiplicative inverse is always exactly 1.

Verification / Alternative check:
Test with a few concrete numbers. If a = 5, then the multiplicative inverse is 1/5, and 5 * (1/5) = 1. If a = -3, the inverse is -1/3, and (-3) * (-1/3) = 1 as well. Even for fractions, such as a = 2/7, its inverse is 7/2, and (2/7) * (7/2) = 1. In every non zero case the product is 1, confirming the general rule.

Why Other Options Are Wrong:

• 0: This would be the result if one factor were zero, but the definition excludes zero because it has no multiplicative inverse.
• -1: This would require the inverse to be the negative of 1/a, which is not the definition of multiplicative inverse.
• Infinity: This is not a meaningful result in standard real number arithmetic for this operation.
• Cannot be determined: In fact, it is completely determined by the definition of multiplicative inverse.

Common Pitfalls:
Some learners confuse multiplicative inverse with additive inverse. The additive inverse of a is the number that sums with a to give 0, namely −a. The multiplicative inverse instead relates to multiplication and gives a product of 1. Keeping these two ideas separate is essential: additive inverse leads to zero, multiplicative inverse leads to one. Remembering that division by a non zero number is multiplication by its reciprocal also reinforces this concept.

Final Answer:
For any non zero number, the product of the number and its multiplicative inverse is 1.