Question

Which of the following statements is false? (A) The reciprocal of any negative number is a negative number. (B) Dividing by a number is the same as multiplying by the reciprocal of the number. (C) The reciprocal of any positive number is a positive number. (D) The reciprocal of any number is greater than zero and less than 1

Short answer

The false statement among the given options is D: The reciprocal of any number is greater than zero and less than 1.

Step-by-step solution

Evaluate Statement A

The reciprocal of a number is defined as \(1\) over that number. By this definition, the reciprocal of any negative number would be \(-1\) over the number which remains a negative number. Hence, Statement A holds true.

Evaluate Statement B

The mathematical operation of division can also be expressed as multiplication by the reciprocal. Therefore, dividing by a number is the same as multiplying by the reciprocal of that number. Thus, Statement B is also true.

Evaluate Statement C

According to the definition of reciprocal, the reciprocal of any positive number is \(1\) over the number. As stated, the result of this operation is always positive. Therefore, Statement C is true.

Evaluate Statement D

The reciprocal of a number is not always a value that is greater than zero and less than one. For a number that is greater than one, its reciprocal will be less than one, but for a number less than one (but not zero), its reciprocal will be greater than one. This makes Statement D false.