Final answer:
When you multiply a fraction by its reciprocal, the result is always the number 1. This is because the numerator and denominator of the fraction are inverted in the reciprocal, and when multiplied together, they cancel out, leaving a value of 1, demonstrating that reciprocal multiplication leads to unity.
Step-by-step explanation:
Understanding Multiplication and Division with Fractions
When you multiply a fraction by its reciprocal, you are essentially performing a mathematical operation where the numerator (the top number) and the denominator (the bottom number) of the original fraction are switched. By definition, the reciprocal of a fraction is obtained by inverting the fraction. When you multiply a fraction by its reciprocal, the result is always 1. This is because the multiplication of a number by its reciprocal equates to multiplying a number by one over itself, which simplifies to one.
For example, consider the fraction 2/3. The reciprocal of 2/3 is 3/2. When we multiply 2/3 by its reciprocal (3/2), the top numbers (numerators) multiply to 6, and the bottom numbers (denominators) multiply to 6 as well. Since we have the same number on both the numerator and denominator, they cancel each other out, leaving us with a fraction that represents 1. This process aligns with the foundational principle of algebra that states when we perform the same operation on both sides of the equals sign, the expression remains an equality. Hence, any time we have the same quantities in the numerator and the denominator, they cancel out, and we are left with the number 1.
Additionally, understanding exponents helps to illustrate this concept. For instance, a negative exponent denotes a division rather than multiplication, flipping the number or fraction to its reciprocal. If x to the power of -1, written as x-1, this is the same as 1/x. Therefore, when multiplying x by its reciprocal, 1/x, the result is 1, because x times 1/x equals x/x, which simplifies to 1.
In mathematical operations involving units, the same principles apply. When units cancel out correctly, such as in unit conversion, we also utilize the concept of multiplying by the reciprocal to simplify the units. For instance, when converting meters to centimeters and vice versa, we can use conversion factors, like 1m/100cm, which also equates to 1. When units cancel out, we're left with the desired unit.