倒数, pronounced as dǎoshù in Chinese, means reciprocal in English. For any non-zero number x, its reciprocal is 1 divided by x. For example, the reciprocal of 2 is one-half. The reciprocal of negative 3 is negative one-third. When we multiply a number by its reciprocal, we always get 1. This is why reciprocals are also called multiplicative inverses.
Let's explore the properties of reciprocals. First, the reciprocal of a reciprocal returns the original number. For example, the reciprocal of 2 is one-half, and the reciprocal of one-half is 2. Second, the reciprocal of a product equals the product of the reciprocals. Third, to find the reciprocal of a fraction, simply invert it by swapping the numerator and denominator. On the graph, we can see that the function y equals 1 over x forms a hyperbola. This curve never touches the x or y axes, which are its asymptotes. Notice how as x approaches zero, the reciprocal value approaches infinity, and as x gets very large, the reciprocal approaches zero.
Reciprocals have many practical applications. First, division can be rewritten as multiplication by a reciprocal. For example, 8 divided by 4 equals 8 times one-fourth, which is 2. Second, when calculating time from distance and speed, we use the reciprocal of speed. Third, in electrical engineering, parallel resistors combine using reciprocals. The reciprocal of the total resistance equals the sum of the reciprocals of individual resistors. This is why adding resistors in parallel always results in a lower total resistance. Fourth, in optics, the thin lens equation uses reciprocals to relate object distance, image distance, and focal length. These applications demonstrate why reciprocals are fundamental in mathematics and science.
Let's practice calculating with reciprocals. In Example 1, the reciprocal of 4 is one-fourth, which equals 0.25. We can see both values on our number line. In Example 2, to find the reciprocal of two-thirds, we invert the fraction to get three-halves, which equals 1.5. For Example 3, let's solve three-fourths divided by one-half. Division by a fraction is the same as multiplication by its reciprocal. So we multiply three-fourths by 2, which gives us six-fourths, or 1.5. Notice that dividing by a fraction less than 1 actually increases the value, which is why our result is larger than what we started with. This makes sense because dividing by one-half is the same as multiplying by 2.
To summarize what we've learned about reciprocals: First, 倒数 (dǎoshù) refers to the reciprocal of a number, which is 1 divided by that number. For any non-zero number x, its reciprocal is 1/x. A fundamental property is that a number multiplied by its reciprocal always equals 1, which is why reciprocals are also called multiplicative inverses. Reciprocals have important applications in mathematics and science, including division operations, rate calculations, electrical circuit analysis, and optical lens formulas. Graphically, the function y equals 1/x forms a hyperbola with the x and y axes as asymptotes, illustrating how the reciprocal approaches infinity as the input approaches zero, and approaches zero as the input grows larger in magnitude.