Does every rational number have a multiplicative inverse? Explain why or why not. Please help ASAP!!! :(
1 and -1 are the only rational numbers which are their own reciprocals. No other rational number is its own reciprocal. ... We know that there is no rational number which when multiplied with 0, gives 1. Therefore, rational number 0 has no reciprocal or multiplicative inverse.
Yes, every rational number except for zero has a multiplicative inverse.
A rational number can be expressed in the form p/q, where p and q are integers and q is not equal to zero. The multiplicative inverse of a rational number p/q is another rational number q/p.
To verify that the product of a rational number and its multiplicative inverse is equal to 1, we can perform the following calculation:
(p/q) × (q/p) = (p × q) / (q × p) = 1.
Therefore, any non-zero rational number has a multiplicative inverse.
Yes, every non-zero rational number has a multiplicative inverse. To understand why, let's first define what a multiplicative inverse is.
The multiplicative inverse of a number x is another number y, such that when x and y are multiplied together, the result is 1. In mathematical notation, if x is a non-zero number, its multiplicative inverse is represented as 1/x or x^(-1).
Now, let's consider a rational number p/q, where p and q are integers and q is non-zero. To find its multiplicative inverse (1/(p/q)), we need to find another rational number such that when we multiply the two together, we get 1.
To achieve this, we can take the reciprocal of p/q, which is q/p. Now, when we multiply p/q and q/p, we get (p/q) * (q/p) = (p * q) / (q * p) = 1.
This demonstrates that every non-zero rational number has a multiplicative inverse. However, it's important to note that the number 0 does not have a multiplicative inverse because any number multiplied by 0 is always 0, not 1.
To summarize, all non-zero rational numbers have a multiplicative inverse, except for the number 0.