Question

Can a geometric mean be negative?

Answer 1

Hint: To get to the result we need to take a few examples. Here we will use two cases for this. One with positive numbers and the other is with negative numbers. After this we will apply formula \[{{\left( {{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}} \right)}^{\dfrac{1}{n}}}\,\,or,\,\,\sqrt[\dfrac{1}{n}]{{{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}}}\] and the one which passes through this formula will be the right case.

Complete step by step solution:
To understand its concept we need to discuss the term geometric mean first. The geometric is simply the nth root of the terms which are multiplied together. For example, consider the terms ${{X}_{1}},{{X}_{2}},{{X}_{3}}$ and multiply them together. Thus, we get ${{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}}$. Now, we will take its nth root like so, \[{{\left( {{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}} \right)}^{\dfrac{1}{n}}}\,\,or,\,\,\sqrt[\dfrac{1}{n}]{{{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}}}\].
Now, to understand the concept that whether the geometric mean can be negative or not, for this we need to consider two cases.
For the first case we will take all the positive numbers. Consider the numbers 4, 2, 3. Since, there are only three numbers, so, n = 3. Now, let’s find out its geometric mean. By the formula \[{{\left( {{X}_{1}}\cdot {{X}_{2}}\cdot {{X}_{3}} \right)}^{\dfrac{1}{n}}}\] we will get the geometric mean of these numbers as \[{{\left( 4\cdot 2\cdot 3 \right)}^{\dfrac{1}{3}}}\]. Therefore, we further get ${{\left( 12 \right)}^{\dfrac{1}{3}}}=2.289$. And clearly, it is a positive number.
Now, we will consider the case of negative numbers. Let’s say we are supposed to find out the geometric mean of the numbers – 4, - 2, - 3. Thus, we get ${{\left( -4\cdot -2\cdot -3 \right)}^{\dfrac{1}{3}}}={{\left( -12 \right)}^{\dfrac{1}{3}}}$. Here, the cube root of negative number cannot be found out. There is no method to do so. Therefore, we cannot find the geometric mean of - 4, - 2, - 3.
Hence, a geometric mean cannot be negative because we can only use the formula of geometric mean for positive numbers.

Note: We need to focus on the fact that we cannot find out the geometric mean of a negative number also, we cannot find the square root of a negative number. Thus, a square root of a negative number is not defined, so is the case with the cube roots. But if there are even numbers of negative numbers then we are able to find out the geometric mean. Because in that case the numbers will become positive after getting multiplied to each other automatically. This is why before answering any question we need to try to check the numbers as in, whether they are even or not. For example the numbers that we used in this question for negative numbers were odd. If we took only – 4 and – 2, then we could have calculated its geometric mean.