Sum of the Numbers Equals Their Product

I’ve dealt with numbers all my life, of course, and after a while you begin to feel that each number has a personality of its own. A twelve is very different from a thirteen, for example. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won’t think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he’s told; nine is deep and mystical, a Buddha of contemplation.

Paul Auster

Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check out Math1089!

Consider the numbers 2 and 2. Certainly, 2 + 2 = 4 as well as 2 × 2 = 4. Also, 0 + 0 = 0 × 0. What we mean is that, the sum of the numbers (may be equal) is equal to their product. This is the point of discussion here.

Of course, the numbers above are integers. Do we need to concentrate only on integers? No. One or both the numbers can be fraction (or rational numbers) also. Under this condition, we need to find the general solution, which will give us the structure of the numbers. Therefore, the question is

“Do there exist two numbers whose sum is equal to their product?”

Answer is certainly YES. Few such numbers are given below:

To find the numbers (in general), let us consider two numbers as x and y. Their sum is x + y and product xy. According to the symmetry, we have

The last expression is well-defined only when y −1 ≠ 0 or, y ≠ 1. 

Here, the numbers may be positive or negative. The number will be positive if y > 1 and the number will be negative if y < 1. Sign of the numbers does not play any pivotal role here.

From here, we can see (0, 0) and (2, 2) as the only integer solutions.

Rest all the solutions are in fractions. For example:

Here, the value of x is negative and y is positive. But there are examples where x is positive and y is negative. For example, if we take y = − 4, then

An observation. Below we will provide two more pairs of numbers (where n is non-zero) satisfying the above condition.

Let us ask the same question for three numbers. Therefore, the question is “Do there exist three numbers whose sum is equal to their product?”

Obviously, there exist. Consider the first three integers 1, 2 and 3. Clearly, 1 + 2 + 3 = 6 as well as 1 × 2 × 3 = 6 (and the possible combinations). Any other examples? Let’s see.

To find the numbers (in general), let us consider three numbers as x, y and z. Their sum is x + y + z and product xyz. According to the symmetry, we have x + y + z = xyz. Now, from this equation we need to find the values (if possible) of x, y and z. The solution strategy is to fix a variable and then look into the others. For example, let us fix z = m, a constant. Then

The last expression is well-defined only when ym −1 ≠ 0 or, (y, m) ≠ (1, 1) and (−1, −1). 

Here, the numbers may be positive or negative. The number will be positive if ym > 1 and the number will be negative if ym < 1. Sign of the numbers does not play any pivotal role here.

In order to get numeric solutions, we proceed as below:

Let z = m = 1. Then the values of x and y are given by

Proceeding as above, we can find infinite number of solutions to this system of equations. Therefore, few final solutions are

Let z = m = −2. Then the values of x and y are given by

As before, we can find infinite number of solutions to this system of equations. Few of them are

These are just few solutions. If we go on changing the values, we can generate different solutions.

An example from Trigonometry. Consider the following known fact:

If a + b + c = π then tan(a) + tan(b) + tan(c) = tan(a) tan(b) tan(c).

Consider three angles a, b and c of a triangle and consider the terms tan(a), tan(b) and tan(c) [except possibly a, b and c ≠ π/ 2]. Then, their sum equals their product.

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