Tag: Always Equal 2

Mathematics is filled with enchanting surprises, but few are as captivating as arithmetic tricks that appear magical on the surface while being grounded in elegant algebra underneath. Imagine asking a friend to pick any three positive numbers completely at random—integers, large numbers, or even decimals. You guide them through a specific set of fraction operations, and before they even finish typing the final calculation into their calculator, you announce the exact final answer.

That answer will always be 2. Guaranteed.

The result does not depend on which positive integers you choose. Whether they choose small numbers like 1, 2, and 3, completely arbitrary numbers like 4, 5, and 14, the mathematical mechanics behind this formula ensure that every variable cancels out, leaving behind a constant result of 2.

Example Time

Let’s consider a few examples to see what’s happening.

Example 1.

Let’s start with the numbers 1, 2 and 3.

Construct three new fractions

$$\frac{1+2}{3}=1, \frac{2+3}{1}=5, \frac{3+1}{2}=2$$

Now, product of the fractions is

$$1\times5\times2=10$$

Sum of them is

1 + 5 + 2 = 8

Difference = product – sum = 10 – 8 = 2.

It’s fine that we get 2 by considering 1, 2 and 3. What if we consider some other numbers?

Example 2.

Now consider the numbers 2, 3 and 4.

As before, construct three new fractions

$$\frac{2+3}{4}=\frac{5}{4}, \frac{3+4}{2}=\frac{7}{2}, \frac{3+1}{2}=\frac{4}{2}=2$$

Now, product of the fractions is

$$\frac{5}{4}\times\frac{7}{2}\times2=\frac{35}{4}$$

Sum of them is

$$\frac{5}{4}+\frac{7}{2}+2=\frac{27}{4}$$

Difference = product – sum

$$=\frac{35}{4}-\frac{27}{4}=\frac{8}{4}=2$$

The Rule

If we follow the steps shown above, will we always arrive at 2? Yes, for sure. That’s interesting. In fact, the rule is given below:

The Rules are

1. Choose any three positive numbers and label them a, b and c.
2. Form three fractions where each numerator is the sum of two numbers, and the denominator is the remaining third number:

$$\frac{a+b}{c}, \frac{b+c}{a}, \frac{c+a}{b}$$

3. Calculate the product of these three fractions.
4. Calculate the sum of these three fractions.
5. Subtract the sum from the product. 6. The result is always 2

The Algebraic Proof

How can we guarantee the same for every time? What’s the mathematical basis? While testing specific numbers builds confidence, only algebra can provide absolute proof that this relationship holds true for all positive real numbers. Let us prove it rigorously.

Let a, b, and c represent any three positive real numbers.

Construct three fractions

Of course, this proof is independent of the values of a, b and c chosen. Only thing is that, they need to be non-zero.

Reader Challenge!

1. Pick three unusual numbers right now. For example, your birthday day, month, and year. 2. Consider them as a, b and c. 3. Form the fractions 4. Find the sum of these three fractions. 5. Find their product. 6. Did you find the same difference this time as well?

What starts as a seemingly complex arithmetic puzzle resolves into one of the cleanest algebraic identities in basic number theory. By expressing the product and sum of cyclic fractions in variable form, all non-constant terms vanish, proving that

Product − Sum = 2

for any starting numbers.

Beyond . . .

Consider an algebraic identity with the proof involving three variables. Can you use that identity to create an interesting algebra game? Do share it with your friends. Do share it with us as well at math1089.9801@gmail.com.