*Math may not teach me*

*how to add love*

*or subtract hate,*

*but it gives me every reason to hope that*

*every problem has a solution.*

Greetings to all visitors of **Math1089 – Mathematics for All**.

I’m thrilled you’re here. I wanted to express my gratitude for your valuable time spent on **Math1089**. Your visit means a lot to me and I truly appreciate you taking the time from your hectic schedule to check out this blog.

**Valentine’s Day** is celebrated annually on **February 14th**, and is a day dedicated to expressing love and affection towards one’s significant other. While it may seem that mathematics and Valentine’s Day have nothing in common, they are in fact intertwined in a number of interesting ways. For further insight, please refer to the concluding section of this blog post.

If you wish to convey an original valentine greeting to someone, expressing sincere friendship, make use of *number theory*. One of the ideas is by sending a card on which is printed something like this:

**Romeo – 220**

**Juliet – 284**

or you could replace the names with a question mark to keep the sender a mystery. The idea behind using these numbers in your Valentine’s greeting is that they symbolize the idea that the two of you are just *made for each other*, just like the *amicable numbers* 220 and 284. The message is a unique and original way to express your sincere friendship and the sentiment that you and your Valentine were meant to be together.

It’s important to note that this message is based purely on mathematical concepts and a full understanding of it requires a comprehensive understanding of amicable numbers. In order to fully appreciate the sentiment behind the message, one must have a solid grasp of the mathematical idea behind it. Before proceeding with further explanation, let’s take a moment to define the following concepts:

**Factors**. A *factor* of a number is an exact divisor of that number.

For example, 1, 2, 3 and 6 are the factors of 6.

A positive divisor of *n* which is different from *n* is called a *proper divisor* (or an aliquot part) of *n*.

For example, 1, 2 and 3 are the proper divisors of 6.

**Perfect Numbers**. A *perfect number* is a number which equals the sum of its proper divisors.

For example, 28 is a perfect number.

**Amicable Numbers**. *Amicable numbers* (sometimes called, *amicable pair*) are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number.

Returning to the original question, let’s see how the numbers 220 and 284 are meant for each other in this Valentine’s Day message. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and 220, and the factors of 284 are 1, 2, 4, 71, 142, and 284.

The sum of the proper divisors of 220 is

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110

=

284.

Similarly, the sum of the proper divisors of 284 is

1 + 2 + 4 + 71 + 142

=

220.

Hence the numbers 220 and 284 are ** made for each other** in the sense that one’s sum of the proper divisors is equal to the other number.

**Happy Valentine’s Day!**

It’s not surprising that this sentimental connection has been interpreted in mystical ways by some of our ancestors. For instance, in Genesis 32:14, Jacob’s gift of 200 she-goats and 20 he-goats is said by one Bible commentator to have been a *hidden secret arrangement*. This is because 220 is part of the pair of amicable numbers 220 ‒ 284 and Jacob was attempting to secure Esau’s friendship through this means. The philosopher Pythagoras went so far as to say that a friend is *one who is the other I*, *such as are* 220 *and* 284.

Mathematician Ramanujam was a man of very few friends and was not known to have a close friendship with anyone. A friend pointed out this to him and asked him the reason for his hesitation in developing a close friendship with anyone. Ramanujam replied *I do wish to have someone with whom I can share a close friendship. But the problem is I am yet to find a person who possesses the qualities I expect from a close friend*.

The friend was eager to know what were the special qualities expected by him. Ramanujan replied *the numbers* 220 *and* 284 *are symbolic of an exemplary friendship and my wish is to have that kind of friendship with someone, *. . . both the numbers contained within themselves the other one.

We can take one more pair in the name of the legendary lovers

**Heer – 1184**

**Ranjha – 1210**

Like the numbers 220 and 284, the numbers 1184 and 1210 are also a perfect match. They are considered amicable because the sum of the proper divisors of one number is equal to the other number. The proper divisors of 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592, and their sum is 1210. And the proper divisors of 1210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605, and their sum is 1184. These two numbers complete each other, making them truly a pair made for each other!

For Valentine’s Day, you can express your love in a unique and mathematical way by **using the graph of polar curves**. A polar curve is a shape constructed using the polar coordinate system. Polar curves are defined by points that are a variable distance from the origin (the pole) depending on the angle measured off the positive *x*-axis. You could present her with a unique expression of love in the form of a mathematical equation. The equation, written in polar coordinates, depicts a rotated cardioid: *r* = 1 – sin *θ*. The curve created by this equation is shaped like a heart, making it the perfect symbol of love to share on this special occasion.

For a more accurate representation of the heart shape, consider the following polar curve, where the variable *θ*varies from ‒*π* to *π*. This curve captures the true essence of the heart symbol and would make for a wonderful and unique expression of love on Valentine’s Day.

We welcome your ideas with open arms and reverence! Looking forward to seeing you soon on ”**Math1089 – Mathematics for All**” for another fascinating mathematics blog.