Is Love Squared More Love?

June 30, 2017

Yoni Nazarathy

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I sometimes color adjectives, verbs or nouns with the suffix “squared”. If something is very tasty, I’ll say it is “tasty squared”. If something is tough I’ll call it “tough squared”. Then with love, I do the same. I sometimes say “love squared”. But thinking about it more, I ask, what does "love squared" mean?

Basically I'm asking about the effect of squaring. Where does this term come from? I guess it is based on the assumption that the strength of love can be measured by a single variable, say $L$ . So if for example $L = 7$ then by squaring $L$ we get much more love because $L^2 = L \times L = 49$ .

As you probably know, the square of a number is obtained by multiplying it by itself. You start with a side of length $L$ and the area of the square is $L \times L$ . So if the love I start with is a line of length $L = 7$ . Then love squared is a square of area $49$ .

OK, I see what you may be thinking. You probably believe that love comes in many ways and forms and it isn't a number. In particular, love is often without bounds. Hence the magnitude, length, strength or value of $L$ does not exist. You know what? I agree. Love is a very special thing and it really cannot be represented just by a single number. However, for the purpose of this exploration, please bear with me and assume there is some value $L$ that summarizes the strength of the love. The larger $L$ is, the more love we have. So back to the key question.

Does love squared mean more love?

Try $L = 3$ and get $L^2 = 9$ . More love.

Try $L = 1.5$ and get $L^2 = 2.25$ . More love.

Hence for these values of $L$ , squaring it makes love grow. But how about $L = 1$ ? What happens to the love? How about $L = 0$ ? As you see, for these two values squaring the number doesn't do anything. Hence,

Love squared is sometimes more love, but not always.

So as we see, $L^2 > L$ for $L = 1.5$ , $3$ and $7$ . We also see that $L^2 = L$ for $L = 0$ and $L = 1$ . You may now ask: Are there any values of $L$ for which $L^2 < L$ . That is,

Is there ever a case where love squared is less love?

Think about it yourself for a bit.

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Well, how about $L = 0.5$ . What is $L^2$ ? My nine year old daughter and I were discussing "a half times a half" the other day. You clearly know it is a quarter ( $0.25$ ). But how? One way is to think of $L^2$ as a half of a half. For example, a half of half a dollar, is half of $50$ cents which is $25$ cents. Then that is a quarter of a dollar.

So we discovered that for $L = 0.5$ we get $L^2 = 0.25$ . Less love!

Try a few other cases where $L$ is a value between $0$ and $1$ . For example if $L = 0.3$ then $L^2 = 0.09$ . You are probably now tempted to conclude the following:

When $0 < L < 1$ then $L^2 < L$ . That is, a proper fraction of love squared is less love.

And what if $L$ was negative? How do you actually interpret negative love? Maybe it is "hate"? At this point it may be nice to recall the basic rules of multiplying positives and negatives together. Basically a negative times a negative is positive. But if the numbers have opposing signs -- that is, one is positive and the other is negative -- then the result is negative. One of my favorite resources for this is this lovely video from Mathologer. But you can also look at this neat page by Math is Fun.

Now going back to our love squared world, take for example $L = -4$ then $L^2=16$ , because it is a negative times a negative. Perhaps metaphorically you can say,

Squaring hate ( $L < 0$ ) makes love ( $L^2 > 0$ ).

Our exploration from above can now be summarized:

$\begin{array}{rcl} L<0~~\text{or}~~L>1 & : & L^2>L\\ L=0~~\text{or}~~L=1 & : & L^2 = L\\ 0< L < 1 & : & L^2 < L\\ \end{array}$

Wow, that is quite a lot to take in and we just did it by experimentation. Can we obtain this conclusion in a more structured manner and be sure about it? You see, often with mathematical exploration you start with experimentation and examples and then move onto a more organized analysis.

Let's do it. But for the path we'll take, we need a bit of basic algebra, common sense and the multiplication rules of positives and negatives mentioned above.

Keep in mind that we are interested in comparing, love squared ( $L^2$ ) and love ( $L$ ). Let's then write the difference between the two:

$L^2 - L.$

If this difference is positive then it means that $L^2 > L$ . And if it is negative then $L^2 < L$ . And when $L^2 - L = 0$ , it just means that love squared equals love.

Now let's manipulate the difference just a bit. Let's factor out an $L$ and get.

$L(L-1).$

If you now take a few minutes to let it sink, you'll see that this way of writing the difference gives us so much because it exactly shows us when $L^2 - L$ is positive, negative or zero. By writing $L^2 - L$ as $L(L-1)$ we can treat it as a product of two numbers, $L$ , and $L-1$ . Let's now consider the three possible outcomes of love vs. love squared:

Q: When is $L(L-1) = 0$ ? This asks when is $L^2 = L$ .

A: Only when $L=0$ or $L=1$ . To get a product of two numbers to be zero, at least one of them has to be $0$ .

Q: When is $L(L-1) > 0$ ? This asks when $L^2 > L$ .

A: Now using the multiplication rules mentioned above, we see that both $L$ and $L-1$ need to have the same sign (either both positive or both negative). This happens when either $L < 0$ (both are negative) or L > 1 (both are positive).

Q: When is $L(L-1) < 0$ ? This asks when is $L^2 < L$ .

A: Now the two factors need to have opposing signs. One needs to be positive and the other negative. Can you see why this happens for $0 < L < 1$ and not for any other values of $L$ ? Work it through.

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That is it, we got what we wanted, we understood the effect of squaring a number on its size. Continuing the love metaphor these were the rules that we found:

Square zero love or 1 unit of love and stay with the same amount of love.

Square negative love or true love (love > 1) and get more love.

Square a proper fraction of love and get less love.

As I wrote this blog post, I couldn't help but think about the love in my life. I am lucky to have so much of it. Frankly, the metaphor of "squaring love" could only be extended so far. But nevertheless it got me thinking. Thinking about love, thinking about life. And thinking about the neat and simple math that compares $L^2$ and $L$ . I'll be sure to share the latter with my loved ones when the right moment comes along.