Mind Performance Hacks

Hack #36. Make Friends with Numbers

With a little experience, you can learn to recognize many individual numbers by their special properties. Some of these properties can help you with mental arithmetic and memory.

Just like a face in a crowd, a number such as 1,729 probably doesn't mean much to you. But 1,000,000 looks like a friendly face. With some effort, more numbers can look like friends. Here's how to get started making their acquaintance.

In Action

Let's start with some numbers that you're probably already friendly with—10 and its powers: 100, 1,000, and so forth. Because we use a decimal number system [Hack #40], powers of 10 end in zeros. Companies are aware of that and try to get phone numbers that are multiples of a power of 10, because those numbers are easy to remember. For instance, the publisher of this book, O'Reilly Media, has a local phone number of [phone], which is a multiple of 1,000. Multiplying by a power of 10 is easy; you just have to append zeros:

314 × 1000 = 314000

Similarly, dividing by a power of 10 just removes zeros (or moves the decimal point to the left if there aren't enough zeros):

2030 / 100 = 20.3

If we look at the factors of 10, which are 2 and 5, we can come up with other useful rules. (Factors of 10 are also called aliquot parts.) For instance, to multiply a number by 5, first multiply by 10 and then take half the result. This is based on the notion that, for multiplication and division, 2 is a friendlier number than 5. For instance, 386 × 5 = (386 × 10) / 2 = 3,860 / 2 = 1,930.

This idea can be extended to factors of powers of 10. For instance, 100 = 4 × 25, so to divide by 25, double the number twice (which is the same as multiplying it by 4) and divide by 100:

217 / 25 = (4 × 217) / 100 = 868 / 100 = 8.68

If you're estimating, near factors can also be useful: 33 × 3 = 99, which is almost 100, and 17 × 6 = 102, which is just a little more than 100.

How It Works

Unlike some of the hacks in this chapter, this one can't give you a straightforward recipe that's guaranteed to make every integer you encounter unique. After all, the hack involves finding something unusual about the number. If every number you met exhibited the same unusual feature, it obviously wouldn't be unusual. Becoming more familiar with the number system is a gradual process. However, I can give you a few tips:

Start with smallish numbers

Although we'll give an example of hacking a seven-digit phone number later, it might be easier to hack the three-digit and four-digit parts separately, or even split the number into the three-digit part and two two-digit parts.

Try factoring

We already talked about factors of powers of 10 (such as 4 and 25). Factoring in other ways can be useful in mental (or paper-and-pencil) arithmetic. For instance, if you have to multiply 193 by 56, it might be easier to factor 56 into 8 × 7, and then multiply each factor separately:

193 x 56 = 193 x 8 x 7 = 193 x 23 x 7 = 386 x 22 x 7 = 772 x 2 x 7 = 1544 x 7 = 10808

In this example, we multiplied by 8 by doubling three times.

Look for patterns and near patterns

I once had a prescription number that was 66123465. This was easy to remember because of two patterns: all my prescription numbers started with 66 and the rest of the number was almost 123456. Remembering the one difference from the pattern was easier than remembering the whole number.

Associate numbers with uses

If you're a sports fan and you know your favorite player wore number 80, remembering other 80s might be easier by associating them with that player. (This is a variant of the mnemonic techniques presented in Chapter 1.)

Find friends of friends

In arithmetic, this mostly means numbers close to other numbers. For instance, 4 × 24 is easy to compute as 4 × (25 – 1) = 100 – 4 = 96.

In terms of mnemonics, any way you can connect numbers is fine.

In Real Life

We've already talked about arithmetic, so we'll concentrate on mnemonics here. Let's see what we can do with O'Reilly's toll-free number—[phone]—in case you want to order more copies of this book. (What am I saying, "in case"?)

Since many toll-free numbers start with 800, this should not be a problem. Notice that 800 was chosen to be a multiple of 100. Then notice that the 998 almost repeats: not only is the second part 99?8, but the extra digit is a 3, which you can think of as a broken 8.

Depending on your tastes, there are other ways to proceed. For instance, back in college, I remember one time I ran into my friend Ben at the beginning of the school year. He told me his phone number: 436-7062. I didn't bother writing it down. I was sure I could remember it without writing it down. And, 20 years later, I do remember it, even though Ben moved out of that particular dorm room 19 years ago.

How did I do it? I saw that number not merely as a sequence of seven numerals, in which form it seems rather arbitrary, but as a particular seven-digit number, and looked for its special properties as a number. Let's try factoring to start, although we'll use some of the ideas mentioned previously as we go along.

First, checking for divisibility [Hack #37], I saw that 4,367,062 is divisible by 2 (because the last digit is even) and 7 (because +62 – 367 + 4 = –301 and –301 / 7 = –43). I could also see that there were no more small factors. If we divide by the factors I got (that is, 2 and 7) and use techniques similar to those in "Put Down That Calculator" [Hack #35], we obtain the following results:

4367062 / 2 = 2183531
2183531 / 7 =  311933

So, 4,367,062 = 2 × 7 × 311,933.

Now, there's a pattern to the number 311,933: if you multiply the first three digits by 3, you get the last three digits. A little thought (and a nodding acquaintance with a few other integers) tells us that this means:

311933 = 311 x 1003

Here I got lucky: both of those numbers were already friends (so our number 311,933 is a friend of those friends). 311 is prime, and 1,003 is 17 × 59, also both prime. As a mathematician, prime numbers—those numbers with no factors other than 1 and themselves—are interesting and often "friendly," and for various reasons¹ 17 is a very old friend, so I already knew the factorization of 1,003.

The complete factorization is:

2 x 7 x 17 x 59 x 311

For me, all of those numbers are friends, and the factorization is nice, with only one of each prime factor. That was enough for me to remember the phone number.

That information might not help you remember the number. The key is to make friends with numbers in your own way. Everyone will have different ways of doing this. For instance, mathematicians love to tell the story of Srinivasa Ramanujan, a great mathematician active in the early 20th century. His friend and colleague, G. H. Hardy, visiting him in the hospital, was making small talk. Hardy mentioned that the cab that took him to the hospital was number 1729—to Hardy's disappointment, because it was such a boring number. Ramanujan disagreed: 1,729 was quite an interesting number, being the smallest number that can be written as a sum of two positive cubes in two different ways.²

Ramanujan was quite correct: 10³ + 9³ = 1,000 + 729 = 1,729, and 12³ + 1³ = 1,728 + 1 = 1,729. It's straightforward, if a little tedious, to check that no smaller integer has this property. If I told you the property that Ramanujan mentioned, you could figure out the number 1,729 with a lot of tedious, routine computation, but to go in the other direction is rather remarkable. Even most professional mathematicians would have trouble doing that, except for the fact that this story is so popular. J. E. Littlewood, another colleague and frequent collaborator of Hardy and Ramanujan, put it this way: every positive integer was one of Ramanujan's personal friends. A single face in the crowd is recognizable if you know the person. So it was with numbers for Ramanujan.

Of course, even if you can't recognize every face in a crowd, recognizing some faces can still be helpful!

End Notes

1. Lefèvre, Vincent. 1998. "17 (Seventeen) and Yellow Pigs." http://www.vinc17.org/yp17/index.en.html .

2. MathPages.com. "The Dullness of 1729." http://www.mathpages.com/home/kmath028.htm.

Moses Klein and Mark Purtill