The True Meaning of 37
Before entering the data analytics field, I spent four years as a math tutor, where I taught everyone from first-graders to Calculus 3 students. In that time, I noticed one key difference between the students who struggled with math and those who “got it.” The rougher students saw everything as concrete and fixed. They viewed math as a set of rules and laws to memorize while the gifted students understood that it’s all made up. I discussed something similar in my article about statistical significance. If you don’t understand why there’s a square root around the “n,” you don’t understand statistical significance.
Let’s start with the number 5784. What does that mean? In English, we say “five-thousand, seven-hundred, and eighty-four.” This aligns with Arabic numerals. We aren’t some crazy weirdos that swap the last two digits. Thus, Arabic numerals and spoken English use a “base-ten” system. Each corresponds to a different power of ten, with left slots representing higher values. What is 5784? It’s 5 thousands, 7 hundreds, 8 tens, and 4 left over.
Base-ten works, and it’s especially intuitive given our ten fingers. It’s not the only system, though. Take the number 37. Our base-ten system writes this as "3 tens and 7 left over.” Another system, popular in some forms of programming, is base 16. Here, each slot represents a power of 16 rather than 10. There, we’d write “thirty-seven” as 25. Confused? So-called “thirty-seven” is “2 sixteens and 5 left over.” It’s (2*16 + 5).
Writing the number in this matter might seem weird, and don’t recommend doing so outside a programming context. I just want to emphasize that “thirty-seven,” as a concept, exists outside the familiar Arabic numerals. The same applies to written language. There’s no inherent reason the three Latin letters “dog” ought to refer to a domesticated canine. It’s fake. It’s made up. If you struggle to view “thirty-seven” as anything other than “37,” I venture to say that you don’t really understand the essence of “37.” If you want to understand “37,” you should just as easily conceive it as “7*5+2” or “40-3” or “66/2+4.”
In Algebra class, you see a lot of rules. If math didn’t interest you, you might have merely memorized them and treated them as no less arbitrary than the rules of a board game. Some of these rules include:
a(b+c) = a*b + a*c
(a+b)/c = a/c + b/c
a^b * a^c = a ^(b+c)
Most students figured these rules out. Yet, many seem to treat them as something that only applies to letters. Math doesn’t know about the letter/number distinction. “A” is just a symbol from Rome and “3” is just a symbol from India. Math doesn’t care.
What’s 37*8? Let’s stick with our base-ten system. Thirty-seven is (30+7). We can think of this problem as 8(30+7). Again, math doesn’t differentiate letters and numbers! Apply the familiar algebraic notation and turn that into 8*30 + 8*7. that’s 240+56, or 296. Next, let’s simplify 137/5. As I liked to say in my tutoring days: fractions are division problems. We can write 137 as (100 + 30 + 5+ 2). Now divide each of those by 5. We get 20+6+1+(2/5), which is 27 and 2/5. Let’s have some additional fun with this problem. We can also conceptualize 137 as (100+40-3). Divide that by 5 and we get 20 + 8 - 3/5, and 8 is, after all, just 7 + 1. That leaves us with 20 + 7 + 1 - 3/5. One is 5/5, so 1 -3/5 must be 2/5. Again, 27 and 2/5.
Converting to Celsius
What’s 111F in C? The formula for converting F to C is (F-32) * (5/9). Don’t you dare open that calculator app! Remember: Algebra applies to numbers. Let’s go step by step
32 is (30+2), so we have to subtract 30 and 2 from 111. Order doesn’t matter in this context (a - b - c = a - c - b), so let’s subtract 2 first (109) and 30 next (79).
How do we multiply by (5/9)? Say it with me: fractions are division problems. We can re-conceptualize this as 79 * 5 / 9. Order doesn’t matter here either, so let’s write this as 79 / 9 * 5
Now, we have to divide 79 by 9. Seems rough, until you remember that 79 is (72+7). Divide each of those by 9 and you have (8 + 7/9).
It’s multiplication time. Algebra applies to numbers, so 5(8 + 7/9) equals 5*8 + (5*7/9). That’s 40 + 35/9. Another fraction! If only someone had told you that fractions are division problems.
That’s 40 + (27/9) + 8/9 or 43 and 8/9 degrees Celcius.
I’ve left everything as fractions so far, but there’s often an easy way to get decimals with single-digit denominators. In fact, 9 is my favorite denominator. What’s 8/9? that’s 0.888888888888…, repeating infinitely. Similarly, 5/9 is 0.5555555555… This applies to repeated nines as well! 47/99 is 0.4747474747… and 275/999 is 0.275275275…
Side note: Fractions (aka division problems)
For each of the digits, there’s an easy way to convert fractions into decimals.
x/1 isn’t even a fraction, you dummy
x/2 alternates between .5 and .0
x/3 alternates between .3333…, .6666…, and .0
x/4? Well, what does it mean to “divide by four” exactly? We can write (x/4) as (x/2)/2. “Divide by four” means “divide by two… twice!” Half of 500 is 250, so x/4 alternates between .250, .500, and .750
x/5. Quick question what’s 2/2? It’s one, right? And what happens when you multiply a number by 1? Nothing. You can multiply by one as much as you want. The math police won’t stop you. So let’s multiply it by (2/2) to get 2x/10. Every fifth is just two-tenths, so it alternates between .2, .4, .6, .8, and .0
x/6 kinda sucks. You can remember that 2/6 and 4/6 are just 1/3 and 2/3, which you know from the x/3 section. Similarly, 3/6 is just 1/2 and 6/6 is 1. That means you only need to remember these fractions: 1/6 is 0.16666… and 5/6 is 0.83333….
x/8. Remember how “divide by four” meant “divide by two twice.” What if we divided by 2… again? 250/2 is 125. The eighths are .125, .250, .375, .500, .625, .750, .875, and .000
x/9 repeats the top digit. Hence, 8/9 is .888888888… and 5/9 is .555555555… Yes, I repeated what I said above, but that’s how dividing by 9 works.
I left x/7 for last. I live in Vegas, and even I must admit that 7 sucks. Here’s the best I got: I remember that 7 is roughly 0.14. So 5/7 is 5(10) + 5(4) or ~. 7. The actual figure is .714, but I’m not going to remember another digit of 1/7.
KM to Miles (Approximately)
One last problem: let’s convert 3333 KM to miles. Note that one kilometer is 0.621371 miles, but we’ll use 5/8 (or 0.625) to get us close. Thus, this problem amounts to 3333 * 5 / 8 or 3333 / 8 * 5.
To start, how many times does 8 go into a thousand? Remember that 8 is just 2*2*2. (1000/8) is therefore 1000 cut in half thrice. Half of 1000 is 500, half of 500 is 250, half of 250 is 125.
Now let’s break this up into some eight-friendly numbers. 3333 is (3000 + 333). The thousand is nice, but we need something better than 333. Which numbers does 8 go into? Let’s count 8, 16, 24, 32. Bingo! If it goes into 32 four times, it must go into 320 forty times. Let’s write this as (3000+ 320 + 13). Even easier, let’s cut that up into (3000 + 320 + 8 + 5). I bet you can divide that by 8 in your head! That’s
3000/8. Each thousand is 125, so that’s 375
32/8 is 4, so 320/8 must be 40
8/8 is 1
5/8 can stay put
So we have 375+40+1+5/8. That “375+40” could screw me up, so I’ll conceive it as 375+25+15+1+5/8. That’s 400+15+1+5/8 or 416 and 5/8.
We still have to multiply everything by 5. 400*5 is 2000, 10 * 5 is 50, 6*5 is 30, and 5/8*5 is 25/8.
That puts us at 2080 and 25/8. You can simplify the 25/8 if you want, but remember that we’re just estimating here. Thus, 3333 is about 2080 miles, and the exact figure is 2071.