Do you see “6 times negative three”? Do you see “6 times x minus 3”? After all, it could be either. Because multiplication is ridiculous. Yes, I said it. Multiplying: the operation that can be expressed using nothing, or practically any other math symbol already in use elsewhere. Why do we do this to students? Why do we do this to ourselves?
|X marks the spot... of no return.|
The “X” symbol was first used for multiplication by William Oughtred, which was published in his book “Clavis Mathematicae” (1631). However, around that same time, Thomas Harriot was using the elevated dot “ . ” symbol. Gottfried Leibniz preferred that notation... supposedly with the complaint that the “X” resembled an unknown “x” too much. (Even THEN they saw this problem coming!) Incidentally, Leibniz also used the inverted U (“cap”) symbol for multiplication, which now denotes intersection.
Owing to general disagreement, it wasn’t until the 1800s that “X” became popular in arithmetic for denoting multiplication. But the elevated dot is still used too - except where it denotes a decimal point in British textbooks (as Ben Orlin found out). But wait, in senior math for North America, the elevated dot is also used to represent the DOT PRODUCT (or scalar product) of two vectors, not to be confused with the CROSS PRODUCT (or vector product) - that’s the one which uses the X. Though we can write dot product as |a|x|b|x(cosC), where those x’s are (of course) multiplications, not cross products.
Confused yet? No? Okay, then let’s make it worse.
Take a step back now to evaluate: (5)(4)2
If you’re like a LOT of students, you’d write (20)2, not realizing that the expression is really (5)(4)(4). Of course, by the distributive law of multiplication, it still means (20)(20), because the 5 distributes to both 4’s, yeah? (NO! WRONG! Distribute on addition only.) What also doesn’t help is the HORRIBLY inconsistent way brackets are then used in conjunction with exponents in textbooks: (5)4 is actually the same as (54), and used interchangeably, but they’re NOT the same as (-54) because now the (-) is NOT on the base. Granted, I admit I’m guilty of being inconsistent with parentheses that way myself, unless it’s pertinent to the question at hand.
Which brings me to the huge problem: Whether something involving brackets is multiplied or not tends to be inferred from CONTEXT. Which is terrible when we come to f(x), the notation for a function. There it’s NOT multiplication, those parentheses aren’t the same parentheses as before! Perhaps rightly, every year this BAFFLES some students, who invariably see f(x)=5x as some weird multiplying ritual. To find out when it equals 20, they will write 20(x)=5x. Then divide by 20. (Or worse, subtract 20. x = -15, right?) Similarly, ask the student to find “f(3)” and their last step is invariably a division by that 3, unless they learned the context clues.
Okay, so X is confusing, but parentheses are JUST HORRIBLE by comparison. What’s left? Well, context for multiplying can’t be THAT hard, right? How about we express multiplying without any symbol at all! What could go wrong?
Perhaps this “no symbol” idea could work - if we weren’t in a place value system!! Combine that issue with early use of “x” as an unknown, and we’re screwed. After all, why can’t 4x mean 4 in the tens place, and an unknown ones place? (Heck, I suppose it can, if we let x=10, so that the ones place is zero.) Then there’s a typical high school expression like “5-2x”. Is it any wonder students combine unlike terms? There’s a subtraction RIGHT THERE, and no other operations! So 3x, right? (Wrong again!)
|That's a deadly Sin...|
Just to round things out, multiplication can also mean division. Because if you’re dividing by 2, that’s the same thing as multiplying by 0.5 - and we generally want to do this if we end up with a rational or trigonometric expression that has fractions on top of divisions. (Such as [(x+1)/(x+2)]/2 ) Oh, and what do we use to show multiplication of two rational expressions? Often the X symbol.
I mean, students don’t know cross product with vector notation yet, so they can’t get confused, right? And once we reach matrices, they’ll know 3x4 is a 3 by 4 matrix, not a calculation, yeah?
The thing that bugs me the most about all confusion this is that there’s an obvious solution to it. Computer scientists have been using it for decades. It’s the asterisk/star (*) symbol! Which you have probably used yourself online, perhaps even in writing up a mathematical blog post!
And before you argue that * is a recent addition, the choice wasn’t completely arbitrary. The asterisk was (supposedly) used in Germany back in the 1600s. Possibly even by Johann Rahn, the same guy who popularized the obelus symbol for division. (Though for division, Leibniz preferred using the colon. Bringing this article full circle.)
|Enough! We thought WE were your X girlfriends!|
In conclusion, I claim the X isn’t working. So why haven’t we switched to * in this day and age??
Well, aside from the fact that traditions are really hard to break (hey, I’ll admit I’m not using it yet), there’s the problem of needing to reprint all the textbooks. Textbooks that schools can’t afford to buy. And online, we might also lose out on Khan Academy videos like “Why aren’t we using the multiplication sign?”. So I guess we’re stuck. Until the computer uprising.
For more on Mathematical Symbol Origins go to this link! Alternatively, read my prior rants about y=a(x-h)2+k or cross multiplication. Or merely comment below.