What if we spoke about division as we do logarithms?

Definition: The quotient, factor a, of c is b if and only if ab is c. We introduce the following notation:


When we read this notation aloud, we say (for the second example), The quotient, factor 5, of 10 is 2.

The important thing to remember about a quotient is that a quotient is a factor. When we see this equation:

we ask ourselves, What factor goes with 3 to make 12?

Properties of the quotient

The following properties of quotient are derived from corresponding properties of multiplication.

laws of the quotient

There are some important laws pertaining to the quotient. The next section will include algebraic proofs of these laws.

These properties will be important when solving equations involving quotients.

factors for the quotient function

Standard quotient tables and calculator functions involve one of two factors: 10 or u, where u, the factor of the natural quotient, is defined by the series below:

Quotients using this factor can be notated one of two ways:

Quotients factor 10 are so common that they also have special notation, so that the following two equations are equivalent in meaning:

One other convention for the factor 10 quotient is this: If a factor is not indicated, we assume the quotient is factor 10. More commonly, we capitalize the Q in Quot in order to indicate the factor 10 quotient but either notation is accepted.

Many times, we will be able to find a quotient by inspection and use of the associated known multiplication facts. In almost all cases involving relatively small numbers, we can approximate the value of a quotient by use of multiplication facts.

When we require a greater level of precision, we will need the change of factor formula.

Change of factor formula

Applications of the quotient

The quotient function appears so often in computations that some important measures are defined in terms of it. Of these, perhaps the most important is speed.

The speed (r) of an object is defined in terms of the distance (d) it travels over the time period (t) in following way:


10 responses to “What if we spoke about division as we do logarithms?

  1. Do you have a better notation for logarithm?

  2. I usually explain to my students that logarithms were pioneered by a bit of a lunatic, so we got stuck with his ridiculous name for them.

  3. Brilliant.

    One method I have used is to ask students: if multiplication is repeated addition:
    7*3 = 7+7+7
    3*7 = 3+3+3+3+3+3+3
    then what is the next step? What is “supermultiplication”? To which many of them say, repeated multiplication. I ask them to make up their own symbols for this:
    7#3 = 7*7*7
    3#7 = 3*3*3*3*3*3*3 etc.
    (It amazes me that when I do this with students who “know” exponentiation already – beginning college students – they very often don’t recognize it at all.) Students quickly see that this new symbol is not commutative. After playing around for a while, and getting them to solve problems like
    2 # x = 8,
    we ask, what is “superdivision”? And they make up their own symbols. But they soon realize that they need two symbols – or a symbol with some sort of directionality to it, an arrow or something – because
    8 superdivide 2
    could mean either
    2 # x = 8
    x # 2 = 8
    which are two different things.

  4. A good symbol would likely be lots better than this crazy-word function notation, but symbols take a while to get used to, too. I think the fact that it’s an inverse function is the bigger part of the problem. But I had lots of fun with P(8)=3, P(32)=5, what’s P(16)?, etc. (After putting a table of values for y=2^x on the board, per jd2718’s suggestion.)

  5. While I enjoy the satirical bent here, I think it cuts more ways than one. Not sure if you intended that or not. In one sense, it makes clear that rigid insistence upon treating kids learning mathematics as if they’re mathematicians in training who need to be moved as soon as humanly possible to the most austere presentation of mathematics imaginable is a pretty stupid idea. But from my personal perspective as a VERY late learner of mathematics who’s still struggling to deepen my understanding at some levels and achieve even minimal understanding of others, I’d be happy if very little mathematics were presented that way except when the audience is indisputably only professional mathematicians or those very well on their way to becoming such (e.g., graduate students in mathematics). I think the rest of us do better with less stark and forbidding writing. But then again, few mathematics authors ever consistently present their thinking in ways I find accessible. I am in no position to insist that they change, however.

    That said, logarithms come a hell of a lot later in math than do quotients, so that one way to look at your presentation of the latter is that if we lowered the age at which this sort of style were foisted upon students, we’d lose nearly everyone, but those we didn’t lose would probably be ready to take on presentations of elementary mathematics by mathematicians. Not exactly a trade I’d be willing to make, but it wouldn’t surprise me to discover that there are folks out there who would be only too happy to see things head in that direction (imagine Bourbaki in charge of elementary mathematics education in this country).

    Then again, I’m not sure that many K-6 teachers speak much more intelligibly about long division than what you’ve written. Sure, they’re not using formalism, but they aren’t generally able to explain what’s going on, either. They present a rote procedure that I suspect most of them don’t understand the workings of.

    And if you’ve taught mathematics for elementary educators, how have you made out trying to get your students to grasp partitive vs. quotative instances of division, whether in the integers or, perish the thought, rational numbers? I’ve seen more than a few eyeballs roll back in heads after asking students to construct a quotative word problem involving division by a proper fraction.

  6. I don’t know Christopher… seems like ‘quot’ is just as complicated at ‘log.’ And then there’s the question of “Just how do you pronounce ‘quot’?” Does it rhyme with ‘hot’ or ‘boat’? And then you’ll need to come up with a good joke that can rival the infamous Noah and the snakes:
    “Go forth and multiply!” he commands.
    The snakes look at each other, and then at Noah. “We can’t, we’re adders”.
    “Yes”, Noah replies, “but, even adders can multiply on a log table”.

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  8. I love this post. It got me thinking about alternate notations for logarithms. What if used the subscript notation for logarithms? As in, 8_2 = 3. Before you dismiss it, write it out — you can practically see 2^3 written there, much in the same way that you can see 2*4 lurking around when you write 8/2 = 4.

    • Yeah. I’ve toyed with that, and with other notations. I’ll write about that in short order. I understand that we’re not gonna change the canonical notation. But as an intermediate step, alternate notation may well be worth investigating as an instructional move. Perhaps a bit of lesson study on this?

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