Four is the frequency of 60-70

]]>Hi, Ryan.

The last sentence is, of course, correct: We can think of a negative number of objects as being owed, as I said: “And you can have a negative number of apples if you *owe* them. This is discussed in How Real Are Negative Numbers?”

But we certainly **can** both add and multiply objects like apples or cars or points: Six points is two times as many as three points. And I demonstrated how to multiply objects by zero, or to multiply zero by a number.

It’s just that zero behaves differently than other numbers when you multiply.

]]>Hi, Scott.

I don’t see why anyone would say that 6 ÷ -(1 + 2) = -18. The only way to get that is to **blindly replace** -(1+2) with -1*(1+2), ignoring how it would change the order of operations in the surrounding expression. I discussed such unthinking substitutions (in a very similar context) in Implied Multiplication 3: You Can’t Prove It. But there is no reason anyone would do that here, because it is already clear what to do.

As I said above in response to Andrew, “you can’t just replace an expression anywhere with an expression that would be equivalent when standing alone, without first considering the order of operations.”

By the way, looking at your site, I find numerous places where I would disagree with you; but I see no value in arguing about this or “trying to confound” people who disagree with me. My goal is peace and mutual understanding, not proving myself correct. Acknowledging ambiguity (even merely *potential* ambiguity) is a central part of that.

I don’t quite agree with you; the obelus and solidus are not “officially” different in meaning, but “colloquially” the latter is thought of more as marking a fraction, and the former as division. As a result, when I want to type an expression containing a fraction (and implicitly treat that fraction as a distinct entity), I will use a solidus (commonly with spaces around it to emphasize that I am seeing it by itself). I don’t think of that as a *rule*, just as a natural way to say what I want. For the same reason, when I *read* such an expression, I tend to guess that the writer intended a fraction, though that is not always true. (It’s harder to type “÷” than “/”, since the latter is on the keyboard, so it is common to type “/” to mean division, without making any distinction. Furthermore, the horizontal fraction bar has exactly the same meaning as both the obelus and the solidus; a fraction *is* a division.)

For example, if I have in mind the formula for the area of a triangle, \(\frac{1}{2}bh\), but am just typing in-line, I will likely start to type “1/2bh”; but then, realizing that is a little ambiguous, I’d instead type either “1/2 bh”, which more clearly suggests the fraction I have in mind, or “(1/2)bh” to make it absolutely clear. If the context is clear enough that any reader would know I am writing the area formula, I would worry less about being misread. And the same is true of reading what someone else has written, especially if it is a student who may not be as aware of ambiguity as I am. But these are not rules, and not everyone necessarily takes it the same way.

Note that this does not necessarily apply when there is a variable present; 1/ab is more likely to be read as \(\frac{1}{ab}\), so if (for some odd reason) I intended \(\frac{1}{a}b\), I would write it as (1/a)b, and not 1/ab or even 1/a b.

You’re not using the term “monomial” in its proper sense; I think what you mean is “a single entity”.

I presume when you say “6 ÷ ½”, you have in mind “6÷1/2”, since what you typed, using a single symbol for 1/2, leaves no doubt as to the meaning. I would say that “6÷1/2” is sufficiently ambiguous (and odd) that it requires clarification, either by writing “6 ÷ 1/2”, using spacing to imply grouping, or explicitly as “6÷(1/2)”.

But all this is informal , not “rules”; as I suggested, it is “colloquial”.

Now, when you bring in mixed numbers, you are going further outside the realm of order of operations. We don’t use mixed numbers in algebraic expressions, precisely because juxtaposition has a different meaning there. Once you know that a mixed number is being used, you know to follow a separate rule, one that is used in arithmetic, not algebra. I wouldn’t extend that rule to algebra.

So, although I probably agree with how you want to read these expressions, I disagree with the idea that that is an absolute rule, and even more that it is provably correct.

]]>6 ÷ -(1 + 2)

Is it 6 ÷ -3 or -6 * 3?

Surprisingly, those who think 6 ÷ 2(1 + 2) = 9 would say that that the answer to my problem is -2! I was taught that a negative sign in front of parentheses without a coefficient is the same as multiplying what is inside the parentheses by -1, so I would have guessed they’d say -18. I believe the answer should be -2, however, just like I believe 6 ÷ 2(1 + 2) = 1.

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