πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±
1/5
Sometimes when I present people with a #Mathematics textbook reference, I get "you know textbooks can be wrong, right?". Actually I do, and you know what?? I know how to check which ones are right and which ones aren't by... doing the #Maths! πŸ™‚ So for this week's (and next week's) #MathsMonday I'm actually going to go through the #Math of a couple of concepts which are fundamental to (you guessed it) order of operations, which some people still aren't accepting. This week is Terms...
Nov 13, 2023 at 9:45amWeb
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2/5
Let's start with a website using the correct definition (see screenshot) - Terms are separated by operators and joined by grouping symbols. They haven't explicitly spelt it out, but we can see they have referred to "2 terms", and the only operator is a division, so yes, division separates Terms.

Terms are inseparable units. For 2ab, I can write 2ba, ab2, or even a2b, and they're all equal (say a=2, b=3, they are all equal to 12) - they're just different ways of writing the same thing...

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3/5
Now let's look at where a textbook has been (perhaps unintentionally) misleading. It says "Terms are separated by + and - signs", which MAY lead one to think (and I've seen this claimed) that Terms are NOT separated by multiply and divide, only plus and minus. Indeed, above that it says "A Term is a collection of numbers, letters, and brackets, all multiplied/divided together" which further gives that impression. But is that even true? Well, let's #DoTheMaths πŸ™‚ ...
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4/5
If I write ab as a*b, well, that still works, so that seems fine so far.

But what if I write 1/ab as 1/a*b - aha! No, that definitely is NOT equal to the same thing (this may look familiar to some. Spoiler alert: 8/2(1+3) vs. 8/2*(1+3)). If a=2, b=3, then 1/ab=1/6, but 1/a*b=1/2*3=1.5.

So right there, by just doing some Maths, we have disproven the claim that Terms are NOT separated by multiply/divide, and shown the textbook to be wrong (perhaps unintentionally).

But, having said that...

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5/5
There IS something such as what the textbook says, but I'm not sure if it has a name (it's definitely not Terms). There is an inseparable unit when it comes to binary operators, since they are associated with TWO Terms - each one on either side - does anyone in #Mathematics #Math #Maths #Teachers #Education #Textbook #Authors know of a collective noun that we use for this group of operator/operand(s)? I'm not sure I've ever seen one for it (other than the demonstrably wrong "Terms").
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1/7
Last week for #MathsMonday we showed that Terms are separated by any operators, not just + and -, contrary to the impression one might get from this #Maths textbook. This week we'll again #DoTheMaths but on binary and unary #Math operators, and deduce for ourselves the #Mathematics properties of them. As far as I'm aware, they don't have a name, so I'm gonna call them TorAnds - the group of an operator and associated operand(s) πŸ™‚

First note what the textbook says about an invisible + ...

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2/7
This is like not writing 1 - e.g. we write a, not 1a - and leaving out un-needed brackets. For clarity I'm going to include that first + sign for now

Recall also an expression is made up of operators and operands. An expression can only start with a unary operator (+ or -), for a binary operator (x or Γ·) MUST be preceded by another term - we can't start an expression with x5, there has to be a Term we're multiplying with the 5! So if there's not a -, the expression must start with +...

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3/7
So if an expression says 2+3, we know that's actually +2+3. This is important, due to left-associativity, if we want to re-arrange the expression. Every Term is associated with the sign (operator) on the left. If in 2+3, we want to put 3 first, what's before the 2? A +, so it's +2+3, re-arranged to +3+2 (3+2). This is important when there's a minus. 2-3(=-1) doesn't re-arrange to 3-2(=+1). Left associativity means +2-3 can be rewritten as -3+2 - both are equal to -1...
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4/7
So for unary operators, a TorAnd is the operand (Term) and the operator (sign) on the left (which, in the case of the first Term in an expression, if not preceded by a minus, is the unwritten +).

For binary operators, the sign is associated with TWO Terms, one on either side of it. For the first Term, this means that it is therefore both left associative AND right associative. i.e. in -2+3x4(=10), the 3 is both associated with the + and the x.+3x4-2=10, but +3-2x4=-5 and -2x4+3=-5,...

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5/7
So this means for a binary operator, the TorAnd consists of the operator and 2 Terms (one either side). e.g. in -2+3x4 our TorAnds are -2 and +3x4. I can rewrite the expression in any way, provided those TorAnds remain intact (otherwise I change the answer).

For 2+3 (unary operators) the TorAnds are +2 and +3 (for if we rewrite the expression, we need to remember there's an unwritten + on the left of the 2, which must be kept with it due to left associativity)...

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6/7
Some say in this case the written + (in +3) is "binary" because of the Term to the left, but what would that mean if so? That would mean the 2 is now associated with TWO plus signs (the written one, plus the unwritten one which it is already associated with) - do I add it twice? Clearly not - that would change the answer! Do I have to treat +2+3 as a TorAnd? Clearly not - I can write +3+2 and it's still equal to the same thing, so no association has been created...
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7/7
So, in other words, having a Term preceding a unary operator fails to fulfil any properties of a binary operator. It's just co-incidental - they can be in a different order and still the same answer. The 2 gets added not because of the written +, but the unwritten +. Starting at 0 on a number-line, there's a +2 jump, and a +3 jump, and can be done in any order.

And binary operators needing to be solved before unary operators is what gives rise to order of operations rule x Γ· before + - πŸ™‚