The Unwritten Grouping in Fractional Forms
Purplemath
Shortly before composing this page of the lesson, I received an email whose author claimed that the order of operations is actually a diabolical plot, recently hatched by a cabal of math teachers in order to destroy students' ability to succeed in science.
The "proof" of this alleged conspiracy was the fact that fractions, when written vertically, don't generally have parentheses bracketing their numerators and denominators; in other words, there are no grouping symbols around their tops and bottoms. Seriously.
So let's handle this alleged issue. First, let's refresh ourselves on the meaning of fractions.
Fractions mean something; specifically, when they display "(all of this stuff on top) over (all of that stuff underneath)", they're telling us that "(all of this stuff on top) is being divided by (all of that stuff underneath)". And the "all" parts mean "all of it goes together", so we can think of the top and bottom expressions has having grouping symbols around them. It's just that, when fractions are typeset vertically (as opposed to typed out sideways (say, in an email), we don't generally include those grouping symbols, because the vertical formatting makes the meaning clear.
With this "understood" grouping in mind, recall the order of operations: We first have to handle any parentheses, simplifying inside them, before we can proceed to the other bits of a given expression. We cannot reach inside a parentheses, "understood" or otherwise, and rip out part of an addition or subtraction, and try to "cancel" that shredded portion with something else. We can cancel only factors, and we can do said cancellation only after having simplified inside the parentheses.
- Simplify the following:
![−9 + (9 + 20)/[3(5)] − (−3)](orderops/order06a.gif)
The middle term of this expression is a fraction. Keeping in mind the fact that fractions mean something, I know that I should not try to cancel the 3 in the denominator of that fraction with the 9 in the numerator. To do that, I'd have to reach inside the "understood" parentheses and rip the 9 off of the "+20", and that would be just wrong. In the same way, I should not try to cancel the 5 in the denominator with the 20 inside the sum in the numerator; this would not be mathematically legitimate.
Instead, I must first simplify the implicit (that is, the unstated and unmarked, but still "understood") groupings which are the top and bottom (that is, the numerator and denonimnator, respectively) of the fraction. The 9 + 20 becomes a 29, and the 3(5) becomes a 15. Then I can proceed as usual: I combine the integers, convert the results to a common-denominator fraction, and simplify to get a single value.
![−9 + [(9 + 20)/(3(5))] − (−3); −9 + [29/15] + 3; −6 + [29/15]; −[90/15] + [29/15]; −61/15](orderops/order06.gif)
To reiterate: By typesetting the fraction vertically, as shown in the original exercise above, we had created groupings; namely, the group of "everything above the line" and the group of "everything below the line". This grouping is implicit, so parentheses are not (generally) used, though the following would mean exactly the same thing as the fractional term in the middle of the previous expression:
![(9 + 20) / [3(5)]](orderops/order13.gif){kind=small}
When we reformat the vertical fraction horizontally (say, for typing it as plain text in an email or a forum posting), we must remember convert the (vertically) implicit grouping into an (horizontally) explicit grouping, or this grouping could be lost or at least misunderstood.
This conversion to explicit typed-as-text form might look like:
(9 + 20) / [3(5)]
Whether we use square brackets, round parentheses, or curly braces, or use some or lots of spacing, is not the point. The point is that we are aware of the "understood" parentheses implicit in vertically-formatted fractions, and that we never violate the order of operations either by trying to reach inside those parentheses in our work or else by making it look like we're breaking the rules because of how we (mis-)formatted our typing.
In other words, when typing horizontally, use sufficient grouping symbols to make your meaning clear.
- Simplify the following:
![[3 + (15 ÷ (−3))] / 16](orderops/order07a.gif)
This is just one fractional term. I'll simplify the top, working from the inside out. This means that I'll do the division first — because it's inside explicit grouping symbols — and then the addition. When I'm finished with that, I'll reduce, if possible.
![(3 + [15 รท (−3)])/16; (3 + [−5])/16; (−2)/16; −1/8](orderops/order07.gif)
When simplifying expressions involving fractional forms, remember to check, when you're done with the order-of-operations stuff, to see if the fraction can be reduced. Don't miss easy points by forgetting.
By the way, the expression for this exercise could be sideways-typed as:
[3 + {15 ÷ (-3)}] / 16
- Simplify the following:
I need to remember that there are "understood" parentheses around the "4 + 8" in the numerator and the "2 + 1" in the denominator. I must not try to "cancel" the 2 with the 4 or the 8. That won't work!
Instead, I'll work bit by bit, simplifying first inside all three sets of parentheses (two "understood", and one explicit).
![[4 + 8]/[2 + 1] − (3 − 1) + 2, [12]/[3] − (3 − 1)) + 2; 4 − (3 − 1) + 2; 4 − 2 + 2; 4](orderops/order08.gif)
Notice how I was careful with the fraction. For instance, I did not try to cancel the original denominator's 2 with anything. I had first to simplify the denominator, adding the 2 and the 1 to get 3 underneath, before I could attempt to do anything with the 4 + 8 on top.
By the way, the original expression above could be typed out horizontally as:
[4 + 8]/[2 + 1] − (3 − 1) + 2
-
Simplify the following:
Can I start by cancelling the 2 in the denominator with the −4 in the numerator, or the 22 = 4 in the denominator with the 16 in the numerator? Of course not; that would be silly (and wrong). First, I must simplify inside the "understood" parentheses; that is, I must first simplify the top and the bottom of the fraction separately. Only then can I consider possible cancellations.
Since 16 − 4 = 12, then:
![6 + [16 - 4]/[2^2 + 2] − 2; 6 + [12]/[4 + 2] − 2; 6 + [12]/[6] − 2; 6 + 2 − 2; 6](orderops/order09.gif)
Do not try to cancel the 4 or 2 into the 12; instead, first add the 4 and 2 to get 6. Only then can you cancel.
By the way, the original expression above could be typed into an e-mail as:
6 + [16 − 4]/[2^2 + 2] − 2
In general, take your time, respect the rules, work from the inside out, be careful with your "minus" signs, and respect all grouping symbols, "understood" or otherwise. If you do this, then you should be very successful with simplifying using the order of operations.
And, when typing mathematics sideways, be generous with grouping symbols. Make your meaning clear.
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