8 ÷ 2(2 + 2) = 1 and Only 1. The Sad Legacy of Calculator Dependence.
Back In the Day
Back in the day, when I attended school, calculators did not exist to become reliant upon. For this reason the math that was learned at school was a practical math that could be applied in simple, real life situations, somewhat like an applied math. It was not simple number crunching to obtain an answer to a problem that was perceived as correct but wasn’t tested for correctness.
Thus we learned things like this –
8 ÷ 2 x (2 + 2)
= 8 ÷ 2 x 4
= 4 x 4
This is a very simple example of how to apply simple ‘rules’ known variously as PEMDAS or BODMAS and similar, which are actually only variable guidelines and not strict rules, and then to follow-up with the left-to-right rule, which is fixed.
We also learned to think beyond the ‘rules’, to ‘think outside the box’, and to adapt PEMDAS/BODMAS guidelines in various situations as necessary.
Thus we also learned this –
8 ÷ 2(2 + 2)
= 8 ÷ 2(4)
= 8 ÷ 8
The practical implications of knowing, realizing, understanding, or at least, accepting, that the PEMDAS/BODMAS ‘rules’/guidelines were to be interpreted and not just simply applied in strict fashion were to become, sadly unnoticeably, far-reaching.
That the P/B element must be intelligently or complexly applied to be ‘wholly or fully evaluated’, and not simply applied to calculate only the parentheses’ contents, enabled math to move from the classroom to practical areas.
That 2(2 + 2) = 8 by whatever interim or extraneous means a person chooses, either the Touching Rule, Juxtaposition Rule, Distributive Property Rule, or my recently-suggested Of Rule, allowed for its use in real-world situations.
Examples or real-world situational usage –
If a teacher has to divide 8 Apples (A) between 2 Classrooms (C) with each Classroom (C) containing or consisting of 2 Girls (G) and 2 Boys (B), how many Apples (A) would each student receive ?
8A divided between 2C, each with 2G and 2B = ?
8A divided between 2C(2G + 2B) = ?
8A ÷ 2C(2G + 2B) = ?
8 ÷ 2(2 + 2) = 1
Imagine, in the heat of a past battle, that a newly-assigned runner was instructed to evenly distribute “that stack” of cartridge boxes among the gun stations or turrets. If he counted 16 in the “stack”, obviously knew that there were 2 sides to the ship, and was then informed that each side had 2 forward and 2 rear turrets, he could use the same calculation and receive 2 as the answer to be given to each turret.
16 ÷ 2(2 + 2)
= 16 ÷ 2(4)
= 16 ÷ 8
This would clearly be far quicker and easier for him than having to run to each turret, drop off one cartridge box, and then continue distributing, one at a time, until the stack was cleared.
Imagine a young nurse being handed the key to the medicine cabinet cart/trolley and instructed to evenly distribute the pills in the storage container labelled “afternoons”, for example, to each bed in the wards that she was responsible for. If she counted the pills as 8 total, knew that 2 wards were in the instructions and that each ward had 2 beds down each side, she could use the same calculation and receive 1 each as the answer.
8 ÷ 2(2 + 2)
= 8 ÷ 2(4)
= 8 ÷ 8
These were three simple examples of math being put to practical use and of all users happy that they learned something useful in their math lessons after all.
Now imagine that all three people in the examples used the incorrect calculator-era method to obtain an incorrect answer. Instead of answers of 1, 2, 1, they would incorrectly obtain answers of 16, 32, 16, and would be aghast that the math they learned was impractical and would be left wondering why they wasted their time learning number crunching with no practical value.
Enter the Calculator
The history of the calculator is interesting. The first solid-state calculators appeared in the early 1960s with the first pocket calculators launching in the early 1970s. With the arrival of integrated circuits, pocket calculators were affordable and already fairly commonplace during the late 1970s.
Some early calculators were programmed to calculate 2(2+2) as =8 which agreed with the pre-calculator manual method.
Then, inexplicably, calculators began to surface which would strangely separate a keyed-in input of “2(2+2)“, i.e. "2(no-space)(...", and would replace it with “2x(2+2)“, i.e. "2(times-sign)(...", and would then clearly produce an incorrect answer.
The clue to the different answer outputs is whether the calculator inserts a multiplication sign or not.
If it does not insert a "x-sign", then the answer will be correct.
If it does so, then the input will need to use an extra set of parentheses known as nested brackets, as shown here: (2x(2+2)), to force the desired output.
Calculators and computers are actually only as good as their input, the numbers and symbols that are keyed in. This phenomenon has been known for decades, among programmers in the computer science fraternity. The term used is GIGO which stands for Garbage-In, Garbage-Out and which is a subtle way of saying that, to obtain a correct output, the inputted data must be in an acceptable format.
I sincerely believe that we should rethink the teaching methods of the generations of so-called “modern math”, as some YouTubers refer to it, but what they are actually meaning is “calculator-era math”. Allowing them, and previous graduates, to believe that 16 is the correct answer, will possibly have some semi-serious repercussions for STEM students and graduate future designers, and will have a knock-on effect for the general public, as is already happening.
Questions & Answers
© 2019 Stive Smyth