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8 ÷ 2(2 + 2) The Viral Equation Has Only One Answer—It's 1, Not 16

Stive has a 1st Class Honors Electronics Engineering degree including a high level math module and also masters level modules, from the O.U.

Equations can be a lot more confusing than you'd think!

Equations can be a lot more confusing than you'd think!

A Challenge

My arguments and proofs below are in reality a challenge to most of the calculator manufacturers and spreadsheet programmers who, for too long, have assumed that "2()" can be always evaluated to "2 x ()". This is true in simple equations but in complex equations, which call for the PEMDAS/BODMAS, is true only when the "2()" is the first item.

They have failed the general public and allowed them to believe that the assumption is true and have failed to instruct them, in the user manuals, on the necessary use of nested brackets when inputting complex equations.

The USA PEMDAS mnemonic stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The UK(+) BODMAS mnemonic stands for Brackets, Orders or Of, Division, Multiplication, Addition, Subtraction.

P and B mean the same thing. The P is for "Parentheses" because parentheses are the usual and most common brackets seen in equations. B for "Brackets" allows the inclusion of any major types of brackets such Parentheses (curved Brackets), Square Brackets ( [ ] ), and Braces or Curly Brackets ( { } ) which are also used.

E and O mean the same thing. The E for "Exponents" is equivalent to O for either "Orders" as in "To the Order Of" or "Of" as in "To the Power Of" which both mean exponents.

Calculators can be complex

Calculators can be complex

Basic Math

Those who understand basic math will acknowledge the following to be true...

That 8 ÷ 2 x (2 + 2)

= 8 ÷ 2 x 4

= 4 x 4

= 16

Next Level Math

The following can also be proven to be true.

That 8 ÷ 2(2 + 2)

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= 8 ÷ 2(4)

= 8 ÷ 8

= 1

My argument revolves around the fact that the 2(4) is an expression consisting of inseparable numbers and is not the same as "2 x 4" which are two separate, individual number values which can be worked-on separately.

Basic math operators

Basic math operators

Check Your Answer (Proof #1)

In my first argument I will discuss earlier math from the mid to late 20th century.

Anybody who can recall the, dreaded by some, algebra, from those glorious school days, will probably remember the phrase "check your answer".

Having solved an equation, for example, for a value for x, it was then necessary to check the value obtained by inserting it into the original equation and testing for the correct result.

Similarly, in the pre-calculator days of the slide rule, we were instructed to perform a rough calculation of the equation, to ensure that our answer was in the right ball park and that the decimal point was not in the wrong position.

And similarly again, in the equation under discussion, 8 divided by something, must reveal an answer of 1 or less unless the rest of the equation is a fraction.

Hence 8 divided by something, cannot give a result of 16 unless the rest of the equation can be shown to be a fraction, which a 2, a 4 and a set of parentheses, clearly are not.

In the YouTube (incorrect) attempts at "proof", most of the narrators state, "In modern math, the answer is 16". Modern math is actually more than 100 years old so they are apparently referring to 'calculator-era' math and they are incorrectly applying a left to right rule without including either the simple "touching" rule or the juxtaposition rule or essential nested brackets which are all discussed later.

Math formulas

Math formulas

Fully Evaluate the Parentheses - Don't Calculate Only the Values 'Within" (Proof #2)

The Parentheses SHOULD be and MUST be Fully and Completely EVALUATED and not simply solved by calculating only the values within the parentheses.

In our problem, this means that 2(2+2) = 2(4), and to complete the evaluation, = 8, as the finished article. This is because, calling on the simple "touching" rule as an extra aid, the 2 touching the parentheses (in contiguous position), without a multiplication sign, is an inclusive and inseparable part of the parentheses function.

The intermediate result cannot be left as 2(4) to be later, incorrectly, separated into "2 x 4" as two independent, separable numbers.

As an After-Thought, I will suggest that the expression 2() actually means "2 of ()" or "2 of these ()", which could be a 'new' 'OF' rule, and should always be interpreted and calculated as such and hence must never ever be separated into 2 x 4 as two independent numbers.

Calculators are only as good as the input.

Calculators are only as good as the input.

Juxtaposition Rule (Proof #3)

In the Juxtaposition Rule, the general consensus among many math fraternity members is that "multiplication by juxtaposition" or "multiplying by putting things next to each other" so that they are contiguous, as opposed to utilizing a times or "×" sign, indicates that the juxtaposed values must be multiplied together before calculating or processing any other operations with the exception of exponents on the juxtaposed values.

This means that, even if we incorrectly disregard the Fully Evaluate Proof#2, the 2(4) expression would still need to be multiplied out before using the final left to right rule.

This rule would essentially necessitate that PEMDAS/BODMAS be adapted to be PJEMDAS/BJODMAS but would still leave inherent problems with any exponents on J values so adaptation is disregarded.

PEMDAS/BODMAS are Guidelines Not Strict Rules

Mnemonics are aide-memoires and are not meant to be strictly followed to the letter without deviations, for example, the trigonometry SOHCAHTOA mnemonic only applies three of the nine symbols per usage.

Similarly PEMDAS/BODMAS are sets of guidelines to be applied in conjunction with other important rules (Touching or Juxtaposition) and are not strict rules to be applied whilst disregarding other mathematical rules, and are often applied circularly.

Even more math formulas

Even more math formulas

There is Only One Answer to An Equation – Distributive Property Rule (Proof #4)

There can ultimately only be a single answer to a mathematical equation problem, no matter how many different, correct, methods are used to arrive at the final answer.

In our given problem the 2(2 + 2) portion can be calculated,

EITHER, using the Touching or Juxtaposition rules,

as 2(2 + 2) = 2(4) = 8

OR, using the Distributive Property Rule,

as 2(2 = 2) = (4 + 4) = 8

As can be easily seen, BOTH methods reveal an answer of 8 for the equation after the divide sign.

Hence both the above methods are then successfully calculated to completion as

8 ÷ 8 = 1.

Nested Brackets (Proof#5)

Now that we are aware that 2(4) must = 8, and that 8 ÷ 2(4) must = 1, we can clearly see that calculators and spreadsheets mishandle n(m) expressions in complex equations.

To counter this problem we must use Nested Brackets, sadly, to force the calculators to provide us with the correct answer.

Thus we must input 8 ÷ (2(2+2)) to receive an answer = 1.

There are some arguments that say that 8 ÷ 2(2+2) is ambiguous or is not correctly written down but they are nonsense. It actually is correct for all who understand either the new OF rule or the Touching or the Juxtaposition rules and that PEMDAS/BODMAS is only a guideline..

Pyramids Joke

Pyramids Joke


Ultimately, taking a problem back to basics can be revealing.

If 8 Apples (A) are divided between 2 Classrooms (C) with each Classroom (C) containing 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

The 2() is But Is a Symbol with Value 2 – Change My Mind

I will suggest that the outside 2 in the 2(2 + 2) part of the equation is not a numerical 2 but is merely a symbol with a value of 2 much the same as the 2 in H2O and should be evaluated similarly.

Thus we could write 2(2 + 2) which would mean 2 items but by no means it would mean an individual, removable 2, such that we would interpret it as ((2+2)+(2+2)) or as Double(2+2), or Dbl(2+2), or D(2+2).

As can be seen, the three "D" expressions would not work in calculators or spreadsheets and the ((2+2)+(2+2)) is cumbersome.

Hence we use the shorter, more manageable version of 2(2+2), still with an immovable outside 2, which must be made forced-immovable in calculators and spreadsheets by encapsulating it thus (2(2+2)).

© 2019 Stive Smyth

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