8divided By 2 2 2

Stive has a 1st Class Honors Electronics Engineering caste 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 disruptive than yous'd think!

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A Challenge

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

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

The United states PEMDAS mnemonic stands for Parentheses, Exponents, Multiplication, Segmentation, Improver, Subtraction. The Britain(+) BODMAS mnemonic stands for Brackets, Orders or Of, Division, Multiplication, Addition, Subtraction.

P and B hateful the same affair. 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.

Eastward and O mean the same matter. 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.

Basic Math

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

That 8 ÷ 2 ten (2 + 2)

= 8 ÷ 2 x 4

= 4 x 4

= 16

Next Level Math

The post-obit can also be proven to exist true.

That 8 ÷ ii(2 + 2)

Coil to Continue

Check Your Answer (Proof #one)

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

Everyone who can recall the, dreaded by some, algebra, from those glorious school days, will probably remember the phrase "cheque your respond".

Having solved an equation, for example, for a value for 10, it was and then necessary to bank 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 dominion, we were instructed to perform a crude calculation of the equation, to ensure that our answer was in the right ball park and that the decimal signal was not in the wrong position.

And similarly over again, in the equation under word, viii 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 requite a result of 16 unless the rest of the equation tin be shown to exist a fraction, which a ii, a 4 and a fix of parentheses, clearly are non.

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

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

The Parentheses SHOULD be and MUST be Fully and Completely EVALUATED and not just solved past calculating only the values inside the parentheses.

In our problem, this means that 2(2+2) = two(4), and to complete the evaluation, = viii, equally the finished article. This is because, calling on the uncomplicated "touching" rule every bit 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 consequence cannot exist left as two(four) to exist later, incorrectly, separated into "2 ten 4" as two contained, separable numbers.

As an After-Thought, I will suggest that the expression 2() actually ways "two of ()" or "2 of these ()", which could exist a 'new' 'OF' rule, and should always be interpreted and calculated as such and hence must never ever be separated into 2 10 iv as ii contained numbers.

Calculators are only as good as the input. — Calculators are merely every bit proficient as the input.

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Juxtaposition Dominion (Proof #3)

In the Juxtaposition Rule, the full general consensus among many math fraternity members is that "multiplication by juxtaposition" or "multiplying past putting things side by side 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 earlier calculating or processing any other operations with the exception of exponents on the juxtaposed values.

This means that, even if we incorrectly condone the Fully Evaluate Proof#2, the ii(4) expression would still need to be multiplied out earlier using the concluding left to correct rule.

This rule would essentially necessitate that PEMDAS/BODMAS be adapted to be PJEMDAS/BJODMAS just would notwithstanding go out inherent problems with any exponents on J values so adaptation is overlooked.

PEMDAS/BODMAS are Guidelines Not Strict Rules

Mnemonics are adjutant-memoires and are not meant to exist strictly followed to the letter of the alphabet without deviations, for case, the trigonometry SOHCAHTOA mnemonic only applies iii of the nine symbols per usage.

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

Even more math formulas — Even more than math formulas

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

There can ultimately only be a unmarried 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(two + 2) portion can be calculated,

EITHER, using the Touching or Juxtaposition rules,

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

OR, using the Distributive Holding Dominion,

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

As tin can be easily seen, BOTH methods reveal an answer of 8 for the equation later on the separate sign.

Hence both the to a higher place methods are then successfully calculated to completion as

8 ÷ viii = ane.

Nested Brackets (Proof#5)

Now that we are aware that 2(4) must = 8, and that 8 ÷ 2(four) must = one, nosotros can conspicuously meet that calculators and spreadsheets mishandle due north(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 respond.

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

At that place are some arguments that say that eight ÷ ii(ii+2) is ambiguous or is not correctly written downwardly but they are nonsense. It really 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..

Ultimately

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

If 8 Apples (A) are divided between 2 Classrooms (C) with each Classroom (C) containing two Girls (Thou) 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) = ?

viii ÷ 2(2 + ii) = one

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

I volition suggest that the outside two in the 2(2 + 2) part of the equation is not a numerical two simply is merely a symbol with a value of ii much the aforementioned equally the 2 in H₂O and should exist evaluated similarly.

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

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

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