8 ÷ 2(2 + 2) The Viral Equation Has Only One Answer and That Is 1 Not 16

Updated on November 12, 2019
Stive Smyth profile image

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

Gear Head
Gear Head | Source

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 | Source

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

Mathematics Word Cloud
Mathematics Word Cloud | Source

Next Level Math

The following can also be proven to be true.

That 8 ÷ 2(2 + 2)

= 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 | Source

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 | Source

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.

Math Formulas II
Math Formulas II | Source

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.

Math Formulas III
Math Formulas III | Source

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.

Math in Technology
Math in Technology | Source

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 | Source


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


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://maven.io/company/pages/privacy

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)