It seems like American English is a mish-mash of other languages and rules with English.  When you add in the quickly evolving slang, it makes me glad it is my first language.  It might be tough as a second language.

English is crazy:

What you send in a car (truck) is a Shipment
and what send in a ship is a Cargo

There is a book called "Crazy English" by Richard Lederer that is devoted to confusing words in English.  (E.g. A parkway is where we drive but a driveway is where we park.)

Proper context is always needed and helpful.  If I said, "Richard consults me," there would be no confusion about whether he is offering or asking for advice (because the word "consult" can mean either).  If I said "biweekly paychecks," we would automatically know it's every two weeks because we never get them twice a week.  So, context, personal experience, implicit understanding in a particular field, subject matter, or situation, and/or other peripheral knowledge often help us obtain a clear meaning, despite the imperfection of the language.

Landlord: I rented this apartment because I expected you to pay on time.

Tenant: I rented it because I didn't know you would raise the rent twice a year.

Neither person would ever be confused by the conflicting, opposite meaning of the phrase "I rent."

How about "next Friday" when you're talking on Monday? Is that four or eleven days away?
The crazy thing here in Colombia, when they mean in a week (that's Monday to Monday.), they say "in eight days".
As in, "I'll see you and bring your order with me in eight days." Me: '''How many days in a week here???
At first, it wasn't obvious to me that they count today as day 1.

@Matt, I tutor English to kids that went to private English schools here in Colombia. Most of their teachers were Spanish speakers to start with. I have a lot of work to do getting them ready for USA university entrance exams.

I prefer programming languages and math over English. No ambiguity.

It's like this stupid meme that bounces around Facebook from time to time... how do you solve this math problem:

6 / 2(1 + 2)

Some people say 9, others say 1. The people who say 9 are correct. The people who say 1 don't understand PEMDAS (or BODMAS for those of you in Europe).

From ChatGBT

PEMDAS is an acronym used to remember the order of operations in mathematics. It stands for:

Parentheses
Exponents (including roots, such as square roots)
Multiplication
Division
Addition
Subtraction

I say:
According to this Parentheses fiest (1+2) = 3,
Then Multiplication 2* (1+2) = 6
Then Division 6/6 = 1

Conflicting meanings of English words or phrases can have deadly consequences, as in an old murder case in England involving the phrase "Let him have it." Two criminals were confronted by the police, who asked them to drop their guns.  One criminal said to the other, "Let him have it." The remark was intended to mean that they should surrender their guns (let the cops have the guns).  Instead, the other guy misinterpreted it to mean that they should fight back, and let the cops "have it" (as in, attack them).  That guy opened fire, killed a cop, and they were tried and executed.  This and several other cases prompted the death penalty to be abolished in the UK.

You're right to do the parentheses first, so you get:

6 / 2(3)

Now at this point the parens don't matter. It's a simple multiplication which can be written as:

6 / 2 x 3

And at this point, you calculate left to right:

3 x 3

9

Let's see how GPT would answer it (out of curiosity):

GPT

The expression shown is:

6 / 2(1 + 2)

Let's break it down step by step following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right):

1. Parentheses first: 1 + 2 = 3, so the expression becomes 6 / 2(3)

2. Now handle the multiplication and division from left to right: 6 / 2 = 3, so now the expression is 3(3)

3. Final step: 3 x 3 = 9

Therefore, the solution to the expression is 9.

Some people arrive at 1 because of a different interpretation of the order of operations. The main point of confusion lies in how to handle the multiplication when it's directly next to parentheses, and whether to process it differently than normal division.

Here's the alternative reasoning:

1. Start with the expression: 6 / 2(1 + 2)
2. Parentheses first: 1 + 2 = 3, so the expression becomes 6 / 2(3)
3. Some people interpret this as 6 / (2 x 3), treating the multiplication by parentheses as having higher priority. So it becomes 6 / 6.
4. Finally, 6 / 6 = 1.

This interpretation comes from viewing 2(3) as one unified term and dividing by that entire term, rather than processing multiplication and division strictly left to right. However, the standard approach to order of operations dictates left-to-right calculation for division and multiplication, which leads to the answer 9.
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A lot of people think that because there are still parens there that they take priority, but they don't. It's the stuff INSIDE the parens that take priority. I remember arguing with my 5th grade math teacher over this one. LOL.

And also, PEMDAS should be written like this:

P
E
MD
AS

Parens go first. Exponents second. Multiplication & Division AT THE SAME TIME, left to right. Addition & Subtraction AT THE SAME TIME, left to right.

Parentheses dictate that the operations inside them should be solved first, but once the parentheses are gone, multiplication and division, as well as addition and subtraction, are handled in the order they appear, from left to right. Many people mistakenly think multiplication automatically takes priority over division because it's listed first, but they are actually on the same level. The same goes for addition and subtraction.

I feel the need to make a video about this now. LOL.

While we're on the topic, I asked GPT "are there any other less common rules about the order of operations in math that I should know about?"

GPT

Here are a few additional important rules and clarifications about the order of operations in math that can help avoid common misunderstandings:

Implied Multiplication: Sometimes multiplication is implied without the use of the "x" or "*" symbol, especially when a number is written next to parentheses, like in the expression 2(3). Although some interpret implied multiplication as having a higher precedence than division, it should still be handled according to the left-to-right rule for multiplication and division.

Fractions as Grouping Symbols: In some expressions, a fraction bar acts as a grouping symbol, meaning everything in the numerator (top) and everything in the denominator (bottom) should be treated as if they were in parentheses. For example:

(3 + 5) / (2 + 4) = 8 / 6, so the operations in both the numerator and denominator must be completed before performing the division.

Nested Parentheses or Brackets: If there are multiple layers of parentheses, brackets, or braces, work from the innermost parentheses outward. For example, in:

2 * [(3 + 4) * (6 / 2)]
You would first solve the inner parentheses (3 + 4) and (6 / 2), then perform the multiplication.

Exponents and Roots: Exponents and roots should be handled after parentheses, but before multiplication, division, addition, and subtraction. Also, remember that exponents apply only to the immediate number or term to their left, unless grouped by parentheses. For example:

-3^2 = -(3^2) = -9 (the exponent only applies to 3, not the negative sign).

Exponents in Parentheses: If an exponent is applied to a value in parentheses, it applies to the entire expression inside the parentheses. For example:

(3 + 2)^2 = 5^2 = 25.

Factorials: Factorials (!), like 5!, are not as common but should be handled immediately after the number they follow before applying any further operations. Factorials involve multiplying all whole numbers from 1 up to that number. For instance, 5! = 5 x 4 x 3 x 2 x 1 = 120.

By keeping these additional rules in mind, you can tackle a wide range of expressions more confidently and correctly!
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9= (6 / 2) * (1+2) 'When in doubt, I write it like this. So many programming languages and calculators get it wrong.
or
1= 6 / (2 * (1+2))

Richard If you think English is ambiguous, then you want to stay away from studying law.
It's such a mess, and (my pet peeve) the Supreme Court refuses to take up cases where different subordinate courts arrive at different opinions based on similar facts. It's crazy.

Yeah, a lot of math is how it's written. People need to learn how to WRITE math equations too. I wish the rules were the same as for programming. ONE WAY to do things.

@Richard One Way! Yes, that's how my old programming languages used to be. Keeps them lean. And then along comes Microsoft, five different ways to do anything, and many times, one or two have bugs.

The problem with Richard's equation is not the order of operations, but rather the ambiguity of using a slash for a division sign.  

In higher formal mathematics division is ALWAYS converted to multiplication by a fraction.  Only horizontal lines are used (i.e., never a "/").  clear demarcation is made to where the fractor (or quotient) line begins and ends.  

For very rudimentary arithmetic, one can occasionally use a standard keyboard; however, even for basic algebra that is not the case.  This is because of the commutivity property of multiplication which states that A*B = B*A and the A*B*C= A*C*B=B*A*C=B*C*A=C*A*B=C*B*A.  In other words, it is a fundamental principle that multiplication can be done in any order without changing the results.

Without additional parenthesis, it is impossible to say what the equation intends.

It should also be noted that the Left-to-right concept is entirely false.  Both multiplication and addition are commutative.  Division is merely multiplication by a fraction, and subtraction is merely the addition of a negative number.  Therefore, all arithmetic can be re-written using only multiplication and addition.  Following the principal of commutativity, the only order that matters is that multiplication is done prior to addition.

The let to right concept is only found in the false linearity of computers which cannot see the equation except in that strict linear manner.  Sadly, many programmers learned that false rule and use it in their code instead of properly and formally including the REQUIRED parenthesis to remove ambiguity.  It is a lazy way of programming that has caused and still causes many computing errors.

When using "/" to denote a fraction, BOTH the numerator and the denominator MUST be placed within parenthesis in order to know where each begins and ends.  Left as Richard initially wrote the equation, it is impossible to answer because we do not know what the denominator actually is.  PEMDAS has no bearing because the equation is syntactically incomplete.

Getting back to the bi vs semi question, it is a matter of understanding context before applying meaning.

Semi refers to one-half (1/2) while bi only implies two possibilities.  Therefore, semi-monthly mean every one-half month or roughly 15 days.  Bi-monthly depends upon context.  It could mean two times a month, or it could mean every two months.  Both offer a two point possibility.  Generally, bi- should be used for whole values (i.e., bi-monthly meaning every two months), but again it is entirely context driven so it is imperative that the context is made unambiguously clear.  Even in higher maths we have to be careful and clear when using bi-.  Generally, we either used a very well established term, e.g., bi-nomial, or we avoid its use altogether.  

Semi- on the other hand is always "to divide into 2, not necessarily equal, pieces".  Hence you often see things described as semi-solid (highly viscous gels etc.), semi-precious (stones of higher value but not precious gem stones), etc.  That is why large cargo trucks are called semi-tractor-trailers.  The two pieces are very different in size shape, purpose, etc.; yet, they are each a complement to the other.  Two halves of one whole if you will.  Thus we have Semitrucks - the truck is "half" of the larger total vehicle.

No one uses "false rules" by using "/".  It's done simply for practicality -- a computer keyboard simply doesn't have and can't have every single math symbol.  So in computing coding, math expressions have to be written differently.  This is scarcely worth mentioning, and nothing to protest about.  Tons of other math symbols can't be typed on a computer either: sigma, d/dx, integral sign, etc.

All very interesting points. It's late, and I'm tired, so I asked GPT what it thought...

Michael raises a few points, but there are some misunderstandings in his argument.

1. Division Sign and Slash:
   It's true that in higher-level mathematics, formal notation typically uses horizontal fraction bars instead of slashes ("/") to avoid ambiguity. However, the slash is widely accepted in basic arithmetic and algebra, especially in non-professional or keyboard-limited contexts. The use of a slash doesn't inherently introduce ambiguity as long as the expression is written clearly and properly follows the order of operations.

2. Ambiguity and Parentheses:
   Michael mentions that without additional parentheses, it's impossible to know what the denominator is, but in the equation 6 / 2(1 + 2), the use of parentheses to indicate multiplication (2(3)) is standard and clear. The confusion arises not from the notation but from a misunderstanding of how the order of operations applies. As mentioned before, the correct interpretation following the order of operations is to evaluate the expression left to right.

3. Left-to-Right Rule:
   Michael argues that the left-to-right rule is false and claims multiplication and addition are commutative, meaning their order doesn't matter. While multiplication and addition are commutative (A B = B A and A + B = B + A), this doesn't apply to division and subtraction, which are not commutative. In the expression 6 / 2(1 + 2), division needs to be handled first, moving from left to right, before multiplication.

   The left-to-right rule is not about linearity for computers; it is a basic principle of arithmetic for handling division and multiplication, or addition and subtraction, when they are on the same level. Computers do follow this rule because it's the mathematical convention.

4. Rewriting Arithmetic:
   Michael is correct that division can be expressed as multiplication by a fraction and subtraction as the addition of a negative number. But that doesn't eliminate the need for the left-to-right rule when dealing with division or subtraction in their usual forms. For instance, 6 / 2 is not the same as 2 / 6, so the order in which division occurs does matter.

5. Ambiguity in Richard's Equation:
   The argument that the equation is ambiguous without parentheses is not quite right. In Richard's original equation, 6 / 2(1 + 2), the parentheses clarify the operation inside them (1 + 2), and after resolving that, the rest of the operations are handled left to right. If Michael's issue is with using the slash instead of a horizontal fraction bar, it's more about style and clarity, but not a fundamental problem with the order of operations.

6. PEMDAS Application:
   Michael claims PEMDAS doesn't apply because the equation is incomplete, but in fact, it does apply here. After resolving the parentheses (P), the multiplication and division (MD) are handled from left to right.

In conclusion, while Michael brings up valid points about formal mathematical notation, the crux of the issue is not the notation but the correct application of the order of operations. The left-to-right handling of division and multiplication is well-established and applies here, leading to the correct answer of 9.

I will reiterate that I am NOT a mathematician. I only formally took math up thru first year college Calculus. So I'm not a math whiz. I learned the rest of what I know thru computer programming books. So if GPT and I are wrong, and you are a mathematician, please let me know.

Actually, I am a mathematician. My second degree is in Applied Mathematics from the University of Houston. My first is in Electrical Engineering.

Richard, you are NOT wrong

Kevin Yip states that, "a computer keyboard simply doesn't have and can't have every single math symbol.  So in computing coding, math expressions have to be written differently."  This is in fact correct.  However, that means we must be unambiguously clear in the annotation.  Sadly, Richard's original statement, "6 / 2(1 + 2)" is not unambiguous.

It could be either 6/ (2(1+2))  0r it could be (6/2) (1+2).  BOTH are valid interpretations, and without the second set of parenthesis, there is no way of knowing what is intended as we do not know what the denominator is meant to be. Back in the 1970s and early 1980s is was actually quite difficult to write/print mathematical notation and would often require multiple lines:

2                2
x  + 2ab + y
-----------------  = Z  <an arbitrary equation> :  Modern> ((x^2) + 2ab + (y^2)) / ((x^2) - (Y^2)) = Z
   2     2
x   - y

Every parenthesis in the "Modern" equation is required to avoid ambiguity.  Therefore we could write Richard's statement as:

    6                    6
---------      or      --  (1+2)     Again both would be valid interpretations because it is unclear as to what the denominator is
2(1+2)                2               meant to be.

While I have found some textbooks listing the "Left to right" rule, many others do not.  The reason being that it is a modern oversimplification (similar to always rounding "5" upwards) of proper mathematical notation and the order of operations.  once division is defined by clearly established fractions, left to right is no longer applicable.

While the slash is now commonly used for basic arithmetic's that does not negate the fact that it does introduce ambiguity as shown above.  With the advent of modern computers, the requirement of additional parenthesis when writing mathematical statements has become a major point of contention -- to the point that there have been several major court cases around the globe.  In every case, the courts have found that the additional parenthesis are REQUIRED to eliminate ambiguity and prevent harm.  This is why when papers are typeset for formal scientific and mathematical works the slash is not used.

I should point out that I am also an Engineer and Mathematician.  This subtle issue is one I have often taught.  While many of my colleagues think such subtleties not worth consideration, others have shown many examples of real world practice where serious damage or harm has occurred because of ignoring such subtleties.  Therefore I stand by my original analysis - the problem is not a matter of the order of operations but the ambiguity of the original statement.

I forgot to address Richard's point 4. Rewriting Arithmetic above.

I agree the 6/2 is not the same as 2/6.  These are two independent and unequal fractions.  However

6            1
-- = 6 *  --  which is properly re-written as multiplication by a fraction and
2            2

1                1
-- * 6 = 6 * --   This properly represents my statement that once written as multiplication by a fraction it is in fact commutative.
2                2

Again, it is subtle and requires meticulous application of all rules.  Maths do have many efficiencies, but there are no "short cuts".  Every rule must be properly taken into account (i.e. content and context) every time.  It is a corner stone of the science.  It is what allows Proofs to be possible.  It also prevents Maths from dealing with the more esoteric questions.  For example, "What is life?"; "Whom am I referring to when I say 'My arm', 'My life', etc.?"; "What is the concept 'Blue'?"  Granted these are both oversimplified examples and under generalized philosophy questions.  I use them only to show that Maths must have a means of measure or they can do nothing.

I agree that how the math equation is written is vitally important. This is just simply a dumb meme that goes around on Facebook all the time with different equations that are poorly written. And I'd like to think that math has one simple straightforward method for solving even problems like this that are written badly by the author. That's where I think PEMDAS and then left to right is the way to go and it removes ambiguity. But that's just my opinion... but again I'm NOT a mathematician. LOL

By the way, thanks everyone for a well spirited discussion.  Excellent defense of your position with virtually no "mud slinging".  Our presidential candidates could learn much from all of you.

Richard, nothing but respect here.  Both the initial semi- vs bi- and the math statement are fantastic examples of how tricky and complex communication can be.  I am constantly seeking clarification to ensure I understand what my clients want, what constraints they are imposing, and what the scope of their project is.  Even then I don't always get it "right".  So I appreciate these discussions.  You and your team always help me grow.

Well said Michael :)

Oh, I love a good, spirited, mature, adult, friendly, rational debate. And if I'm wrong, I'm the first to admit it. You're right... All politicians should follow suit.