The Order of Operations: Why 3 – 2 + 1 = 2 not 0

For Mathematical Minds

The Order of Operations: Why 3 – 2 + 1 = 2 not 0

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When people look at a mathematical expression it must mean the same thing to them or they won’t be able to talk about it, work together effectively, or even agree on puzzle solutions! To have that common understanding, they must agree what order will be used to execute operations.

“4 + 6” is no problem! Everyone knows what numbers mean and that “+” means addition, so everyone will see that 4 + 6 means adding 4 and 6, and the result will be 10. There’s only one thing to do, so there’s no choice of what to do first. But if an expression has more than one operation, as it can in Mathler and HardMathler, it might make a difference which operation is done first.

Example 1: 20 – 4 + 3

A. If you do the subtraction first: 20 – 4 = 16; then add 16 + 3, you get 19 BUT

B. If you do the addition first: 4 + 3 = 7; then subtract 20 – 7 you get 13 !

Example 2: 12 + 4 *6

A. If you do the addition first: 12 + 4 = 16; then multiply 16 * 6 you get 96 BUT

B. If you do the multiplication first: 4 * 6 = 24 and then add 12 + 24 you get 36

Example 3: 6 / 3 * 2

A. If you do the division first, 6 / 3 = 2, then multiply 2 * 2 you get 4. BUT

B. If you do the multiplication first 3 * 2 = 6, then divide 6 / 6 you get 1.

If you already know the standard order of operations, just for fun choose whether A or B is the correct way to evaluate each example. (Answers in discussion below.)

The “Order of Operations” for Mathler Puzzles

  • First, if there is an expression inside parentheses evaluate that.
  • Second, evaluate multiplications and divisions from left to right.
  • Third, evaluate additions and subtractions from left to right.

Do you think you need to change any of your choices of how to do the examples above? (Last chance before the solutions!)

In Example 2 the correct answer is B: there are no parentheses; the multiplication is in the next level of precedence, so do the multiplication, then the addition:

12 + 4 * 6
= 12 + 24
= 36

Your author, a member of the MathlerHelper team at Arundel Mobile Professionals. taught prealgebra to adults for many years. She found that many adults mis-remember or had been allowed to mis-learn, or had even been mis-taught (!) the order of operations, and as a result mistakenly thought all multiplications are done before any divisions and all additions before any subtractions. This can get in the way of finding a Mathler solution and even make a solver think that Mathler is making mistakes.

This confusion was caused, or made worse, by teaching the order of operations using “PEMDAS” (and its all too memorable memory aid: “Please Excuse My Dear Aunt Sally”.) PEMDAS is sometimes described or taught in classrooms and on the internet (including by people who you’d think would know better) as describing six levels of priority, one for each letter in “PEMDAS”, as follows:

INCORRECT ORDER OF OPERATIONS Six levels of precedence:

  • First, evaluate expressions in Parentheses,
  • Second, evaluate Exponents (not used in Mahler),
  • Third, evaluate Multiplications,
  • Fourth, evaluate Divisions,
  • Fifth, evaluate Additions,
  • Sixth, evaluate Subtractions.

This misunderstanding of the order of operations is all too common and is a real roadblock to student success in math classes and STEM subjects (perhaps even more important than making it hard to solve Mathler!) If you learned/were taught/remember it that way, please excuse my dear Aunt Sally and those who have used her name in vain, and re-learn it correctly as follows.

CORRECT ORDER OF OPERATIONS: Four levels of precedence:

  • First, evaluate expressions in Parentheses,
  • Second, evaluate Exponents (not used in Mahler),
  • Third, evaluate Multiplications and Divisions left to right,
  • Fourth, evaluate Additions and Subtractions left to right.

In olden days, before Aunt Sally appeared on the scene, this was called the order of precedence or hierarchy of operations.j. If you must use PEMDAS to remember this, you might try writing it as P E MD AS, and read it as PLEASE– Excuse–M’Dear–AuntSally.

In Example 1 the correct answer is A. There are no parentheses and no multiplications or divisions. Do additions and subtractions left to right. The subtraction is left of the addition, so do the subtraction before the addition:


20 – 4 + 3
= 16 + 3
= 19

Referring back to the title, the reason “Why 3 – 2 + 1 = 2 not 0” is that subtraction and addition are done left to right, the subtraction is to the left of the addition, so the subtraction is done first! If you think/thought the addition should be done first, and that the value of 3 – 2 + 1 is 0, you have/had the Aunt Sally problem! This is not your fault — you may well have learned it wrong because of a lack of good practice in textbooks or even because of bad explanations from teachers. But please — now is the time to get it right!

3 – 2 + 1
= 1 + 1
= 2


In Example 3 the correct answer is also A. There are no parentheses, the multiplication and division are both in the next level of precedence and are done left to right. The division is left of the multiplication so do the division before the multiplication:

6 / 3 * 2
= 2 * 2
= 4

NOTES:

1 . Checking order of operations

A good way to see whether you’ve used the Order of Operations correctly is to compare your evaluation against the evaluation by a “scientific calculator” like the TI-83 or the scientific version of an on-line calculator. A “four-function calculator” (hardware or on-line) that does not allow you enter the whole expression before evaluating it will not work.

Some students who don’t know the correct order of operations survive math and STEM classes by using a scientific calculator. TI-83 is probably most common calculator in use basic algebra.

2. Order of operations at the same level of precedence

Addition and subtraction are evaluated left to right at the same precedence. When you learn about negative numbers you learn you can rewrite either addition or subtraction as the other by changing the operator and replacing the number after the operator by its negative. Similarly, when you learn about fractions, you learn you can rewrite a multiplication or a division as the other by changing the operator and replacing the number after the operator by its “reciprocal”. Examples: Using brackets [] around negative numbers and fractions for clarity:

  • 8 – 3 = 8 + [-3] = 5
  • 11 + 7 = 11 – [-7] = 18
  • 16 / 4 = 16 * [1/4] = 4
  • 6 * 5 = 6 / [1/5] = 30

So :

  • 8 – 3 + 10. is evaluated left to right to have the same value as 8 + [-3] + 10. Both 15!
  • 16 /4 * 2 is evaluated left to right to have the same value as 16 * [1/4] * 2

3. Order of operations for advanced math

More advanced math uses additional operations that have to be incorporated Into the order of operations. Math classes often approach this by following textbook examples, but calculators (for example) have to be very explicit about it. You might be interested in seeing the full “order of operations” for the TI-83, which Texas Instruments calls the EOSTM (Equation Operating System.) Unless you’ve taken a LOT of math beyond algebra you won’t recognize all the notation — don’t worry about that — you will find the Mathler operations are there in the correct relative order.

Equation Operating System (Texas Instruments)


4. For the seriously Mathematically Minded:

MUCH more can be said on the subjects of notation and order of operations. Related Wikipedia articles:

Mathematical Notation (and its “See also” links)

Calculator Input Methods

Order of Operations (and its “See also” links, especially Reverse Polish Notation (really!) )

 

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