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I ran across the following math problem where there is arithmetic involved: $$9-4+1$$ Supposedly the answer is $6$? I entered into my computer and calculator and got the same result so I realized there was something strange in this math problem because it does not follow the PEMDAS pattern.

Can someone explain to me why the answer is equal to $6$ and not $4$?

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closed as off-topic by Simply Beautiful Art, Andrés E. Caicedo, Namaste, José Carlos Santos, Xander Henderson Oct 23 '18 at 13:23

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  • 3
    $\begingroup$ PE(MD)(AS) is the actual rule. $\endgroup$ – Randall Oct 23 '18 at 1:13
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    $\begingroup$ Can you be more specific?. Can you show some examples?. $\endgroup$ – ac1002 Oct 23 '18 at 1:14
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    $\begingroup$ The order of addition and subtraction goes from left to right. In this case subtraction is first then addition. $\endgroup$ – Phil H Oct 23 '18 at 1:21
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    $\begingroup$ how is this supposed to equal 4? $\endgroup$ – user13267 Oct 23 '18 at 7:41
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    $\begingroup$ How did you get 4? If you would calculate using the substraction as the first precedence, I get it to: -4 + 9 + 1 = 6. $\endgroup$ – Cenderze Oct 23 '18 at 8:19
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Your confusion lies in thinking that Addition supersedes Subtraction. The actual precedence set by "PEMDAS" is

  • Parenthetical terms first
  • Exponents
  • Any multiplication OR division (equal precedence!)
  • Any addition OR subtraction (equal precedence!)

This is why I suggested you think of it as PE(MD)(AS).

This is nothing more than a convenient agreement and not some mysterious law of nature. It allows us to write $3x-y$ without ambiguity, as it means $(3x)-y$ and not $3(x-y)$. If we mean the latter, we have to say it.

So, for example, $$ 5\cdot 3 -2= 15-2=13 $$ and NOT $5 \cdot 3-2=5$. If I wanted the latter I have to bracket it off to make it respect PEMDAS: $5\cdot (3-2) = 5 \cdot 1 =5$.

In general, when a "tie" in precedence comes, we operate left to right. For your case, $9-4+1=5+1=6$ since the subtraction is done first. You gave the second (addition) precedence, when really the subtraction gets done first, since they have equal precedence and the left-most one comes first.

To be honest, even this is sort of rigid. I can even do your addition first, as long as I respect the minus sign the right way. For instance $$ 9 - 4 + 1 = 9 + (-4+1) = 9 + (-3) = 9-3 =6. $$

For some other examples, $$ 7+4-5+6= 11-5+6=6+6=12 $$ and $$ 2\cdot 5 - 7 \cdot 3 + 4 = 10 - 21 + 4 = -11 + 4 = -7. $$

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  • 8
    $\begingroup$ An exception to the left-to-right rule is exponentiation. $\endgroup$ – Dan Oct 23 '18 at 2:14
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    $\begingroup$ Ha, holy smokes, you're right. I give up. $\endgroup$ – Randall Oct 23 '18 at 2:21
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    $\begingroup$ @ArcanistLupus: No, that's more like the difference between prefix, infix, and postfix. $\endgroup$ – Kevin Oct 23 '18 at 6:14
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    $\begingroup$ Instead of PEMDAS, in German, there is the proverb "Punktrechnung vor Strichrechnung" (engl. "Dot calculation before dash calculation"), meaning: Perform the operators drawn with dots, that is $\cdot$ and $:$ before those drawn with dashes, that is $+$ and $-$. This makes it clear the addition and subtraction do not supersede each other. $\endgroup$ – rexkogitans Oct 23 '18 at 6:48
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    $\begingroup$ @Ister With multiplication and division, it's not always so clear-cut. At least when you have implicit multiplication, like in $R = C/2\pi$. Suddenly explicit and implicit multiplication are two different operations with different precesence rules? I think it's better to just say it's ambiguous, and tell people to use fractions or parentheses. $\endgroup$ – Arthur Oct 23 '18 at 8:13
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Probably getting too abstract... but subtraction doesn't really exist. What we are really doing when we subtract is adding a negative number.

$9 + (-4) + 1 = 6$

Once we have it in the this form, then we can do our addition in any order.

$(9 + (-4)) + 1 = 5+1 = 6\\ (9 + ((-4) + 1) = 9+(-3) = 6$

That is addition is associative.

And we can even swap it around. Addition is commutative.

$9 + 1 + (-4) = 10+(-4) = 6$

In many ways "PEMDAS" creates more confusion and problems than it is worth.

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    $\begingroup$ Can you show me an example on how you could do the problem with multiplication and division?. $\endgroup$ – ac1002 Oct 23 '18 at 1:32
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    $\begingroup$ Multiplication distributes over addition. Keep that straight and the rest takes care of itself. $\frac {5(2 + 4)}{2} - 6\cdot \frac {7}{2} + 13 = 5(3) + 7(3) + 13 = 12(3) + 13 = 49$ $\endgroup$ – Doug M Oct 23 '18 at 1:59
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    $\begingroup$ @ac1002 division is multiplication by an inverse. $2 \div 3 \times 6 = 2 \cdot \frac{1}{3} \cdot 6$. Similarly with exponents vs radicals: $\sqrt{2} = 2^\frac{1}{2}$. $\endgroup$ – jaxad0127 Oct 23 '18 at 3:14
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"PEMDAS" does not mean addition comes before subtraction. In fact addition and subtraction are done from left to right (unless modified by brackets).

So for your problem, first do $9-4=5$, then $5+1=6$.

This is an excellent illustration of why people should not rely on memorisation without understanding!

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  • $\begingroup$ So you solve the math problem from the left to right only when addition and subtraction are involved, or always despite the arithmetic symbol?. $\endgroup$ – ac1002 Oct 23 '18 at 1:17
  • $\begingroup$ Same thing goes for multiplication and division - left to right. However multiplication and division come before addition and subtraction. For example $$5+4\times3\div2-1=5+12\div2-1=5+6-1=11-1=10\ .$$ It is often a good idea to insert brackets for clarity, even if they are not strictly necessary. The above could be written $$5+(4\times3\div2)-1\ .$$ $\endgroup$ – David Oct 23 '18 at 1:23
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The misconception here is that addition and subtraction are somehow different operations (and similarly with multiplication and division in some similar questions). The expression $$ a - b $$ means exactly $$ a + (-b) $$ where $ -b$ is the unique number such that $b + (-b) = 0$. So, in the statement $$ 9 - 4 + 1 $$ one should read this as $$ 9 + (-4) + 1$$ The associativity of addition means that this can be resolved in any order. In all cases, the result will be $6$.

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in germany we have no PEMDAS memorization but we have:
"Punkt- vor Strich-Rechnung" (dot calculation before line calulation).

That depends on our notation: multiplication is indicated by a middot ·
division is written with :
addition and subtraction as + and -

there is no priority rule between division and multiplication and also addition and subtraction as they are on the same level. Being on the same level means operation from left to right.
Of course you have the usual rules:
parenthesis first,
exponantion before dot-calculation (as it is a shortcut of multiplication)
and last dot-calculation before line calulation (as multiplication is a shortcut of addition).

so your example has serial caclulation (parenthesis indicating the next caculation)

 9 − 4 + 1 =
(9 - 4)+ 1 =
   5   + 1 =
       6

other examples:

 2 + 3 · 4 - 5 =
 2 +(3 · 4)- 5 =
 2 +   12  - 5 =
(2 +   12) - 5 =
    14     - 5 =
         9

.

 6 : 2 + 3 · 4 - 5 =
(6 : 2)+(3 · 4)- 5 =
   3   +   12  - 5 =
  (3   +   12) - 5 =
       15      - 5 =
            10
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Working from the left to the right,$$9-4+1= (9-4)+1=5+1=6$$

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  • $\begingroup$ You always work from the left to the right?. $\endgroup$ – ac1002 Oct 23 '18 at 1:15
  • $\begingroup$ when it comes to addition and subtraction, yes, from the left to the right. $\endgroup$ – Siong Thye Goh Oct 23 '18 at 1:18
  • $\begingroup$ Only for addition and subtraction?. $\endgroup$ – ac1002 Oct 23 '18 at 1:19
  • $\begingroup$ Do things in Parentheses First, (Powers, Roots) before Multiply, Divide, Add or Subtract, (Multiply or Divide) before you (Add or Subtract), Otherwise just go left to right. That's what PEMDAS says. $\endgroup$ – Siong Thye Goh Oct 23 '18 at 1:21
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There is another easy way to remember this... BODMAS According to Bodmas rule, if an expression contains brackets ((), {}, []) we have to first solve or simplify the bracket followed by of (powers and roots etc.), then division, multiplication, addition and subtraction from left to right.

B - Brackets

O - powers and roots

D - division

M - multiplication

A - addition

S - substraction

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