When all the operations of addition, subtraction, multiplication and division are present together as a mathematical expression we have mixed operations.
The operations of addition and subtraction are opposite, or inverse, of each other. Therefore, these two operations are of the same order. However, subtraction is more complex because, it leads to the idea of negative numbers. Subtraction, thus, takes precedence over addition in mixed operations.
Multiplication is "compressed addition." One multiplication consists of several additions. Therefore, multiplication is a higher order operation than addition. Since addition and subtraction are of the same order, multiplication takes precedence over both subtraction and addition.
Division is opposite, or inverse, of multiplication. Therefore, multiplication and division are operations of the same order of. However, division is more complex because it leads to the idea of fractions. Division, thus, takes precedence over multiplication in mixed operations.
Multiplication and division, when grouped together, are referred to as terms. Numbers and operations within parentheses are also called terms. The terms are separated from each other by plus or minus.
Terms are always reduced first before reducing rest of the mathematical expression. Thus, the concept of terms automatically enforces the precedence of multiplication and division over addition and subtraction.
Understanding the logic involved in reducing mixed operations is very important. Only when this logic is understood, do the "formulas" about precedence of operations make sense.
Vinaire
From: Vinaire <vinaire@yahoo.com>
To: Math4u@yahoogroups.com
Sent: Wednesday, December 12, 2007 10:55:40 AM
Subject: Re: [Math4u] Sequence of Operations
6. We may total the minuends and subtrahends before subtraction.
(a) 10 – 5 + 8 – 4 = (10 + 8) – (5 + 4)
= 18 – 9
= 9
(b) 9 – 5 + 3 – 4 – 6 + 7 + 9 – 8 = (9 + 3 + 7 + 9) – (5 + 4 + 6 + 8)
= 28 – 23
= 5
From: Vinaire <vinaire@yahoo. com>
To: Math4u@yahoogroups. com
Sent: Tuesday, December 11, 2007 9:55:50 PM
Subject: Re: [Math4u] Sequence of Operations
5. We may replace all the subtrahends by their total without changing the final outcome.
(a) 12 – 5 – 3 = 12 – (5 + 3) = 12 – 8 = 4
We may check the above by carrying out the operations from left to right.
12 – 5 – 3 = 7 – 3 = 4
(b) 17 – 5 – 7 – 2 = 17 – (5 + 7 + 2) = 17 – 14 = 3
We may check the above by carrying out the operations from left to right.
17 – 5 – 7 – 2 = 12 – 7 – 2 = 5 – 2 = 3
Vinaire
From: Vinaire <vinaire@yahoo. com>
To: Math4u@yahoogroups. com
Sent: Tuesday, December 11, 2007 2:21:29 PM
Subject: Re: [Math4u] Sequence of Operations
- The numbers may be moved around within the expression, but only if their operational signs are moved with them.
(a) When we add 0 to the whole expression, the outcome remains the same; but we note that the "no sign" in front of the first number is actually a "plus" sign.
10 – 5 + 8 – 4 = 0 + 10 – 5 + 8 – 4
(b) The numbers in an expression may be moved around as long as they carry their + and – signs with them.
0 + 10 – 5 + 8 – 4 = 0 – 5 + 8 – 4 + 10 = 0 – 4 – 5 + 10 + 8
Regards,
From: Rick <rcastrap@yahoo. com>
To: Math4u@yahoogroups. com
Sent: Tuesday, December 11, 2007 12:09:31 PM
Subject: Re: [Math4u] Sequence of Operations
Vinaire,
The problem isn't necessarily that your "rule" is incorrect. It is that
it is too complicated. What does the beginning algebra student do if
there is more than one consecutive subtraction? I think strict left
association is easier to teach and easier to remember.
-- Rick
--- Vinaire <vinaire@yahoo. com> wrote:
> When addition and subtraction are present together they may be
> carried out from left to right.
> 10 – 5 + 8 – 4 = 5 + 8 – 4 =
> 13 – 4 = 9
> 3 + 9 – 4 + 7 = 12 – 4 + 7 =
> 8 + 7 = 15
> When addition and subtraction are present together they may also be
> carried out safely with subtraction first.
> 10 – 5 + 8 – 4 = 5 + 4 =
> 9
> 3 + 9 – 4 + 7 = 3 + 5 + 7 =
> 15
> Mistakes are likely if addition is carried before subtraction.
> 10 + 8 – 6 + 3 = 18 – 9 =
> 9 (incorrect)
> 10 + 8 – 6 + 3 = 10 + 2 + 3 =
> 15 (correct)
>
>
> Can anybody come up with an exception to the above?
>
> Vinaire
>
>
>
>
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