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Integers : Precedence Order PEMA / BOMA

what you'll learn...

overview

This page redefines the precedence order in arithmetics with PEMA or BOMA. As part of integers, the revised version is in handling subtraction -- handle subtraction as inverse of addition.

recap

Integer expressions are simplified with the following rules.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for multiple operations at same precedence level.

In this the sequence order is not desirable. If the sequence order is to be strictly followed, then the following requires unnecessary computation.

$3+4-3$$3 + 4 - 3$ has to be simplified to $=7-3=4$$= 7 - 3 = 4$, which involves two steps.

$4×7÷4$$4 \times 7 \div 4$ has to be simplified to $28÷4=7$$28 \div 4 = 7$, which involve two steps.

As part of studies on integers, the expressions involving subtraction is analyzed and the rules of simplification are redefined.

As part of studies on fractions, the expressions involving division will be analyzed and the rules of simplification will be redefined.

Numerical expressions involving subtraction are simplified with the redefined rule

•  subtraction is the inverse of addition. And, all subtractions are converted to addition of negative of subtrahend.

•  Expression is simplified using precedence order PEMA. That is, Parentheses, Exponents (which are not introduced yet), Multiplication ( handling division will be explained in fractions), and Addition. Note, the precedence order does not have subtraction, as the subtractions are converted into addition of inverse.

The advantage of this is explained in this simple example.

$3+4-3$$3 + 4 - 3$

$=3+4+\left(-3\right)$$= 3 + 4 + \left(- 3\right)$

No need to follow the sequence order. Since all are addition, add any number to any other in the sequence that suits.
$=3+\left(-3\right)+4$$= 3 + \left(- 3\right) + 4$

$=0+4$$= 0 + 4$ $=4.$$= 4.$

Simplify $32-7+47-40$$32 - 7 + 47 - 40$.
Note that it has $47$$47$ which equals $-7$$- 7$ and $-40$$- 40$.

The answer is "both the above". Note that the flexibility to change the order of operation can be an advantage in calculations.

There are good reasons to adapt to PEMA. Students at 6th or 7th level may skip this explanation.

When learning properties of numbers, we learn the commutative property of addition. $a+b=b+a$$a + b = b + a$. But the subtraction does not follow the commutative property. What this means is, the position of numbers cannot be modified to simplify the problem. The redefined precedence order PEMA converts the subtraction into addition and allows commutative property to be used to ones advantage. $a-b=a+\left(-b\right)=-b+a$$a - b = a + \left(- b\right) = - b + a$

Another property of numbers is associative property of addition. $a+\left(b+c\right)=\left(a+b\right)+c$$a + \left(b + c\right) = \left(a + b\right) + c$. But the subtraction does not follow the associative property. What this means is, the sequence of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the subtraction into addition and allows associative property to be used to ones advantage. $a-\left(b-c\right)=a+\left(\left(-b\right)+c\right)=\left(a+\left(-b\right)\right)+c$$a - \left(b - c\right) = a + \left(\left(- b\right) + c\right) = \left(a + \left(- b\right)\right) + c$.

Algebra is extensively based on the numerical expressions and the properties of numerical arithmetic. In an algebraic expression of multiple terms, with subtraction in some of them, simplification is possible only if the sequence order is not required to be followed. For example, ${x}^{2}-x+2{x}^{2}-4-7x$${x}^{2} - x + 2 {x}^{2} - 4 - 7 x$. This expression is intuitively understood to be ${x}^{2}+\left(-x\right)+2{x}^{2}+\left(-4\right)+\left(-7x\right)$${x}^{2} + \left(- x\right) + 2 {x}^{2} + \left(- 4\right) + \left(- 7 x\right)$. Then it can be simplified into ${x}^{2}+2{x}^{2}+\left(-x\right)+\left(-7x\right)+\left(-4\right)$${x}^{2} + 2 {x}^{2} + \left(- x\right) + \left(- 7 x\right) + \left(- 4\right)$ which equals, $3{x}^{2}-8x-4$$3 {x}^{2} - 8 x - 4$.

summary

Numerical Arithmetics Precedence Order : PEMA / BOMA
PEMA = Parentheses, Exponents, Multiplication and Addition

BOMA = Brackets, Order, Multiplication and Addition

Note : Division is converted into multiplication by multiplicative inverse of divisor. And, Subtraction is converted into addition with additive inverse of subtrahend.

Outline