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Integers: Simplification of Expressions

what you'll learn...

overview

•  Numerical expression,

•  precedence order,

•  sequence order.

evaluating an expression

Let us quickly revise what is numerical expression, and precedence order, sequence in simplifying the numerical expressions. This was introduced in whole numbers. It takes very little time to revise.

Consider $2+4+3$$2 + 4 + 3$. This can be evaluated to $9$$9$.

This is an example of a numerical expression.

The word "expression" means: collection of numbers and arithmetic operations between them, which together represent a quantity.

$2×4×3$$2 \times 4 \times 3$ is an example of a numerical expression. It is evaluated to $24$$24$.

$a+3x$$a + 3 x$ is not a numerical expression. It is not entirely numbers and arithmetic operations. It has letters a and x.

Is $3+4-2+1$$3 + 4 - 2 + 1$ a numerical expression?
Yes, addition and subtraction can be part of an expression.

Is $3+4×2-6÷3$$3 + 4 \times 2 - 6 \div 3$ a numerical expression?
Yes, all arithmetic operations can be part of a numerical expression

is $3$$3$ a numerical expression?
Yes. technically a number can also be considered a numerical expression.

Consider $1+2$$1 + 2$ and $3×1$$3 \times 1$. Note that when evaluated, both result in identical numerical value $1+2=3$$1 + 2 = 3$ and $3×1=3$$3 \times 1 = 3$.

They are two different expressions, evaluating to equal values.

precedence

Consider $9-6÷3$$9 - 6 \div 3$.

To simplify the expression, the division is performed first. It is

$9-6÷3$$9 - 6 \div 3$

$=9-2$$= 9 - 2$

$=7$$= 7$

Division has higher precedence over subtraction.

In a numerical expression the precedence order is given as :

•  Brackets / Parentheses

•  Order / Exponent

•  Division and Multiplication

This is abbreviated as BODMAS or PEMDAS.

sequence

Consider $20-4-3$$20 - 4 - 3$.

It is wrong to do $20-4-3$$20 - 4 - 3$ $\ne 20-1$$\ne 20 - 1$ $=19$$= 19$.

The correct order of simplification is $20-4-3$$20 - 4 - 3$ $=16-3$$= 16 - 3$ $=13$$= 13$

The two subtractions are in the same precedence level. This is to be handled from left to right sequence.

Consider $36÷6÷3$$36 \div 6 \div 3$

It is wrong to simplify as $36÷6÷3$$36 \div 6 \div 3$ $\ne 36÷2$$\ne 36 \div 2$ $=18$$= 18$

The correct order of simplification is $36÷6÷3$$36 \div 6 \div 3$ $=6÷3$$= 6 \div 3$ $=2$$= 2$.

The two divisions are in same precedence level. This is to be handled from left to right sequence. .

Rule of sequence is, when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

Consider $6÷3×2$$6 \div 3 \times 2$.

The division and multiplication are of same precedence, so it is simplified from left to right.

$6÷3×2$$6 \div 3 \times 2$

$=2×2$$= 2 \times 2$

$=4$$= 4$

bracket

Consider $6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

The bracket has higher precedence, and so the expression inside bracket is simplified first.

$6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

$=6÷6$$= 6 \div 6$

$=1$$= 1$

The rule of brackets or parentheses in numerical expression is, the subexpression within a bracket or parentheses has the highest precedence

In a numerical expression, the precedence order is:
•  Parentheses or Brackets is the highest precedence order
•  division and multiplication in same level same level of precedence
•  addition and subtraction in same level of precedence.

This is abbreviated as BODMAS (Brackets, Division, Multiplication, Addition, Subtraction) or PEMDAS (Parentheses, Multiplication, Division, Adddition, Subtraction).

When multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

All these were studied as part of whole numbers. The same applies for integers.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for same precedence

Let us simplify some expressions of integers.

examples

Simplify $4+\left(-6\right)÷3×\left(-2\right)$$4 + \left(- 6\right) \div 3 \times \left(- 2\right)$

The answer is "$8$$8$".

$4+\left(-6\right)÷3×\left(-2\right)$$4 + \left(- 6\right) \div 3 \times \left(- 2\right)$

The division and multiplication are of higher precedence over addition. In that, the division and multiplication are of same precedence, so it is simplified from left to right.
$=4+\left(-2\right)×\left(-2\right)$$= 4 + \left(- 2\right) \times \left(- 2\right)$

Multiplication is of higher precedence
$=4+4$$= 4 + 4$

$=8$$= 8$

Simplify $10-\left(-3\right)-2×\left(-3\right)$$10 - \left(- 3\right) - 2 \times \left(- 3\right)$.

The answer is "$19$$19$".

$10-\left(-3\right)-2×\left(-3\right)$$10 - \left(- 3\right) - 2 \times \left(- 3\right)$

The multiplication is of higher precedence over subtraction
$=10-\left(-3\right)-\left(-6\right)$$= 10 - \left(- 3\right) - \left(- 6\right)$

the two subtraction are in the same precedence level and so they are simplified in the left to right sequence.
$=13-\left(-6\right)$$= 13 - \left(- 6\right)$

$=19$$= 19$

Simplify $\left(-1+3-1\right)×2+\left(-2\right)-1$$\left(- 1 + 3 - 1\right) \times 2 + \left(- 2\right) - 1$

The answer is "$-1$$- 1$"

$\left(-1+3-1\right)×2+\left(-2\right)-1$$\left(- 1 + 3 - 1\right) \times 2 + \left(- 2\right) - 1$

Bracket is of the highest precedence
$=1×2+\left(-2\right)-1$$= 1 \times 2 + \left(- 2\right) - 1$

multiplication is higher in precedence
$=2+\left(-2\right)-1$$= 2 + \left(- 2\right) - 1$

left to right sequence for operations of same precedence
$=0-1$$= 0 - 1$

$=-1$$= - 1$

What is the value of $-3-\left(\left(-1\right)-2×2\right)$$- 3 - \left(\left(- 1\right) - 2 \times 2\right)$?

The answer is "$2$$2$".

$-3-\left(\left(-1\right)-2×2\right)$$- 3 - \left(\left(- 1\right) - 2 \times 2\right)$

brackets take precedence and within the brackets, the multiplication is higher precedence.
$=-3-\left(\left(-1\right)-4\right)$$= - 3 - \left(\left(- 1\right) - 4\right)$

brackets have higher precedence
$=-3-\left(-5\right)$$= - 3 - \left(- 5\right)$

subtraction of negative is positive
$=-3+5$$= - 3 + 5$

simplifying
$=2$$= 2$

What is the value of $-3-\left(-1\right)-1$$- 3 - \left(- 1\right) - 1$?

The answer is "$-3$$- 3$".
$-3-\left(-1\right)-1$$- 3 - \left(- 1\right) - 1$

bracket takes the highest precedence
$=-3+1-1$$= - 3 + 1 - 1$

left to right sequence
$=-2-1$$= - 2 - 1$

$=-3$$= - 3$

Simplify $4+6÷\left(-3×2\right)$$4 + 6 \div \left(- 3 \times 2\right)$

The answer is "$3$$3$".

$4+6÷\left(-3×2\right)$$4 + 6 \div \left(- 3 \times 2\right)$

The expression inside bracket is simplified first.
$=4+6÷\left(-6\right)$$= 4 + 6 \div \left(- 6\right)$

The division is of higher precedence over addition.
$=4+\left(-1\right)$$= 4 + \left(- 1\right)$

$=3$$= 3$.

Simplify $\left(10-3-2\right)×\left(-3\right)$$\left(10 - 3 - 2\right) \times \left(- 3\right)$.

The answer is "$-15$$- 15$".

$\left(10-3-2\right)×\left(-3\right)$$\left(10 - 3 - 2\right) \times \left(- 3\right)$

The expression inside bracket is simplified first. The two subtraction are in the same precedence level and so they are simplified in the left to right sequence.
$=\left(7-2\right)×\left(-3\right)$$= \left(7 - 2\right) \times \left(- 3\right)$
completing the expression inside bracket
$=5×\left(-3\right)$$= 5 \times \left(- 3\right)$

$=-15$$= - 15$

summary

Simplification of Expressions : BODMAS

•  B - Brackets

•  O - Order (exponents, roots, logarithm)

•  D - Division

•  M - Multiplication

•  S - Subtraction

•  And Left to Right sequence for multiple operations of same precedence.

PEMDAS

•  P - Parentheses

•  E - Exponents (roots and logarithm)

•  M - Multiplication

•  D - Division