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mathsIntegersNumerical Expressions with Integers

Integers: Simplification of Expressions

In this page, handling of integers, both positive and negative numbers, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.



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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.

What is `2+4+3`?

  • `9`
  • `9`
  • `243`

The answer is "`9`".

This is an example of a numerical expression.

Which of the following is a meaning for the word "expression"?

  • something done very fast
  • collection of numbers and arithmetic operations between them, which together represent a quantity
  • collection of numbers and arithmetic operations between them, which together represent a quantity

The answer is "collection of numbers and arithmetic operations between them, which together represent a quantity".

What is the term used to refer "collection of numbers and arithmetic operations between them, which together represent a quantity"?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is "numerical expression".

What is `2xx4xx3`?

  • `24`
  • `24`
  • `243`

The answer is "`24`".

This is an example of a numerical expression.

Is `a+3x` a numerical expression?

  • Yes. It has the number 3.
  • No. It is not entirely numbers and arithmetic operations
  • No. It is not entirely numbers and arithmetic operations

The answer is "No. It is not entirely numbers and arithmetic operations".

Solved Exercise Problem:

Is `3+4-2+1` a numerical expression?

  • No, it has both addition and subtraction
  • Yes, addition and subtraction can be part of a numerical expression
  • Yes, addition and subtraction can be part of a numerical expression

The answer is "Yes, addition and subtraction can be part of an expression".

Solved Exercise Problem:

Is `3+4xx2-6-:3` a numerical expression?

  • No, as it has the addition, subtraction, multiplication, division
  • Yes, all arithmetic operations can be part of a numerical expression
  • Yes, all arithmetic operations can be part of a numerical expression

The answer is "Yes, all arithmetic operations can be part of a numerical expression"

Solved Exercise Problem:

is `3` a numerical expression?

  • No. 3 is only a number. Not a numerical expression
  • Yes. technically a number can also be considered a numerical expression
  • Yes. technically a number can also be considered a numerical expression

The answer is "Yes. technically a number can also be considered a numerical expression".

Consider `1+2` and `3xx1`. Note that when evaluated, both result in identical numerical value `1+2=3` and `3xx1 = 3`.

Are these two different numerical expressions?

  • They are two different expressions, evaluating to equal values
  • They are two different expressions, evaluating to equal values
  • They are identical expressions as they evaluate to equal values

The answer is "They are two different expressions, evaluating to equal values".

Simplify `9-6-:3`

  • `7`
  • `7`
  • `1`

The answer is "`7`". Division has higher precedence over subtraction. So

`9-6-:3`

`=9-2`

`=7`

What is the rule of precedence in numerical expressions?

  • multiplication and division have higher precedence over addition and subtraction
  • parentheses or brackets have highest precedence
  • both the above
  • both the above

The answer is "both the above".

What is `20-4-3`?

 •  One student did `20-4-3` `= 16-3` `=13`

 •  Another student did `20-4-3` `=20-1` =`19`.

Which one is correct?

  • `13`
  • `13`
  • `19`

The answer is "`13`".

The two subtractions are in the same precedence level. This is to be handled from left to right sequence. `20-4-3`

`= 16-3` `=13`

 •  It is NOT correct to do `=20-1` =`19`.

Simplify `36-:6-:3`. Which one of the following is correct?

 •  `36-:6-:3` `=6 -:3` `= 2`

 •  `36-:6-:3` `=36-:2` `=18`

  • `2`
  • `2`
  • `18`

The answer is "`2`".

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

`36-:6-:3`

`= 6-:3` `=2`

 •  It is NOT correct to do `=36-:2` `=18`.

What is the rule of sequence in numerical expressions?

  • there is no such thing as rule of sequence
  • when multiple operations of same precedence is to be simplified, the operations are performed from left to right sequence
  • when multiple operations of same precedence is to be simplified, the operations are performed from left to right sequence

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

Simplify `6-:3xx2`

  • `4`
  • `4`
  • `1`

The answer is "`4`".

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

`6-:3xx2`

`=2xx2`

`=4`

Simplify `6-:(3xx2)`

  • `4`
  • `1`
  • `1`

The answer is "`1`".

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

`6-:(3xx2)`

`=6-:6`

`=1`

What is the rule of brackets or parentheses in numerical expressions?

  • there is no such thing as rule of brackets
  • the subexpression within a bracket or parentheses has the highest precedence
  • the subexpression within a bracket or parentheses has the highest precedence

The answer 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.

Solved Exercise Problem:

Simplify `4+(-6)-:3xx(-2)`

  • `8`
  • `8`
  • `5`

The answer is "`8`".





`4+(-6)-:3xx(-2)`

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+(-2)xx(-2)`

Multiplication is of higher precedence
`=4+4`

`=8`.

Solved Exercise Problem:

Simplify `10-(-3)-2xx(-3)`.

  • `19`
  • `19`
  • `7`

The answer is "`19`".

`10-(-3)-2xx(-3)`

The multiplication is of higher precedence over subtraction
`=10-(-3)-(-6)`

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

`=19`

Solved Exercise Problem:

Simplify `(-1+3-1)xx2+(-2)-1`

  • `-5`
  • `-1`
  • `-1`

The answer is "`-1`"

`(-1+3-1)xx2+(-2)-1`

Bracket is of the highest precedence
`=1xx2+(-2)-1`

multiplication is higher in precedence
`=2+(-2)-1`

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

`=-1`

Solved Exercise Problem:

What is the value of `-3-((-1)-2xx2)`?

  • `-8`
  • `2`
  • `2`

The answer is "`2`".

`-3-((-1)-2xx2)`

brackets take precedence and within the brackets, the multiplication is higher precedence.
`=-3-((-1)-4)`

brackets have higher precedence
`=-3-(-5)`

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

simplifying
`=2`

Solved Exercise Problem:

What is the value of `-3-(-1)-1`?

  • `-3`
  • `-3`
  • `-5`

The answer is "`-3`".
`-3-(-1)-1`

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

left to right sequence
`=-2-1`

`=-3`

Solved Exercise Problem:

Simplify `4+6-:(-3xx2)`

  • `3`
  • `3`
  • `5`

The answer is "`3`".

`4+6-:(-3xx2)`

The expression inside bracket is simplified first.
`=4+6-:(-6)`

The division is of higher precedence over addition.
`=4+(-1)`

Then addition is simplified.
`=3`.

Solved Exercise Problem:

Simplify `(10-3-2)xx(-3)`.

  • `-1`
  • `-15`
  • `-15`

The answer is "`-15`".

`(10-3-2)xx(-3)`

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.
`=(7-2)xx(-3)`
completing the expression inside bracket
`=5xx(-3)`

`=-15`

Simplification of Expressions : BODMAS

 •  B - Brackets

 •  O - Order (exponents, roots, logarithm)

 •  D - Division

 •  M - Multiplication

 •  A - Addition

 •  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

 •  A - Addition

 •  S - Subtraction

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

                            
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