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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

This course is designed for students at 9th and 10th grade level. It assumes the following were introduced.

 •  Whole numbers, integers, fractions, decimal numbers.

 •  Addition, subtraction, multiplication, division, exponent, root, logarithm.

 •  Subtraction is inverse of addition

 •  Division is inverse of multiplication

 •  Root and Logarithm are inverses of Exponent

 •  introduction to variables in algebra

 •  Numerical expressions

 •  BODMAS or PEMDAS for precedence order

 •  BOMA or PEMA The objective of this topic is to formalize numerical arithmetic as applicable to algebra.

It is very important to go through these to understand algebra.



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What is `2+4+3`?

  • `9`
  • `9`
  • `243`

The answer is "`9`".

This is an example of a numerical expression.

What is `2-4-3`?

 •  One student did `2-4-3` `= -2-3` `=-5`

 •  Another student did `2-4-3` `=2-1` =`1`.

Which one is correct?

  • `1`
  • `-5`
  • `-5`

The answer is "`-5`".

Subtraction is to be handled as inverse of addition `2-4-3` `=2+(-4)+(-3)`.

In this case both students will get the correct answer.

 •  `2+(-4)+(-3)` `= -2-3` `=-5`

 •  `2+(-4)+(-3)` `=2-7` =`-5`.

What is `2xx4xx3`?

  • `24`
  • `24`
  • `243`

The answer is "`24`".

This is an example of a numerical expression.

Simplify `2-:4-:3`. Which one of the following is correct?

 •  `2-:4-:3` `=2/4 -:3` `= 2/12`

 •  `2-:4-:3` `=2-: 4/3` `=6/4`

  • `2/12`
  • `2/12`
  • `6/4`

The answer is "`2/12`".

Division is to be handled as inverse of multiplication `2-:4-:3` `=2xx 1/4 xx 1/3`.

In this case, both methods will give the correct answer.

 •  `2xx 1/4 xx 1/3` `= 2/4 xx 1/3` `=2/12`

 •  `2xx 1/4 xx 1/3` `=2xx 1/12` `=2/12`.

Simplify `2+4xx3`. Which one of the following is correct?

 •  `2+4xx3` `=6 xx 3` `= 18`

 •  `2+4xx3` `=2 + 12` `=14`

  • `18`
  • `14`
  • `14`

The answer is "`14`".

Multiplication has higher precedence to addition. In `2+4xx3`, the multiplication is to be done ahead of addition.

Which one of the following is correct?

 •  `2xx4^3` `=2 xx 64` `= 128`

 •  `2xx4^3` `=8^3` `=512`

  • `128`
  • `128`
  • `512`

The answer is "`128`".

Exponent has higher precedence to multiplication. In `2xx4^3`, the exponent is to be done ahead of multiplication.

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

  • priority over another; order to be observed
  • priority over another; order to be observed
  • preparation required for a task; fundamental requirement

The answer is "priority over another; order to be observed".

In a numerical expression, the precedence order is:
 •  exponents
 •  multiplication
 •  addition.

Multiplication has higher precedence to addition. In some expressions, addition has to be carried out before multiplication.

For example: Result of `2+4` has to be multiplied by `3`. This cannot be given as `2+4xx3` as the result of this expression does not equal the example.

Which of the following helps to define such expressions?

  • parenthesis or brackets are used
  • parenthesis or brackets are used
  • the precedence is fixed and cannot be modified with parenthesis

The answer is "Parenthesis or brackets are used".

Parenthesis or brackets have higher precedence.

Precedence order is "PEMA" is also known as PEMDAS / BODMAS

LPA - Precedence : Precedence Order in arithmetics is PEMA
PEMDAS or BODMAS
    →  Parenthesis
    →  Exponents
    →  Multiplication
    →  Addition

Note 1: Subtraction is handled as inverse of addition.

Note 2: Division is handled as inverse of multiplication

Note 3: Roots and Logarithm are handled as inverse of exponents

For a number of operations of same precedence, it is prescribed that the operations be carried out from left to right in sequence.
eg: `4-:2xx2` equals `(4-:2)xx2=4` and not `4-:(2xx2)=1`.

With the definition of PEMA, the rule of "left-to-right-sequence" is not required.
eg: `4-:2xx2 = 4xx 1/2 xx 2` and simplify either way and both result in the same correct answer.
`(4xx1/2)xx2 = 4` or
`4xx (1/2 xx 2)= 4`.

this is very important in the context of Algebra, as variables or terms may require to be handled in different order than the prescribed left to right order.

Solved Exercise Problem:

Simplify `(1/2+3.1)xx2+2.1`

  • `9.3`
  • `9.3`
  • `14.76`

The answer is "`9.3`"

                            
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