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Thought-Process to Discover Knowledge

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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.

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In this lesson properties of expressions are explained.

 •  expressions are statement of a value

 •  the value of an expression remains unchanged when the expressions are modified per PEMA / CADI

It is very important to go through this once to understand in-equalities in algebra.



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Which of the following is an example of numerical expression?

  • `2+2+84-2`
  • `23 xx 123 + 8`
  • `(32-23)+43xx2 - 4`
  • all the above
  • all the above

The answer is 'all the above'

What is a numerical expression?

  • a collection of numbers and arithmetic operations between them forming a quantity
  • a statement of a quantity with numbers and arithmetic operations
  • both the above
  • both the above

The answer is 'both the above'

A numerical expression is a statement of quantity with numbers and arithmetic operations.

For example, all the following are `2`.

 •  `1+5-4` (expression involving addition and subtraction)

 •  `130^0 + 123/123` (expression involving exponent and division)

 •  `sqrt(2) xx sqrt(2)` (expression involving multiplication and root)

 •  `2` (`2` is technically an expression and also the simplified form of the expressions given above.)

Consider the number `2` and the equivalent numerical expressions `1+5-4`. Can the numerical expression be used in place of the number `2`?

  • no, though the numerical expression equals the number, it cannot be used in place of the number.
  • yes, the numerical expression and the number can be used interchangeably
  • yes, the numerical expression and the number can be used interchangeably

The answer is "yes, the numerical expression and the number can be used interchangeably"

Consider `1+5-4`. Can the numerical expression be simplified to `1+1`?

  • Yes. It can be simplified as per PEMA rules.
  • Yes. It can be simplified as per PEMA rules.
  • No. It can not be simplified.

The answer is "Yes. It can be simplified as per PEMA rules".

Consider two numerical expressions `3+2` and `1-3`. These represent two quantities and the sum of these two are required. Can the numerical expressions be added to `3+2+1-3`?

  • No, these should to be simplified first to `5` and `-2` respectively and then only these can be added.
  • Yes. These can be added directly to `3+2+1-3`
  • Yes. These can be added directly to `3+2+1-3`

The answer is "Yes, These can be added directly"

Consider two numerical expressions `3+2` and `1-3`. These represent two quantities and the product of these two are required. Can the numerical expressions be multiplied to `(3+2)xx(1-3)`?

  • No, these should to be simplified first to `5` and `-2` respectively and then only these can be multiplied.
  • Yes. These can be multiplied directly to `(3+2)xx(1-3)`
  • Yes. These can be multiplied directly to `(3+2)xx(1-3)`

The answer is "Yes, These can be multiplied directly"

Arithmetics with Numerical Expressions :
Numerical expressions can be

 •  simplified as per PEMA precedence
eg: `3^2 + 3^3 + 4xx3^2` simplified to `5xx3^2 + 3^3`

 •  added
eg: `3+2` added to `1-3` gives `3+2+1-3`

 •  subtracted
eg: `1-3` subtracted from `3+2` gives `3+2 - (1-3)`

 •  multiplied
eg: `3+2` multiplied by `1-3` gives `(3+2)xx(1-3)`

 •  divided
eg: `3+2` divided by `1-3` gives `(3+2)-:(1-3)`

 •  factorized
eg: `(4+2+12+6)` factored gives `(1+3)(4+2)`

Note: Exponents, roots, and logarithm will be explained in higher level mathematics

Which of the following is true for numerical expressions?

  • By closure law of addition and multiplication, any numerical expression simplifies to a number.
  • By closure law of addition and multiplication, any numerical expression simplifies to a number.
  • Not all numerical expressions simplifies to a number.

The answer is "By closure law of addition and multiplication, any numerical expression simplifies to a number"

Consider the numerical expression `2-6//2+2`. Which of the following is true?

  • By commutative law of addition, the expression can be written as `2+2-6//2`
  • By associative law of addition, the expression can be written as `2+(-6//2+2)`
  • By additive identity property, the expression can be written as `2+2-6//2 + 0`
  • all the above
  • all the above

The answer is "all the above"

Laws and Properties of Arithmetics -- Numerical Expressions : The value of a numerical expression remains unchanged in the following.

 •  any subexpression can be considered a number by closure law of addition and multiplication

 •  position of two subexpressions can be changed by commutative laws of addition and multiplication

 •  order of operations can be modified as per associative laws of addition and multiplication

 •  multiplication by a subexpression can be distributed over addition as per the distributive law of multiplication over addition

 •  additive identity can be added or multiplicative identity can be multiplied

 •  additive identity in an expression can be modified into sum of a number and its additive inverse as per the additive inverse property

 •  multiplicative identity in an expression can be modified into product of a number and its multiplicative inverse as per the multiplicative inverse property.

These laws and properties are called CADI properties of multiplication.

Important note: How the CADI properties are relevant for algebra?
The algebraic expressions are modified as per the exact same laws and properties given above. For example: `3(x+2) = 3x+6` as per distributive law of multiplication over addition.

                            
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