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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, the laws and properties of multiplication are revised. The properties are Closure, Commutative, Associative, Distributive, Multiplicative Identity, Multiplicative Inverse.

The division is handled as inverse of multiplication and in that case, the properties mentioned above are applicable.
eg: Division is not commutative `a-:b !=b-:a`
But Division as inverse of multiplication `a-:b = a xx (1/b) = 1/b xx a`



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



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In this lesson, the laws and properties of multiplication is revised. It is very important to go through this once to understand algebra.

For any real numbers `p,q`, will `p xx q` be a real number?

  • yes, always a real number
  • yes, always a real number
  • for some values of `p` or `q`, `p+q` is not a real number

The answer is "yes, always a real number".

Closure Property of Multiplication: Given `p,q in RR`. `p xx q in RR`.closure property illustration Closure Property applied to Division: Given `p,q in RR`. `p-:q in RR`.

Proof:
Given `p,q in RR`
`1/q in RR` as per Multiplicative Inverse Property
`pxx(1/q) in RR` as per Closure property of Multiplication
`=> p-:q in RR`

Using Closure Property: Given `p,q,r,s in RR`, `pxxq-:rxxs`, the subexpression `pxxq-:r` is a number and can be considered as a single number for any other property.

For example as per commutative property `pxxq-:rxxs = sxxpxxq-:r`, in which `pxxq-:r` is considered to be a single real number.

Given `p,q in RR`. which of the following equals `pxxq`?

  • `qxxp`
  • `qxxp`
  • `pxxq` is not equal to `qxxp`

The answer is "`qxxp`".

Commutative Property of Multiplication: Given `p,q in RR`, `pxxq=qxxr`.commutative property illustration Commutative Property applied to Division: `p-:q=1/qxxp`.
Note: Division has to be handled as inverse of multiplication, `p-:q = pxx(1/q)` and then commutative property can be used.

Using Commutative Property: Given `p,q,r in RR`, the expression `pxxqxxp-:qxxr-:pxxr^2` is simplified to `pr^3`. students may work this out to understand.

Given `p,q,r in RR`. Which of the following equals `(pxxq)xxr`?

  • `pxx(qxxr)`
  • `pxx(qxxr)`
  • it does not equal `pxx(qxxr)`

The answer is "`pxx(qxxr)`"

Associative Property of Multiplication: Given `p,q,r in RR`. `(pxxq)xxr = pxx(qxxr)`. associative property illustration Associative Property applied to Division:

`(p-:q)-:r = pxx(1/q xx 1/r)`.
Note: Division has to be handled as inverse of multiplication, `(p-:q)-:r = (pxx 1/q)xx 1/r` and then associative property can be used.

Given `p,q,r in RR`. Which of the following equals `(p+q)xxr`?

  • `pxxr + q xx r`
  • `pxxr + q xx r`
  • it does not equal `pxxr + q xx r`

The answer is "`pxxr + q xx r`"

What does 'distribute' mean?distributive property illustration

  • to share; to spread
  • to share; to spread
  • to restrict; to arrest

The answer is 'to share; to spread'

Distributive Property of Multiplication: Given `p,q,r in RR`. `(p+q)xxr = pxxr + q xx r`.distributive property illustration Distributive Property applied to Division: There are 3 possible scenarios. In any such scenario, convert the division to inverse of multiplication, and distributive property applies.

 (1)  `(p+q)-:r = p/r + q/r`.

 (2)  `r-:(p+q) = r/(p+q)`. The multiplication does not distribute in this case.

 (3)  `t-:r xx (p+q) = t xx (p/r + q/r)`.

In algebra, division is always given as `p/q` and not `p-:q`. Thus commutative property, associative property, and distributive property can be used without any unintended errors.

Given `p in RR`. What is `pxx1`?

  • `p`
  • `p`
  • `1`

The answer is "`p`"

Multiplicative Identity Property: For any `p in RR`, there exists `1 in RR` such that `pxx1=p`.

Multiplicative Identity applied to Division: `p-:1 = p`

Note: `1/1=1` and so `p-:1=pxx(1/1)=pxx1 = p`.

Given `p in RR`. What is `pxx 1/p`?

  • `p`
  • `1`
  • `1`

The answer is "`1`"

Multiplicative Inverse Property: For any `p in RR` and `p!=0`, there exists `1/p in RR` such that `pxx 1/p=1`.

The properties together are named as CADI properties of multiplication. The abbreviation CADI is a simplified form of the first letters of Closure, Commutative, Associative, Distributive, Inverse, and Identity properties.

Note: Distributive property is shared with addition.

The CADI properties of Addition and CADI properties of Multiplication are together referred as "CADI properties".

LPA: CADI properties of Multiplication
 •  Closure Property
      if `x,y in RR`, then `x xx y in RR`

 •  Commutative Property
      `x xx y = y xx x`

 •  Associative Property
      `(x xx y) xx z = x xx (y xx z)`

 •  Distributive Property Over Addition
      `(x + y)xx z = x xx z + y xx z`

 •  Multiplicative Identity Property
      `1 in RR`, such that `x xx 1 = x`

 •  Multiplicative Inverse Property
      `1/x in RR` for any `x in RR` (except for `0`) such that `x xx (1/x) = 1`

 •  Division is to be handled as inverse of multiplication for the properties

This is important as algebra extensively involves these properties.

      Commutative property involving division : `x -: y` is given as `x xx 1/y = 1/y xx x`

      Associative property involving division : `(x-:y)-:z` is given as `(x xx 1/y) xx 1/z = x xx (1/y xx 1/z)`

      Distributive property involving division : `p-:q xx (x+y)` is given as `p xx 1/q xx (x+y) = p xx (x/q + y/q)`

                            
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