firmfunda
  maths > algebra

Laws and Properties of Arithmetics : Multiplication


    what you'll learn...

Overview

CADI Properties of Multiplication

 »  Closure Property
    →  if x,y, then x×y

 »  Commutative Property
    →  x×y=y×x

 »  Associative Property
    →  (x×y)×z=x×(y×z)

 »  Distributive Property Over Addition
    →  (x+y)×z=x×z+y×z

 »  Multiplicative Identity Property
    →  1, such that x×1=x

 »  Additive Inverse Property
    →  1x for any x (except for 0) such that x×(1x)=1

 »  Division is to be handled as multiplicative inverse for the properties
this is important as algebra extensively involves these properties
    →  Commutative property involving division : x÷y is given as x×1y=1y×x
    →  Associative property involving division : (x÷y)÷z is given as (x×1y)×1z=x×(1y×1z)
    →  Distributive property involving division : p÷q×(x+y) is given as p×1q×(x+y)=p×(xq+yq)

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

closure means within

For any real numbers p,q, p×q is a real number.

closure property illustration

Closure Property of Multiplication: Given p,q. p×q.

Closure Property applied to Division: Given p,q. p÷q.

Proof:
Given p,q
1q as per Multiplicative Inverse Property
p×(1q) as per Closure property of Multiplication
p÷q

Using Closure Property: Given p,q,r,s, p×q÷r×s, the subexpression p×q÷r is a number and can be considered as a single number for any other property.

For example as per commutative property p×q÷r×s=s×p×q÷r, in which p×q÷r is considered to be a single real number.

forward and backward

Given p,q. p×q=q×p

commutative property illustration

Commutative Property of Multiplication: Given p,q, p×q=q×r.

Commutative Property applied to Division: p÷q=1q×p.
Note: Division has to be handled as inverse of multiplication, p÷q=p×(1q) and then commutative property can be used.

Using Commutative Property: Given p,q,r, the expression p×q×p÷q×r÷p×r2 is simplified to pr3. students may work this out to understand.

with this or that

Given p,q,r. (p×q)×r=p×(q×r)

associative property illustration

Associative Property of Multiplication: Given p,q,r. (p×q)×r=p×(q×r).

Associative Property applied to Division:

(p÷q)÷r=p×(1q×1r).
Note: Division has to be handled as inverse of multiplication, (p÷q)÷r=(p×1q)×1r and then associative property can be used.

spread across

Given p,q,r. Which of the following equals (p+q)×r=p×r+q×r

The word 'distribute' means 'to share; to spread'

distributive property illustration

Distributive Property of Multiplication: Given p,q,r. (p+q)×r=p×r+q×r.

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=pr+qr.

 (2)  r÷(p+q)=rp+q. The multiplication does not distribute in this case.

 (3)  t÷r×(p+q)=t×(pr+qr).

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

one

Given p. What is p×1=p

Multiplicative Identity Property: For any p, there exists 1 such that p×1=p.

Multiplicative Identity applied to Division: p÷1=p

Note: 11=1 and so p÷1=p×(11)=p×1=p.

inverse

Given p. What is p×1p?

Multiplicative Inverse Property: For any p and p0, there exists 1p such that p×1p=1.

summary

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, then x×y

 •  Commutative Property
      x×y=y×x

 •  Associative Property
      (x×y)×z=x×(y×z)

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

 •  Multiplicative Identity Property
      1, such that x×1=x

 •  Multiplicative Inverse Property
      1x for any x (except for 0) such that x×(1x)=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×1y=1y×x

      Associative property involving division : (x÷y)÷z is given as (x×1y)×1z=x×(1y×1z)

      Distributive property involving division : p÷q×(x+y) is given as p×1q×(x+y)=p×(xq+yq)

Outline

The outline of material to learn "Algebra Foundation" is as follows.

Note: click here for detailed outline of Foundation of Algebra

    →   Numerical Arithmetics

    →   Arithmetic Operations and Precedence

    →   Properties of Comparison

    →   Properties of Addition

    →   Properties of Multiplication

    →   Properties of Exponents

    →   Algebraic Expressions

    →   Algebraic Equations

    →   Algebraic Identities

    →   Algebraic Inequations

    →   Brief about Algebra