__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical Arithmetics: Laws and Properties for Algebra__### Laws and Properties of Arithmetics : Addition

In this lesson, the laws and properties of addition are revised. The properties are Closure, Commutative, Associative, Additive Inverse, Additive Identity.

The subtraction is handled as inverse of addition and in that case, the properties mentioned above are applicable.

eg: Subtraction is not commutative `a-b !=b-a`

But Subtraction as inverse of addition `a-b = a+(-b) = -b+a`

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

*click on the content to continue..*

Consider the numbers `2` and `3`. Is `2+3` a real number?

- yes, a real number
- yes, a real number
- no, not a real number

The answer is "yes, a real number".

For any real numbers `p,q`, is `p+q` 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".

What does "closure" mean?

- closed
- not open
- both the above
- both the above

The answer is "both the above".

Closure Property of Addition: Given `p,q in RR`, `p+q in RR`. Closure Property applied to Subtraction: Given `p,q in RR`. `p-q in RR`.

Proof:

Given `p,q in RR`

`-q in RR` *as per Additive Inverse Property*

`p+(-q) in RR` *as per Closure property of Addition*

`=> p-q in RR`

Using Closure Property: Given `p,q,r,s in RR`, `p+q-r+s`, the subexpression `p+q-r` is a real 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.

Consider the numbers `2` and `3`. Which of the following equals `2+3`?

- `3+2`
- `3+2`
- `3+2` does not equal to `2+3`

The answer is "`3+2`".

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

- `q+p`
- `q+p`
- `p+q` does not equal to `q+p`

The answer is "`q+p`".

What does "commute" mean?

- to go to and fro on a regular basis
- to go to and fro on a regular basis
- It is not an English word.

The answer is "to go to and fro between two places on a regular basis".

Commutative Property of Addition: Given `p,q in RR`, `p+q=q+r`. Commutative Property applied to Subtraction: `p-q=-q+p`.

Note: Subtraction has to be handled as inverse of addition, `p-q = p+(-q)` and then commutative property can be used.

Using Commutative Property: Given `p,q,r in RR`, the expression `p+q+p-q+r-p+2r` is simplified to `p+3r`. *students may work this out to understand.*

Consider `2`, `3`, and `7` Which of the following equals `(2+3)+7`?

- `2+(3+7)`
- `2+(3+7)`
- `(2+7)+(3+7)`

The answer is "`2+(3+7)`"

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

- `p+(q+r)`
- `p+(q+r)`
- `(p+r)+(q+r)`

The answer is "`p+(q+r)`"

What does "associate" mean?

- to connect with; to join
- to connect with; to join
- to separate

The answer is 'to connect with; to join'.

Associative Property of Addition: Given `p,q,r in RR`. `(p+q)+r = p+(q+r)`. Associative Property applied to subtraction: `(p-q)-r = p+(-q-r)`.

Note: Subtraction has to be handled as inverse of addition, `(p-q)-r = (p+(-q))+(-r)` and then associative property can be used.

Given `p in RR`. What is `p+0`?

- `p`
- `p`
- `0`

The answer is "`p`"

Additive Identity Property: For any `p in RR`, there exists `0 in RR` such that `p+0=p`. Additive Identity applied to Subtraction: `p-0 = p`

Note: `-0=0` and so `p-0=p+(-0)=p+0 = p`.

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

- `p`
- `0`
- `0`

The answer is "`0`"

Additive Inverse Property: For any `p in RR`, there exists `-p in RR` such that `p+(-p)=0`.

The properties together are named as **CADI** properties of addition. 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 multiplication and is explained in the next page. *

**LPA: CADI properties of Addition **

• Closure Property

if `x,y in RR`, then `x+y in RR`

• Commutative Property

`x+y = y+x`

• Associative Property

`(x+y)+z = x+(y+z)`

• Additive Identity Property

`0 in RR`, such that `x+0 = x`

• Additive Inverse Property

`-x in RR` for any `x in RR` such that `x+(-x) = 0`

Subtraction is to be handled as additive inverse and properties of addition applies to subtraction in the form of addition.

This is important as algebra extensively uses these properties.

→ Commutative property involving subtraction : `x-y` is given as `x+(-y) ``= (-y) + x`

→ Associative property involving subtraction : `(x-y)-z` is given as `(x+ (-y))+(-z) ``= x+ ((-y)+(-z))`

*slide-show version coming soon*