__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical expressions, equations, identities, and in-equations for Algebra__### Numerical Expressions

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.

*click on the content to continue..*

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.

*slide-show version coming soon*