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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.

Read in the blogs more about the unique learning experience at nubtrek.

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

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

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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