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Thought-Process to Discover Knowledge

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mathsFoundation of Algebra with Numerical ArithmeticsNumerical Arithmetics: Revision for Algebra

### Numbers and Arithmetic Operations

This page revises the numbers quickly. It is important to understand the following concepts from this lesson, Like Laws and Properties of Arithmetic : Numbers and Operations :

•  Ordinal property of numbers

•  Comparison (greater, equal, or lesser)

•  division (inverse of multiplication)

•  exponent (repeated multiplication)

•  root (one inverse of exponent)

•  logarithm (another inverse of exponent)

click on the content to continue..

What are numbers?

• a value of measure or count
• a representation of quantity or amount
• both the above
• both the above

The answer is "both the above".

which of the following is the fundamental property of numbers?

• numbers are in ordered sequence representing the magnitude of quantity represented by them
• numbers are in ordered sequence representing the magnitude of quantity represented by them
• numbers are not ordered sequence

The answer is "numbers are in ordered sequence". The ordered sequence represents the magnitude of quantity represented by the numbers.

The whole numbers are in the ordered sequence

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,  etc. This is the ordinal property of numbers.

Which of the following is a meaning for the word "ordinal"?

• relating to order or series
• relating to order or series
• a person in authority to pass orders

The answer is "relating to order or series".

What is the term used to refer "relating to order in a sequence"?

• Pronunciation : Say the answer once

What are "integers"?

• negative numbers and whole numbers
• negative numbers and whole numbers
• inside a house made of timber

The answer is "negative numbers and whole numbers".

Integers are also called directed numbers.

Consider a girl and her brother sharing candies. She takes 2 candies one day and on another day her brother takes 3 candies from her.

This represents a directed transaction, "She takes from him" or "He takes from her".
The same can be given as "She takes" or "She gives".

This form of directed representation of numbers is captured with positive and negative numbers under "integers".

Does integers have ordinal property?

• No, with negative numbers the ordered sequence is not maintained
• Yes, the sequence is ...,-2, -1, 0, 1, 2, ...
• Yes, the sequence is ...,-2, -1, 0, 1, 2, ...

The answer is "Yes, the sequence is ...,-2, -1, 0, 1, 2, ..."

What are "fractions" and "decimals"?

• they represent counting parts of whole
• they represent counting parts of whole
• they represent natural numbers

The answer is "they represent counting parts of whole"

Fractions and Decimals are two forms of numbers representing "part of whole".

Consider a girl and her brother sharing candies. She split a large candy bar into 4 pieces and gave 3 pieces to her brother. The pieces are part of whole with place value 1/4 and the count 3 pieces. The number is represented as 3/4. The count 3 is the numerator. The place value 1/4 is given by the denominator 4.

The same can be represented with standardized place-value to 1/10 and the place value need not be specified. The number is represented as 0.75.

These two forms of "part-of-whole" representation of numbers are fractions and decimals respectively.

Do fractions and decimals have the ordinal property?

• No, the ordered sequence is not maintained when split into whole
• Yes, the ordered sequence is maintained with numerator when denominators are made equal
• Yes, the ordered sequence is maintained with numerator when denominators are made equal

The answer is "Yes, the ordered sequence is maintained with numerator when denominators are made equal"

Ordinal Property of numbers is used in comparison of numbers.

4>2 means 4 is positioned higher in the ordinal to 2

4=IV means 4 & IV are in the same ordinal position

3<7 means 3 is positioned lower in the ordinal to 7

Which operation represent "putting-together" two quantities?

• division

What is 3+2?

• 5
• 5
• 32

The answer is "5".

Addition is one of the arithmetic operations.

Which operation represent "taking away" part of a quantity?

• multiplication
• subtraction
• subtraction

What is 5-2?

• 52
• 3
• 3

The answer is "3".

Subtraction is one of the arithmetic operations. Subtraction 5-2=3 is the inverse of addition 3+2=5.

Which operation represent "repeated addition" of a quantity?

• multiplication
• multiplication
• division

What is 3xx2?

• 32
• 6
• 6

The answer is "6".

Multiplication is one of the arithmetic operations. Multiplication 3xx2 is repeated addition 3+3.

Which operation represent "splitting" a quantity?

• division
• division

The answer is "division". Division is inverse of multiplication.

What is 6-:2?

• 3
• 3
• 2

The answer is "3".

Division is one of the arithmetic operations. Division 6-:2=3 is inverse of multiplication 3xx2 = 6

Which operation represent "repeated multiplication" of a quantity?

• exponent
• exponent

What is 3^2?

• 6
• 9
• 9

The answer is "9".

Exponent is one of the arithmetic operations. Exponent 3^2=9 is repeated multiplication 3xx3=9

What is the inverse operation of exponent?

• root
• logarithm
• logarithm
• both the above

The answer is "both the above". The two forms of inverse of exponents are root and logarithm.

eg:
Exponent form 3^4 = 81
Inverse of exponent given the power root(4)(81) = 3
Inverse of exponent given the base log_3(81) = 4

What is sqrt(9)?

• 9
• 3
• 3

The answer is "3".

Root is one of the arithmetic operations. Root 9^(1/2)=3 is one form of inverse of exponent 3^2=9

What is log_3 9?

• 3
• 2
• 2

The answer is "2".

log is one of the arithmetic operations. Logarithm log_3 9=2 is one form of inverse of exponent 3^2=9

Note that inverse of addition is subtraction and inverse of multiplication is division. But inverse of exponent has two forms: roots and logarithms.

This is because 3+2=2+3 = 5 so inverse to get the left of the operator or right of the operator is defined by a single inverse operator.

But 3^2 != 2^3, and so,
inverse to get the left of exponent is root sqrt(9)=3 and
inverse to get the right of exponent is logarithm log_3 9 = 2.

Laws and Properties of Arithmetic : Numbers and Operations :

•  Ordinal property of numbers

•  Comparison (greater, equal, or lesser)