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Thought-Process to Discover Knowledge

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mathsIntegersInteger Multiplication & Division

### Integer Division : First Principles

The first principles of division is, splitting a quantity into a number of parts and count or measure one part. This page explains the same for integers, which are directed whole numbers.

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In whole numbers, what does 6-:2 mean?

• division is just division, no meaning
• dividend 6 is split into 2 equal parts and one part is put in
• dividend 6 is split into 2 equal parts and one part is put in

The answer is "dividend 6 is split into 2 equal parts and one part is put in"

In integers what do 2 and -2 mean?

• no meaning for the numbers
• text(received:)2=2 and text(given:2)=-2
• text(received:)2=2 and text(given:2)=-2

The answer is "text(received:)2=2 and text(given:2)=-2". It is also called text(aligned:)2=2 and text(opposed:2)=-2

Integers are "directed" whole numbers.

A whole number division represents splitting the dividend into divisor number of parts and one part is put-in.

In integers, what do a positive or a negative divisor represent?

• positive divisor represents: one part is put-in
• negative divisor represents: one part is taken-away
• both the above
• both the above

The answer is "both the above". This is explained with an example in the coming pages.

A girl has a box of candies. The number of candies in the box is not counted. But, she maintains a daily account of how many are received or given.

6 received is split into 2 equal parts. In the box, one part of that is put-in. (To understand this : 6 candies received is shared with her brother and only her part is put in the candy-box.)

The numbers in the integer forms are text(received:)6=6 and text(received:)2=2. How many candies are received?

• 6-:2=3
• text(received:)6=6 is split into 2 parts and one part text(received:)3=3 is put-in
• both the above
• both the above

The answer is "both the above".

Considering the box of candies and the daily account of number of candies received or given.

6 given is split into 2 equal parts. In the box, one part of {2 equal parts of 6 given} is put-in. (Her brother and she gave 6 candies and only her part is reflected for her number.)

The numbers in the integer forms are text(given:)6=-6 and text(received:)2=2. How many candies are received?

• 6-:2=+3
• text(given:)6=-6 is split into 2 parts and one part text(given:)3=-3 is put-in
• text(given:)6=-6 is split into 2 parts and one part text(given:)3=-3 is put-in

The answer is "-3". This is explained in the next page.

Considering division of (-6)-:2. The numbers are given in integer form. The numbers in directed whole numbers form are text(given:)6 and text(received:)2.

The Division is explained as
text(given:)6=-6 is the dividend
text(received:)2 is divisor

Division is dividend split into divisor number of parts and one part is put-in.
-6 split into 2 parts is -3 and -3. One part of that is -3.
Thus the quotient of the division is =text(given:)3.

The same in integer form
=(-6)-:2
=-3

Considering the box of candies and the daily account of number of candies received or given.

6 received is split in 2 equals part of which one part is to be taken-away. From the box, one part of {2 equal part of 6 received} is taken-away. (Her brother and she returned 6 candies that was received earlier and only her part is reflected for her number.)

The numbers in the integer forms are text(received:)6=6 and text(given:)2=-2. How many candies are received?

• 6-:2=+3
• text(received:)6=6 is split into 2 parts and one part text(received:)3=3 is taken away, which is text(given:)3=-3
• text(received:)6=6 is split into 2 parts and one part text(received:)3=3 is taken away, which is text(given:)3=-3

The answer is "-3". This is explained in the next page.

Considering division of 6-:(-2). The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form text(received:)6 and text(given:)2.

The Division is explained as
text(received:)6=6 is the dividend
text(given:)2 = -2 is divisor

Division is dividend split into divisor number of parts and one part is taken away since divisor is negative.
6 split into 2 parts is 3 and 3. One part of that is 3. Since divisor is negative, one part 3 is taken-away. text(received:)3 taken away is text(given:)3.
Thus the quotient of the division is =text(given:)3.

The same in integer form
=6-:(-2)
=-3

Considering the box of candies and the daily account of number of candies received or given.

6 given is split into 2 equal parts of which one part is to be taken-away. In the box, a part of {2 equal part of 6 given} is taken-away. (Her brother and she got back 6 candies which were given earlier and only her part is reflected for her number.)

The numbers in the integer forms are text(given:)6=-6 and text(given:)2=-2. How many candies are received?

• (-6)-:(-2)=-3
• text(given:)6=-6 is split into 2 parts and one part text(given:)3=-3 is taken-away, which is text(received:)3=+3
• text(given:)6=-6 is split into 2 parts and one part text(given:)3=-3 is taken-away, which is text(received:)3=+3

The answer is "3". This is explained in the next page.

Considering division of (-6)-:(-2). The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form text(given:)6 and text(given:)2.

The Division is explained as
text(given:)6=-6 is the dividend
text(given:)2=-2 is divisor

Division is dividend split into divisor number of parts and one part is is taken away since the divisor is negative.
-6 split into 2 parts is -3 and -3. One part of that is -3. Since divisor is negative, one part -3 is taken-away. text(given:)3 taken away is text(received:)3.
Thus the quotient of the division is =text(received:)3.

The same in integer form
=(-6)-:(-2)
=3

The summary of integer division illustrative examples:

•  6-:2 = 3
6 received split into 2 parts and one part is put-in = 3 received

•  (-6)-:2 = -3
6 given split into 2 parts and one part is put-in = 3 given

•  6-:(-2) = -3
6 received split into 2 parts and one part is taken-away = 3 given

•  (-6)-:(-2) = 3
6 given split into 2 parts and one part is taken-away = 3 received

The above is concise form to capture the integer division in first principles.

What is the result of the division 7-:3?

By first principles, text(received:)7 is split into 3 equal parts and one part is put-in (positive divisor). The remainder is what is remaining in text(received:)7.

• 3
• quotient 2 and remainder 1
• quotient 2 and remainder 1

The answer is "quotient 2 and remainder 1".

This is verified with 2xx3 + 1 = 7 (quotient multiplied divisor + remainder = dividend )

What is the result of the division (-7)-:3?

By first principles, text(given:)7 is split into 3 equal parts and one part is put-in (positive divisor). The remainder is what is remaining in text(received:)7.

• quotient -2 and remainder +1
• quotient -2 and remainder -1
• quotient -2 and remainder -1

The answer is "quotient -2 and remainder -1".

This is verified with (-2)xx3 + (-1) = -7

What is the result of the division 7-:(-3)?

By first principles, text(received:)7 is split into 3 equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in text(received:)7.

• quotient -2 and remainder +1
• quotient -2 and remainder +1
• quotient -2 and remainder -1

The answer is "quotient -2 and remainder 1".

This is verified with (-2)xx(-3) + 1 = 7

What is the result of the division (-7)-:(-3)?

By first principles, text(given:)7 is split into 3 equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in text(received:)7.

• quotient 2 and remainder +1
• quotient 2 and remainder -1
• quotient 2 and remainder -1

The answer is "quotient 2 and remainder -1"

This is verified with 2xx(-3) + (-1) = -7

The summary of integer division illustrative examples:

•  7-:2 = 3 with 1 remainder
7 received split into 2 parts and one part is put-in = 3 received and remainder 1 received

•  (-7)-:2 = -3 with -1 remainder
-7 given split into 2 parts and one part is put-in = 3 given and remainder 1 given

•  7-:(-2) = -3 with 1 remainder
7 received split into 2 parts and one part is taken-away = 3 given and remainder 1 received

•  (-7)-:(-2) = 3 with -1 remainder
7 given split into 2 parts and one part is taken-away = 3 received and remainder 1 given.

Remainder takes the sign of the dividend.

Integer Division -- First Principles: Directed whole numbers division is splitting the dividend into divisor number of equal parts with direction taken into account.

If the divisor is positive, then one part is put-in.
If the divisor is negative, then one part is taken-away.
Remainder is that of the dividend retaining direction information.

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