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Thought-Process to Discover Knowledge

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mathsIntegersInteger Multiplication & Division

### Integer Multiplication : First Principles

The first principles of multiplication is, repeatedly combine a quantity a number of times and count or measure the combined quantity. This page explains the same for integers, which are directed whole numbers.

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In whole numbers, we had studied the following.

Multiplication - First Principles : Two numbers are considered, each of which represents a count or measurement. One of the amount is repeated the number of times given by the second amount to form the result representing the collective amount from such repetition.

eg: 12 xx 3 = 36

12 is the multiplicand

3 is the multiplier

36 is the product

In whole numbers, what does 3xx2 mean?

• multiplication is just multiplication. No meaning
• multiplicand 3 is repeatedly put-in multiplier 2 times
• multiplicand 3 is repeatedly put-in multiplier 2 times

The answer is "multiplicand 3 is repeatedly put-in multiplier 2 times"

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 referred as text(aligned:)2=2 and text(opposed:2)=-2

Integers are "directed" whole numbers. A whole-number multiplication represents repeating the multiplicand multiplier number of times.

In integers, What do a positive or a negative multiplier represent?

• positive multiplier represents repeatedly putting-in
• negative multiplier represents repeatedly taking-away
• both the above
• both the above

The answer is "both the above".

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.

In the box, 2 received is put in 3 times. The numbers in the integer forms are text(received:)2=2 and text(received:)3=3.

How many candies are received?

• 2xx3=6
• text(received:)2 xx text( received:)3 = text(received:)6
• 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.

In the box, 2 given is put in 3 times. The numbers in the integer forms are text(given:)2=-2 and text(received:)3=3.

How many candies are received?

• 2xx3=6
• text(given:)2 xx text(received:)3 = text(given:)6 = -6
• text(given:)2 xx text(received:)3 = text(given:)6 = -6

The answer is "-6". This is explained in the next page.

Considering multiplication of -2 and 3. The numbers are given in integer form. The numbers in directed whole numbers form are text(given:)2 and text(received:)3.

The multiplication is explained as
text(given:)2=-2 is the multiplicand
text(received:)3 is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
-2 put-in 3 times is given as -2+(-2)+(-2) = -6.
Thus the product of the multiplication is =text(given:)6.

The same in integer form
=(-2)xx3
=-6

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

In the box, 2 received is taken-away 3 times. The numbers in the integer forms are text(received:)2=2 and text(given:)3=-3 (equivalent of multiplier taken-away).

How many candies are received?

• 2xx3=6
• text(received:)2 xx text(given:)3 = text(given:)6 = -6
• text(received:)2 xx text(given:)3 = text(given:)6 = -6

The answer is "-6". This is explained in the next page.

Considering multiplication of 2 and -3. The numbers are given in integer form. The numbers in directed whole numbers form are text(received:)2 and text(given:)3.

The multiplication is explained as
text(received:)2 is the multiplicand
text(given:)3 = -3 is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
Putting in text(given:)3 times is equivalently taking away text(received:)3 times. This was explained in lesson handling signs

2 taken away 3 times is given as -2-2-2 = -6.
Thus the product of the multiplication is =text(given:)6.

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

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

In the box, 2 given is taken away 3 times. The numbers in the integer forms are text(given:)2=-2 and text(given:)3=-3 (equivalent of multiplier taken-away).

How many candies are received?

• two negative numbers cannot be multiplied
• text(given:)2 xx text(given:)3 = text(received:)6 = 6
• text(given:)2 xx text(given:)3 = text(received:)6 = 6

The answer is "6". This is explained in the next page.

Considering multiplication of -2 and -3. The numbers are given in integer form. The numbers in directed whole numbers form are text(given:)2 and text(given:)3.

The multiplication is explained as
text(given:)2=-2 is the multiplicand
text(given:)3=-3 is the multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
Putting in text(given:)3 times is equivalently taking away text(received:)3 times. This was explained in lesson handling signs

-2 put-in -3 times is given as -(-2)-(-2)-(-2) = 6.
Thus the product of the multiplication is =text(received:)6.

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

The summary of integer multiplication illustrative examples:

•  2xx3 = 6
2 received is put in 3 times = 6 received

•  (-2)xx3 = -6
2 given is put in 3 times = 6 given

•  2xx(-3) = -6
2 received is taken-away 3 times = 6 given

•  (-2)xx(-3) = 6
2 given is taken-away 3 times = 6 received

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

Integer Multiplication First Principles : Directed whole numbers multiplication is repeating the multiplicand the multiplier number of times with direction taken into account.

Repeating positive number of times is represented as repeatedly putting-in
Repeating negative number of times is represented as repeatedly taking-away

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