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### Integer Addition : First Principles

The first principles of addition is, combine two quantities 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.

Addition - First Principles : Two numbers are considered, each of which represents a count or measurement. The quantities represented by the numbers are combined to form a result representing a combined count or measurement. The combined count or measurement is the result of addition.

eg: 20 and 13 are combined together 20+13 = 33.

20 is an addend

13 is also an addend

33 is the sum

Integers are directed whole numbers.

3 is understood as text(received:) 3 or aligned to the direction.

-3 is understood as text(given:)3 or opposed to the direction.

A girl has a box of candies. The number of candies in the box is not counted. But she keeps track of how many candies she receives or how many she gives away. She maintains a daily account of how many are received or given.

She received 3 candies yesterday. The number of candies received is positive 3.

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

She gave 3 candies today. The number of candies received today is -3

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

She made two transactions, received 3 candies and later received another 2 candies. The numbers in integer forms are text(received:)3=3 and text(received:)2=2.

The total candies received is 3+2=5

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

She made two transactions, received 3 candies and given away 2 candies. The numbers in integer forms are text(received:)3=3 and text(given:)2=-2.

The total candies received is, 3+(-2) = 1

Considering addition of 3 and -2. The numbers are given in integer form. To understand first principles of addition, let us convert that to directed whole numbers form text(received:)3 and text(given:)2.

text(received:)3 is the addend
text(given:)2 is put-in with the addend
=text(received:)1 is the result.

The same in integer form
=3+(-2)
=1

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

She made two transactions, given 5 candies and then received 2 candies. The numbers in the integer forms are text(given:)5=-5 and text(received:)2=2.

On combining, the total number of candies is, (-5)+2 = -3

Considering addition of -5 and 2. The numbers are given in integer form. To understand first principles of addition, let us convert that to directed whole numbers form text(given:)5 and text(received:)2.

text(given:)5 is the addend
text(received:)2 is accepted with the addend
=text(given:)3 is the result.

The same in integer form
=(-5)+2
=-3

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

She gave 4 candies and another time, given away 5 candies. The numbers in integer forms are text(given:)4=-4 and text(given:)5=-5.

On combining, the total number of candies is (-4)+(-5)= -9

Considering addition of -4 and -5. The numbers in the directed whole numbers form are text(given:)4 and text(given:)5.

text(given:)4 is the addend
text(given:)5 is put-in with the addend
=text(given:)9 is the result.

The same in integer form
=(-4)+(-5)
=-9

The summary of integer addition illustrative examples:

•  5+3 = 8
text(received:)5 and text(received:)3 result in text(received:)

•  3+(-2) = 1
text(received:)3 is more than text(given:)2. When received is larger, the addition result in text(received:)

•  (-5)+3 = -2
text(given:)5 is more than text(received:)3; When given is larger, the addition result in text(given:)

•  (-5)+(-3) = -8
text(given:)5 and text(given:)3 result in text(given:)8

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

Integer Addition -- First Principles : Directed whole numbers addition is combining the two amounts with direction information taken into account.

Examples are:
•  text(received:)3 + text(received:)2 = text(received:)5
3+2 = 5

•  text(received:)3 + text(given:)2 = text(received:)1
3+(−2) = 3−2 = 1

• text(given:)3 & text(received:)2 = text(given:)1
-3+2 = -1

• text(received:)3 & text(given:)5 = text(given:)2
3+(−5) = 3−5 = −2

• text(given:)3 + text(given:)2 = text(given:)5
−3+(−2) = −5

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