maths > integers

Integer Addition : First Principles

what you'll learn...

overview

In whole numbers, we had studied that addition is combining two quantities represented by the numbers to form a result representing a combined count or measurement.

The integers, both positive and negative integers, are considered and addition is explained for different possibilities.

combine them

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$20$ and 13$13$ are combined together 20+13=33$20 + 13 = 33$.

20$20$ is an addend

13$13$ is also an addend

33$33$ is the sum

Integers are directed whole numbers.

$3$ is understood as $\textrm{\left(r e c e i v e d\right\rangle} 3$ or aligned to the direction.

$- 3$ is understood as $\textrm{\left(g i v e n\right\rangle} 3$ or opposed to the direction.

receive and give

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 $3$.

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

She made two transactions, received $3$ candies and later received another $2$ candies. The numbers in integer forms are $\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$ and $\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$.

The total candies received is $3 + 2 = 5$

receive more, give less

She made two transactions, received $3$ candies and given away $2$ candies. The numbers in integer forms are $\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$ and $\textrm{\left(g i v e n\right\rangle} 2 = - 2$.

The total candies received is, $3 + \left(- 2\right) = 1$

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 $\textrm{\left(r e c e i v e d\right\rangle} 3$ and $\textrm{\left(g i v e n\right\rangle} 2$.

The addition is explained as
$\textrm{\left(r e c e i v e d\right\rangle} 3$ is the addend
$\textrm{\left(g i v e n\right\rangle} 2$ is put-in with the addend
$= \textrm{\left(r e c e i v e d\right\rangle} 1$ is the result.

The same in integer form
$= 3 + \left(- 2\right)$
$= 1$

receive less, give more

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 $\textrm{\left(g i v e n\right\rangle} 5 = - 5$ and $\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$.

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

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 $\textrm{\left(g i v e n\right\rangle} 5$ and $\textrm{\left(r e c e i v e d\right\rangle} 2$.

The addition is explained as
$\textrm{\left(g i v e n\right\rangle} 5$ is the addend
$\textrm{\left(r e c e i v e d\right\rangle} 2$ is accepted with the addend
$= \textrm{\left(g i v e n\right\rangle} 3$ is the result.

The same in integer form
$= \left(- 5\right) + 2$
$= - 3$

give and give

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 $\textrm{\left(g i v e n\right\rangle} 4 = - 4$ and $\textrm{\left(g i v e n\right\rangle} 5 = - 5$.

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

Considering addition of $- 4$ and $- 5$. The numbers in the directed whole numbers form are $\textrm{\left(g i v e n\right\rangle} 4$ and $\textrm{\left(g i v e n\right\rangle} 5$.

The addition is explained as
$\textrm{\left(g i v e n\right\rangle} 4$ is the addend
$\textrm{\left(g i v e n\right\rangle} 5$ is put-in with the addend
$= \textrm{\left(g i v e n\right\rangle} 9$ is the result.

The same in integer form
$= \left(- 4\right) + \left(- 5\right)$
$= - 9$

integer addition

The summary of integer addition illustrative examples:

•  $5 + 3 = 8$
Addition of $\textrm{\left(r e c e i v e d\right\rangle}$ and $\textrm{\left(r e c e i v e d\right\rangle}$ result in $\textrm{\left(r e c e i v e d\right\rangle}$

•  $3 + \left(- 2\right) = 1$
$\textrm{\left(r e c e i v e d\right\rangle} 3$ is more than $\textrm{\left(g i v e n\right\rangle} 2$. When received is larger, the addition of $\textrm{\left(r e c e i v e d\right\rangle}$ and $\textrm{\left(g i v e n\right\rangle}$ result in $\textrm{\left(r e c e i v e d\right\rangle}$

•  $\left(- 5\right) + 3 = - 2$
$\textrm{\left(g i v e n\right\rangle} 5$ is more than $\textrm{\left(r e c e i v e d\right\rangle} 3$; When given is larger, the addition of $\textrm{\left(r e c e i v e d\right\rangle}$ and $\textrm{\left(g i v e n\right\rangle}$ result in $\textrm{\left(g i v e n\right\rangle}$

•  $\left(- 5\right) + \left(- 3\right) = - 8$
Addition of $\textrm{\left(g i v e n\right\rangle}$ and $\textrm{\left(g i v e n\right\rangle}$ result in $\textrm{\left(g i v e n\right\rangle}$

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

summary

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

Examples are:
•  $\textrm{\left(r e c e i v e d\right\rangle} 3$ + $\textrm{\left(r e c e i v e d\right\rangle} 2$ = $\textrm{\left(r e c e i v e d\right\rangle} 5$
$3 + 2 = 5$

•  $\textrm{\left(r e c e i v e d\right\rangle} 3$ + $\textrm{\left(g i v e n\right\rangle} 2$ = $\textrm{\left(r e c e i v e d\right\rangle} 1$

• $\textrm{\left(g i v e n\right\rangle} 3$ & $\textrm{\left(r e c e i v e d\right\rangle} 2$ = $\textrm{\left(g i v e n\right\rangle} 1$
$- 3 + 2 = - 1$

• $\textrm{\left(r e c e i v e d\right\rangle} 3$ & $\textrm{\left(g i v e n\right\rangle} 5$ = $\textrm{\left(g i v e n\right\rangle} 2$

• $\textrm{\left(g i v e n\right\rangle} 3$ + $\textrm{\left(g i v e n\right\rangle} 2$ = $\textrm{\left(g i v e n\right\rangle} 5$

Outline