This page extends the addition in first principles into a simplified procedure for addition of integers, which is called *sign property of integer addition*.

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In whole numbers, we had studied the following.**Addition by Place-value with regrouping** -- simplified procedure : Two numbers are added as per the following procedure:

• the place-value positions are arranged units under units, `10`s under `10`s, etc.

• the units are added and if the result has `10` numbers, then it is carried to the `10`s place.

• The addition is continued to the higher place-value position. The carry over is the simplification of combining `10` of a place value to a higher place value.

What does the terminology "received" and "given" are used in this course on integers?

- they refer only to candies received and given
- they symbolically refer to the positive and negative numbers of all applications
- they symbolically refer to the positive and negative numbers of all applications

The answer is "they symbolically refer to the positive and negative of all applications".

For example, temperature `3^@` above `0` and `2^@` below `0` are referred as `text(received:)3` and `text(given:)2` respectively. The same in integers is `3` and `-2`.

Integers are directed whole numbers.

`text(received:)3` is also referred as `text(aligned:)3`, meaning the magnitude given by the number is aligned to the chosen direction.

`text(given:)2` is also referred as `text(opposed:)2`, meaning the magnitude given by the number is opposed to the chosen direction.

For example, temperature `3^@` above `0` and `2^@` below `0` are referred as `text(aligned:)3` and `text(opposed:)2` respectively. The same in integers is `3^@C` and `-2^@C`.

What is the sum of `4` and `3`?

By first principles, to `text(received:)4` another `text(received:)3` is put-in.

- `43`
- `7`
- `7`

The answer is "`7`".

What is the sum of `4` and `-3`?

By first principles, to `text(received:)4` another `text(given:)3` is put-in.

- `+1`
- `+1`
- `-1`

The answer is "`1`". Note that the sum of the given two numbers is the difference of the absolute values of the numbers.

Absolute value of the numbers `|4| = 4` and `|-3|=3`.

Difference of absolute values `4-3=1`

What is the sum of `4` and `-7`?

By first principles, to `text(received:)4` another `text(given:)7` is put-in.

- `+3`
- `-3`
- `-3`

The answer is "`-3`"

What is the sum of `-4` and `-3`?

By first principles, to `text(given:)4` another `text(given:)3` is put-in.

- `+7`
- `-7`
- `-7`

The answer is "`-7`".

Summary of integer addition illustrative examples:

• `4+3 = 7` : To `text(received:)4` another `text(received:)3` is put-in

• `4+(-3) = 1` : To `text(received:)4` another `text(given:)3` is put-in

• `-4 + 3 = -1` : To `text(given:)4` another `text(received:)3` is put-in

• `4+(-7) = -3` : To `text(received:)4` another `text(given:)7` is put-in

• `-4+(-3) = -7` : To `text(given:)4` another `text(given:)3` is put-in

Based on this, the addition is simplified as**sign-property of addition**

• +ve + +ve is whole number addition of absolute values

• +ve + -ve is subtraction as given below

absolute values of the two numbers are compared

subtract the number of smaller absolute value from the number of larger absolute value. The difference is the absolute value of the result.

the sign of result is the sign of the number with larger absolute value

• -ve + -ve The result is sum of absolute values with negative sign.

The addition or subtraction of absolute values in the procedure are detailed in *addition by place-value* and *subtraction by place-value*.

**Addition of Integers Simplified Procedure** : Two numbers with signs positive or negative are added as follows. **sign-property of addition**

• +ve `+` +ve is whole number addition

• +ve `+` -ve is subtraction as given below

absolute values of the two numbers are compared

subtract the number of smaller absolute value from the number of larger absolute value. The difference is the absolute value of the result.

the sign of result is the sign of the number with larger absolute value

• -ve `+` -ve is addition of absolute values with negative sign.

The addition or subtraction of absolute values in the procedure are detailed in *addition by place-value* and *subtraction by place-value*

*Solved Exercise Problem: *

Find the sum of `-6` and `0`.

- `-6`
- `-6`
- `+6`

The answer is "`-6`"

*Solved Exercise Problem: *

Find the sum of `11` and `-140`

- `-30`
- `-129`
- `-129`

The answer is "`-129`"

*Solved Exercise Problem: *

Find the sum of `2` and `-2`

- `0`
- `0`
- `4`

The answer is "`0`"

*Solved Exercise Problem: *

Find the sum of `-2345` and `-889`

- `-3234`
- `-3234`
- `-1456`

The answer is "`-3234`"

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