Addition & Subtraction of Integers
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers consisting of the natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, -2, -3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... -2, -1, 0, 1, 2, ...}. For example, 65, 7, and -756 are integers; 1.6 and 1½ are not integers. In other terms, integers are the numbers you can count with items such as apples or your fingers, and their negatives, including 0.
Integers are denoted with the following symbol:
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Addition
Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection. Repeated addition of the number one (1) is the most basic form of counting.
Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equal sign. For example,
Subtraction
Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with.. Subtraction is denoted by a minus sign, "-," in infix notation.
Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition:
.
From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:
.
Now, imagine a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.
To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction.
The solution is to consider the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …). From 3, it takes 4 steps to the left to get to −1, so
.
Addition and subtraction
For purposes of addition and subtraction, one can think of negative numbers as debts.
Adding a negative number is the same as subtracting the corresponding positive number:
- (if you have $5 and acquire a debt of $3, then you have a net worth of $2)
Subtracting a positive number from a smaller positive number yields a negative result:
- (if you have $4 and spend $6 then you have a debt of $2).
Subtracting a positive number from any negative number yields a negative result:
- (if you have a debt of $3 and spend another $6, you have a debt of $9).
Subtracting a negative is equivalent to adding the corresponding positive:
- (if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).
Also:
- (if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).
Quick Rules To Remember
- When subtracting, use "keep it, change it, opposite."
- If the numbers have the same sign, add them.
- If the numbers have different signs, subtract them.
- Always use the sign of the bigger number.
