Integers
What are Integers?
Integers are whole numbers that do not have fractional or decimal parts. They can be positive, negative, or zero:
- Positive Integers: Numbers greater than zero (1, 2, 3, 4, …)
- Negative Integers: Numbers less than zero (-1, -2, -3, -4, …)
- Zero (0): Neither positive nor negative, but still an integer.
Difference Between Integers and Whole Numbers
While integers include both positive and negative numbers as well as zero, whole numbers are only the non-negative integers, meaning they start from zero and include positive numbers without negatives.
- Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
- Whole Numbers: {0, 1, 2, 3, …}
Why Integers are Easy to Work With
Integers are often seen as simple and familiar because they follow straightforward counting rules. Positive integers move forward on a number line, while negative integers move backward, offering a clear way to visualize and understand them.
- Counting Up (Positive Integers): 1, 2, 3, and so on.
- Counting Down (Negative Integers): -1, -2, -3, and so on.
Integers are essential in everyday math, making them a fundamental concept that underpins many mathematical operations and problem-solving scenarios. Whether dealing with positive gains or negative losses, integers provide a clear and structured way to navigate numbers.
Is 12.75 an Integer?
Decimal numbers are not integers because they contain decimal or fractional parts. For example, in the number 12.75, the “.75” represents a decimal part, which means the number is not whole.
Integers are defined as numbers without any fractional or decimal components, meaning they must be whole numbers like -3, 0, 7, etc. Since 12.75 has a decimal part, it does not meet the criteria of being an integer, which requires the absence of any value beyond the decimal point.
Is 8.0 an integer?
Answer: Yes, 8.0 is considered an integer because, although it has a decimal point, the decimal part is zero. This means it represents the whole number 8, which fits the definition of an integer.
In mathematics, the presence of “.0” does not change the fact that it is a whole number. It is only in certain fields, such as computer science, where the distinction between 8 and 8.0 might matter due to data types, but mathematically, they are treated as the same integer.
Zero is an Integer
Zero is considered an integer because it can be expressed as a whole number without any fractional or decimal parts. It fits the definition of integers, which include all whole numbers, both positive and negative, as well as zero itself.
Although zero is neither positive nor negative, it still belongs to the set of integers because it represents a whole value. Zero acts as the neutral point on the number line, separating positive integers from negative integers, and plays a unique role in mathematics as an integer without any direction or value beyond itself.
Addition of Integers
Adding Integers and Zero
Adding zero to any integer is the simplest form of addition because zero has no effect on the value of the number. When zero is added, it’s like adding nothing at all. Simply ignore the zero and consider the remaining numbers to find the sum.
Key Concept: Adding zero to any integer leaves the integer unchanged.
Example 1: What is the sum when 37 and 0 are added together?
Solution:
Since zero does not change the value, the sum is the other number, 37.
37 + 0 = 37
Example 2: Evaluate −18 + 0
Solution:
Zero does not affect the integer, so the sum remains the same.
−18 + 0 = −18
Example 3: Evaluate −7 + 2 + 0
Solution:
Remove zero from the equation, and add the remaining integers:
−7 + 5 + 0 = – 2
In this case, the zero is irrelevant. What matters is understanding that adding zero has no impact on the overall sum.
This basic principle is crucial when working with integers, as it simplifies calculations by eliminating the zero and focusing on the non-zero values.
Addition of Two or More Positive Integers
When adding positive integers, think of each number as a step to the right on a number line, starting from zero. Every time you add a positive number, you move further to the right. This is the simplest and most straightforward addition—just combine the values as you would with regular numbers.
Key Concept: Adding positive integers involves a simple sum, just as you would add any whole numbers.
Example 1: Calculate 22 +38
Solution:
Add the numbers directly.
22 + 38 = 60
Example 2: What is the sum when 71 and 29 are added together?
Solution:
Add the numbers as usual.
71 + 29 =100
Example 3: Evaluate the following equation: 22+ 9 + 38 Solution:
You can add them step-by-step for simplicity:
22 + 9 + 38 = 69
Summary:
Adding positive integers is just like regular addition. Visualize it as moving steps to the right on a number line, and simply combine the values for the total sum.
Addition of Two or More Negative Integers
When adding negative integers, you can simplify the process by temporarily ignoring the negative signs. Treat the numbers as if they were positive, perform the addition, and then apply the negative sign to the final sum.
Key Concept: Add the numbers normally without considering the negative signs, then make the final result negative.
Steps to Add Negative Integers:
- Ignore the Negative Signs: Treat the numbers as positive.
- Add the Numbers: Sum the numbers as you normally would.
- Apply the Negative Sign: Prefix the result with a negative sign.
Addition of Positive and Negative Numbers Together
Identify the Larger Magnitude: Determine which integer has the larger absolute value (magnitude) and note its sign.
Subtract the Magnitudes: Subtract the smaller magnitude from the larger magnitude, ignoring the signs.
Apply the Sign: The result will take the sign of the integer with the larger magnitude from step 1.
| Problem | Solution |
|---|---|
| Addition of Integer and Zero | Disregard zero from the equation |
| Addition of two positive integers | Add the numbers together |
| Addition of two negative integers | Disregard the negative signs, add the numbers, put the negative sign |
| Addition of integers with different signs | Identify the larger number, subtract the smaller number from the larger number, use the sign of the larger number |
Multiplication of Integers
Integer Properties of Multiplication
A. Commutative Property
The order of the factors does not affect the product.
Example :
4 x 5 = 5 x 4
20 = 20
B. Associative Property
The grouping of the factors does not affect the product.
Example: (2 x 3) x 5 = 2 (3 x 5)
6 x 5 = 2 x 15
30 = 30
C. Identity Property
Any integer except zero (0) multiplied by one (1) has its product as itself.
Example:
4 x 1 = 4
4 = 4
4. Zero Property
Any integer multiplied by zero (0) has its product as zero.
Example: 8 x 0 = 0
0 = 0
5. Distributive Property
When an integer is multiplied by the sum or difference of two integers, it can be distributed and multiplied separately to each term within the sum or difference.
Example
4 x (8 + 2) = (4 x 8) + (4 x 2)
4 x 10 = 32 + 8
40 = 40
6. Negation Property
Any integer except zero (0) multiplied by -1 changes its sign to the other.
Example:
5 x (-1) = -5
-5 = -5
or (-3) x (-1) = 3
3 = 3
| Integer Property of Multiplication | Meaning |
|---|---|
| Commutative Property | Order of factors does not affect the product |
| Associate Property | Grouping of factors does not affect the product |
| Identity Property | any integer except 0 multiplied by 1 has itself as the product |
| Zero Property | any multiplied by 0 has 0 as the product |
| Distributive Property | any integer can be distributed to a sum or difference |
| Negation Property | any integer except 0 multiplied -1 has a changed sign as the product |
Division of Integers
Integer Properties of Division
1. Identity Property of Division
When any integer, except zero, is divided by one, the quotient remains the same as the original integer. This property highlights that dividing by one does not alter the value of the integer.
Example 1
5 ÷ 1 = 5
5 = 5
Example 2
-4 ÷ 1 = -4
-4 = -4
2. Self Property of Division
When any non-zero integer is divided by itself, the result is always one. This property emphasizes that dividing a number by itself simplifies the value to one, as each unit of the number perfectly matches the other
Example 1.
-7 ÷ -7 = 1
1 = 1
3. Zero Property of Division
When zero is divided by any non-zero integer, the quotient is always zero.
Example
0 ÷ 5 = 0
0 = 0
4. Undefined Property of Division
Dividing any integer by zero is undefined because it has no meaningful quotient. This is because dividing by zero implies splitting something into no parts, which is mathematically impossible and lacks a defined result.
Example:
5 ÷ 0 = Undefined
Dividing Integers with the Same Signs.
Positive ÷ Positive: When a positive integer is divided by another positive integer, the result is a positive quotient. For example, 12 ÷ 3 = 4
Negative ÷ Negative: When a negative integer is divided by another negative integer, the result is also a positive quotient. This is because the two negative signs cancel each other out. For example, −12 ÷ −3 = 4
In both cases, the rule is that dividing two integers with the same signs results in a positive quotient.
Dividing Integers with Different Signs
Positive ÷ Negative: When a positive integer is divided by a negative integer, the result is a negative quotient. For example, 12 ÷ −3 = −4 The positive and negative signs do not cancel out, so the quotient takes a negative sign.
Negative ÷ Positive: When a negative integer is divided by a positive integer, the result is also a negative quotient. For example, −12 ÷ 3 = −4. Here, the division results in a negative because the signs are different.
In both scenarios, when dividing integers with different signs, the quotient is always negative.
| Property | Meaning |
|---|---|
| Identity Property | any integer except 0 divided by 1 the quotient remains the same as the original integer |
| Self Property | any integer except 0 divided by itself the quotient is always 1 |
| Zero Property | zero divided by any non-zero integer, the quotient is always zero. |
| Undefined Property | any integer cannot be divided by 0 |
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