Integers

In the world of mathematics, numbers play an important role. As students, you may already be familiar with whole numbers and natural numbers. In this lesson, we will explore a more advanced group of numbers known as "integers."

What are integers?

Integers are a set of numbers that includes all whole numbers and their negative counterparts. They can be defined as follows:

Integers = {..., -3, -2, -1, 0, 1, 2, 3, ...}

As you can see from the definition, integers include:

• All positive numbers: 1, 2, 3, 4, 5, ...
• All negative numbers: -1, -2, -3, -4, -5, ...
• Zero: 0

Integers are represented by the letter "Z" which stands for "Zahlen", which is the German word meaning "numbers".

Visual representation of integers

Visualizing numbers often makes them easier to understand. Below is a simple representation of integers on a number line.

Examples of integers in real life

Understanding integers is not only important for math classes, but also helpful in many real-life situations. Here are some examples:

• Temperature: If the weather report says the outside temperature is -5°C, you know the temperature is below freezing point. Similarly, 32°C is a positive integer, indicating a warm or hot day.
• Bank accounts: If you have $500 in debt, this can be thought of as a negative integer, -500. Conversely, if you have $500 in savings, this is represented as a positive integer, +500.
• Altitude: Climbing a mountain can take you up to +2000 meters above sea level, while boarding a submarine can take you down to -200 meters below sea level.

Integer operations

We perform various mathematical operations with integers, just as we do with other numbers. The basic operations include addition, subtraction, multiplication, and division.

Add

Adding two integers can give a positive or negative number. When both integers are positive:

3 + 5 = 8

If both integers are negative, then the sum will also be negative:

(-3) + (-5) = -8

If one integer is positive and the other is negative, the result depends on which integer is larger:

5 + (-3) = 2

Subtraction

Subtraction of integers is the same as addition. Consider this expression:

5 - 3 = 2

Subtracting a negative number is equivalent to adding a positive number:

5 - (-3) = 5 + 3 = 8

Multiplication

Multiplying integers is simple. The rules involve the signs of the integers:

• Positive x Positive = Positive
• Negative x Negative = Positive
• Positive x Negative = Negative
• Negative x Positive = Negative

For example:

(-4) x 5 = -20 (-3) x (-6) = 18

Division

Division follows the same sign rules as multiplication:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

For example:

20 ÷ (-4) = -5 (-18) ÷ (-3) = 6

Properties of integers

Integers have special properties that help us do things efficiently:

1. Closing property

The result of adding, subtracting, or multiplying two integers will always be another integer. For example:

3 + (-5) = -2 4 - (-7) = 11 (-4) x 3 = -12

2. Exchangeable assets

This property applies to addition and multiplication, where the order of the integers does not change the result:

5 + (-3) = (-3) + 5 6 x (-2) = (-2) x 6

3. Associative property

The way the integers are grouped in addition and multiplication has no effect on the result:

(2 + 3) + 4 = 2 + (3 + 4) (2 x 3) x 4 = 2 x (3 x 4)

4. Additive identity

The number 0 is the additive identity. Adding 0 to an integer leaves it unchanged:

6 + 0 = 6

5. Qualitative identification

The number 1 is the multiplicative identity. Any integer multiplied by 1 remains unchanged:

(-7) x 1 = -7

6. Distributive property

This property allows us to multiply sums by multiplying each term separately and then adding the products:

3 x (4 + 2) = (3 x 4) + (3 x 2)

The negative of the negative

A unique aspect of integers is how negative signs interact. The negative of a negative integer is a positive one:

-(-5) = 5

This rule may seem confusing, but the concept of "moving in the opposite direction" may make it easier to understand.

Absolute value

The "integer value" of an integer is simply the distance of that number from zero on the number line, regardless of direction. This is represented by the vertical bars:

|-3| = 3 |5| = 5 |0| = 0

The absolute value is always a non-negative number.

Integer sequences

Integers can form various sequences. Here are some examples:

• Even integers: ...,-4, -2, 0, 2, 4, ...
• Odd integers: ...,-3, -1, 1, 3, 5, ...
• Consecutive integers: ...,-2, -1, 0, 1, 2, 3,...

Conclusion

Integers are a fundamental component of mathematics that extends our understanding of whole numbers into the realm of negative values and zero. By mastering the basics of integers, we create a strong foundation for exploring more complex mathematical concepts. Remember, like any new skill, becoming comfortable with integers takes practice and patience.

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