1. Introduction to Integers

Integers are a set of numbers that include all positive numbers, negative numbers, and zero. They do not include fractions or decimals.

• Examples of integers: ... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...
• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero is neither positive nor negative.

2. Representation of Integers on a Number Line

Integers can be represented on a number line. Positive integers are to the right of zero, and negative integers are to the left.

3. Comparing Integers

• On a number line, numbers to the right are greater.
• For any two integers, the one on the right is greater.
• Example: -2 < 0, 3 > -1

4. Addition and Subtraction of Integers

Addition

• Adding two positive integers: Add and keep the sign positive.
  Example: 3 + 5 = 8
• Adding two negative integers: Add and keep the sign negative.
  Example: (-2) + (-4) = -6
• Adding a positive and a negative integer: Subtract the smaller absolute value from the larger and keep the sign of the larger.
  Example: 7 + (-3) = 4, (-7) + 3 = -4

Subtraction

• Subtracting an integer is the same as adding its opposite.
  Example: 5 - 8 = 5 + (-8) = -3
• Example: (-4) - (-7) = (-4) + 7 = 3

5. Properties of Addition and Subtraction

• Closure: The sum or difference of any two integers is always an integer.
• Commutative Property (Addition): a + b = b + a
• Associative Property (Addition): (a + b) + c = a + (b + c)
• Subtraction is not commutative or associative.
• Zero is the identity for addition: a + 0 = a

6. Word Problems

1. The temperature was -3°C in the morning. It rose by 5°C by noon. What is the temperature at noon?
   Solution: -3 + 5 = 2°C
2. A submarine is at a depth of -120 m. It rises 30 m. What is its new depth?
   Solution: -120 + 30 = -90 m
3. Calculate: (-7) - (-2) = ?
   Solution: (-7) + 2 = -5

7. Practice Exercises

1. Represent the following integers on a number line: -5, 0, 3, -2, 4
2. Find the sum: (-8) + 6
3. Find the difference: 5 - (-3)
4. Which is greater: -4 or 2?
5. Solve: (-10) + (-5)

8. Common Mistakes

• Forgetting to change the sign when subtracting a negative integer.
• Confusing the rules for addition and subtraction of integers.
• Placing negative numbers to the right of zero on the number line.

9. Summary

• Integers include positive numbers, negative numbers, and zero.
• They can be represented on a number line.
• Learn the rules for addition and subtraction of integers.
• Practice helps avoid common mistakes.