All About Integers

• Integers are numbers that can be positive, negative, or zero, but without any fractional or decimal parts.
• Examples of integers: -3, -2, -1, 0, 1, 2, 3.
• Non-integers: 1.5, -2.7, 3/4.

Number Line Representation

• Integers can be visually represented on a number line.
• Zero is in the middle, positive integers extend to the right, and negative integers extend to the left.
• The number line helps in understanding the order and relative position of integers.

Absolute Value

• The absolute value of an integer is its distance from zero on the number line.
• Absolute value is always non-negative.
• Denoted by |x|, where x is an integer.
• Example: |-5| = 5 and |5| = 5.

Addition of Integers

• Adding two positive integers results in a positive integer.
• Example: 3 + 5 = 8.
• Adding two negative integers results in a negative integer.
• Example: (-3) + (-5) = -8.
• Adding a positive and a negative integer: Find the difference between their absolute values and use the sign of the integer with the larger absolute value.
• Example: (-7) + 4 = -3 (since |-7| > |4| and -7 is negative).
• Example: 7 + (-4) = 3 (since |7| > |-4| and 7 is positive).

Subtraction of Integers

• Subtracting an integer is the same as adding its opposite.
• a - b = a + (-b).
• Example: 5 - 3 = 5 + (-3) = 2.
• Example: 5 - (-3) = 5 + 3 = 8.
• Example: -5 - 3 = -5 + (-3) = -8.
• Example: -5 - (-3) = -5 + 3 = -2.

Multiplication of Integers

• Multiplying two positive integers results in a positive integer.
• Example: 3 * 5 = 15.
• Multiplying two negative integers results in a positive integer.
• Example: (-3) * (-5) = 15.
• Multiplying a positive and a negative integer results in a negative integer.
• Example: (-3) * 5 = -15.
• Example: 3 * (-5) = -15.

Division of Integers

• Dividing two positive integers results in a positive integer (if the division is exact).
• Example: 10 / 2 = 5.
• Dividing two negative integers results in a positive integer (if the division is exact).
• Example: (-10) / (-2) = 5.
• Dividing a positive and a negative integer results in a negative integer (if the division is exact).
• Example: (-10) / 2 = -5.
• Example: 10 / (-2) = -5.
• Division by zero is undefined.

Properties of Integer Operations

• Commutative Property:
  • Addition: a + b = b + a.
  • Example: 2 + 3 = 3 + 2.
  • Multiplication: a * b = b * a.
  • Example: 2 * 3 = 3 * 2.
• Associative Property:
  • Addition: (a + b) + c = a + (b + c).
  • Example: (1 + 2) + 3 = 1 + (2 + 3).
  • Multiplication: (a * b) * c = a * (b * c).
  • Example: (1 * 2) * 3 = 1 * (2 * 3).
• Distributive Property:
  • a * (b + c) = a * b + a * c.
  • Example: 2 * (3 + 4) = 2 * 3 + 2 * 4.
• Identity Property:
  • Addition: a + 0 = a.
  • Example: 5 + 0 = 5.
  • Multiplication: a * 1 = a.
  • Example: 5 * 1 = 5.

Integer Operations - Examples

• Evaluate: -8 + 5
  • Solution: -8 + 5 = -3
• Evaluate: 12 - (-6)
  • Solution: 12 - (-6) = 12 + 6 = 18
• Evaluate: -4 * 7
  • Solution: -4 * 7 = -28
• Evaluate: -20 / (-5)
  • Solution: -20 / (-5) = 4
• Simplify: 3 * (4 - 6)
  • Solution: 3 * (4 - 6) = 3 * (-2) = -6
• Simplify: -15 + (8 / 2)
  • Solution: -15 + (8 / 2) = -15 + 4 = -11
• Solve for x: x + 5 = 3
  • Solution: x = 3 - 5 = -2
• Solve for y: 2y = -10
  • Solution: y = -10 / 2 = -5

Comparing Integers

• On the number line, integers to the right are greater.
• Positive integers are greater than zero.
• Negative integers are less than zero.
• For two negative integers, the one closer to zero is greater.
• Examples: 5 > 2, -2 > -5, 0 > -3, -1 < 1.

Integer Word Problems

• A submarine is 200 feet below sea level. It descends another 50 feet. What is its new depth?
  • Solution: -200 + (-50) = -250 feet (250 feet below sea level).
• The temperature is -5°C. It rises by 8°C. What is the new temperature?
  • Solution: -5 + 8 = 3°C.
• A person deposits $50 in a bank account and then withdraws $80. What is the net change in their account balance?
  • Solution: 50 + (-80) = -$30 (a decrease of $30).