3.2 The Integers

Integers are the building blocks of mathematics, encompassing whole numbers that can be positive, negative, or zero. They're represented on a number line, with positive numbers to the right of zero and negative numbers to the left.

Understanding integers is crucial for basic arithmetic operations. Addition, subtraction, multiplication, and division of integers follow specific rules based on their signs. These operations form the foundation for more complex mathematical concepts.

Understanding Integers

Integers on number lines

• Integers are whole numbers that can be positive, negative, or zero
  • Positive integers located to the right of zero on a number line (1, 2, 3)
  • Negative integers located to the left of zero on a number line (-1, -2, -3)
  • Zero is neither positive nor negative and is located at the center of the number line

Comparison of integer values

• Comparing integers
  • Integer to the right on the number line is greater than the integer to the left
  • Greater than symbol (>) indicates one integer is larger than another (5 > 2)
  • Less than symbol (<) indicates one integer is smaller than another (-3 < 1)
  • Equal to symbol (=) indicates two integers have the same value (-4 = -4)
• Ordering integers
  • Arrange integers from left to right on the number line to order from least to greatest (-5, -2, 0, 3, 7)
  • Arrange integers from right to left on the number line to order from greatest to least (8, 4, 1, -1, -6)
• Absolute value of an integer is its distance from zero on the number line, regardless of its sign

Arithmetic operations with integers

• Addition
  • Add integers with the same sign (both positive or both negative) by adding their absolute values and keeping the sign $(+5) + (+3) = +8$, $(-2) + (-7) = -9$
  • Add integers with different signs by subtracting the absolute values and keeping the sign of the integer with the larger absolute value $(+4) + (-6) = -2$, $(-3) + (+8) = +5$
• Subtraction
  • Subtract integers by adding the opposite (additive inverse) of the subtrahend (number being subtracted) to the minuend (number being subtracted from)
    1. $(+7) - (+2) = (+7) + (-2) = +5$
    2. $(-5) - (-9) = (-5) + (+9) = +4$
    3. $(+6) - (-3) = (+6) + (+3) = +9$
• Multiplication
  • Multiply integers by multiplying their absolute values and following the sign rules:
    • Same signs (both positive or both negative) result in a positive product $(+3) \times (+5) = +15$, $(-2) \times (-4) = +8$
    • Different signs (one positive and one negative) result in a negative product $(+6) \times (-2) = -12$, $(-7) \times (+3) = -21$
• Division
  • Divide integers by dividing their absolute values and following the sign rules (same as multiplication):
    • Same signs (both positive or both negative) result in a positive quotient $(+15) \div (+3) = +5$, $(-24) \div (-4) = +6$
    • Different signs (one positive and one negative) result in a negative quotient $(+18) \div (-2) = -9$, $(-35) \div (+5) = -7$

Properties of Integers

• Even numbers are integers divisible by 2 without a remainder
• Odd numbers are integers not divisible by 2 without a remainder
• Rational numbers include all integers, as they can be expressed as a ratio of two integers
• Integer exponents represent repeated multiplication of a base number by itself, with negative exponents indicating reciprocals