Mathematics Chapter 2: Operations on Integers

Operations on Integers

• Operations include addition, subtraction, multiplication, and division of signed integers.
• Target skills: perform operations on integers, apply rules in word problems, define prime, composite, factors, and multiples, list prime factors, and find GCF and LCM.

Rules on Signed Numbers: Addition

• Same Sign:
  • Add the absolute values, keep the sign (e.g., 8 + 14 = 22; -5 + -5 = -10).
• Unlike Sign:
  • Subtract the smaller absolute value from the larger, keep the sign of the larger number (e.g., 54 + -12 = 42; -40 + 5 = -35).

Rules on Signed Numbers: Subtraction

• To subtract, change the sign of the subtrahend and follow addition rules (e.g., 56 - 23 becomes 56 + -23).

Rules on Signed Numbers: Multiplication

• Same Sign: Product is positive (e.g., -5 * -5 = 25).
• Unlike Sign: Product is negative (e.g., 8 * -3 = -24).

Rules on Signed Numbers: Division

• Similar to multiplication, the signs determine the result:
  • Same Sign: Result is positive (e.g., -12 ÷ -12 = 1).
  • Unlike Sign: Result is negative (e.g., 81 ÷ -3 = -27).

Prime and Composite Numbers

• Prime Numbers:
  • Natural numbers with exactly two positive factors: 1 and itself (e.g., 2, 3, 5).
• Composite Numbers:
  • Natural numbers greater than 1 with more than two factors (e.g., 35, factors are 1, 5, 7, 35).
• Note: 1 is neither prime nor composite, while 2 is the smallest prime number.

Prime Factorization

• The unique representation of a number as a product of prime numbers.
• All integers greater than 1 can be expressed as products of prime factors (Theorem).
• To find it:
  • Divide by the smallest prime (2), then subsequently by larger primes (3, 5, etc.) until reaching 1.

Recall Points for Operations

• For addition and subtraction, apply the rules depending on the signs. For multiplication and division, ensure to account for the signs properly to determine the result.