1  Review: integer arithmetic

We start with a short review of computing with integers.

1.1 Addition and subtraction

Here are two examples:

  • \(5 + 12 = 17\)

  • \(5-12 = -7\)

Subtraction and negative numbers

Subtraction is the same as the addition of the opposite/negative of a number:

  • \(7 - 4\) is the same as \(7 + (-4)\)

  • \(8 - (-5)\) is the same as \(8 + 5\)

Example 1.1 (Practice: addition and subtraction)  

  1. \(12 + 18=\)

  2. \(-9 + 15=\)

  3. \(7 + (-13)=\)

  4. \(-8 + (-12)=\)

  5. \(-14 + 9=\)

  6. \(11 + (-16)=\)

  7. \(20 - 13=\)

  8. \(-12 - 7=\)

  9. \(18 - (-9)=\)

  10. \(-15 - (-8)=\)

  1. \(12 + 18 = 30\)

  2. \(-9 + 15 = 6\)

  3. \(7 + (-13) = -6\)

  4. \(-8 + (-12) = -20\)

  5. \(-14 + 9 = -5\)

  6. \(11 + (-16) = -5\)

  7. \(20 - 13 = 7\)

  8. \(-12 - 7 = -19\)

  9. \(18 - (-9) = 27\)

  10. \(-15 - (-8) = -7\)

1.2 Multiplication

Multiplication is a short-cut for repeatedly adding. For example: \[ 4\times 3 = 3+3+3+3 = 12. \] Here is another example: \[ 3\times (-5) = (-5)+(-5)+(-5) = -15. \]

Multiplication notation

We have three different ways to write multiplication. Here are three different ways to write “three times four”.

  • Cross notation: \(3\times 4 = 12\)
  • Dot notation: \(3\cdot 4 = 12\)
  • Neighbor notation: \(3(4) = 12\)

Example 1.2 (Practice: multiplication)  

  1. \(7 \cdot 9=\)

  2. \((-5)(8)=\)

  3. \(12 \cdot (-4)=\)

  4. \(6(11)=\)

  5. \((-8) \cdot (-3)=\)

  6. \(9(-7)=\)

  7. \(15 \cdot 4=\)

  8. \((-6)(-9)=\)

  1. \(7 \cdot 9 = 63\)

  2. \((-5)(8) = -40\)

  3. \(12 \cdot (-4) = -48\)

  4. \(6(11) = 66\)

  5. \((-8) \cdot (-3) = 24\)

  6. \(9(-7) = -63\)

  7. \(15 \cdot 4 = 60\)

  8. \((-6)(-9) = 54\)

1.3 Division

Division is the un-doing of multiplication. Sometimes, numbers divide evenly: \[ 48\div 6 = 8. \] Other times, there is a remainder: \[ 26\div 6 = 4\text{ R } 2. \]

Example 1.3 (Practice: division)  

  1. \(72 \div 8=\)

  2. \(45 \div 5=\)

  3. \(63 \div 9=\)

  4. \(56 \div 7=\)

  5. \(84 \div 12=\)

  6. \(90 \div 6=\)

  7. \(50 \div 7=\)

  8. \(65 \div 8=\)

  9. \(58 \div 6=\)

  10. \(77 \div 9=\)

  1. \(72 \div 8 = 9\)

  2. \(45 \div 5 = 9\)

  3. \(63 \div 9 = 7\)

  4. \(56 \div 7 = 8\)

  5. \(84 \div 12 = 7\)

  6. \(90 \div 6 = 15\)

  7. \(50 \div 7 = 7\) R \(1\)

  8. \(65 \div 8 = 8\) R \(1\)

  9. \(58 \div 6 = 9\) R \(4\)

  10. \(77 \div 9 = 8\) R \(5\)

1.4 Homework exercises

Exercise 1.1 Compute.

  1. \(8 + 15\)

  2. \(-7 + 12\)

  3. \(9 + (-14)\)

  4. \(-6 + (-11)\)

  5. \(-13 + 8\)

  6. \(5 + (-9)\)

  7. \(12 - 7\)

  8. \(-8 - 5\)

  9. \(15 - (-6)\)

  10. \(-10 - (-4)\)

  1. \(8 + 15 = 23\)

  2. \(-7 + 12 = 5\)

  3. \(9 + (-14) = -5\)

  4. \(-6 + (-11) = -17\)

  5. \(-13 + 8 = -5\)

  6. \(5 + (-9) = -4\)

  7. \(12 - 7 = 5\)

  8. \(-8 - 5 = -13\)

  9. \(15 - (-6) = 21\)

  10. \(-10 - (-4) = -6\)

Exercise 1.2 Compute.

  1. \(7 \cdot 9\)

  2. \((-5)(8)\)

  3. \(12 \cdot (-4)\)

  4. \(6(11)\)

  5. \((-8) \cdot (-3)\)

  6. \(9(-7)\)

  7. \(15 \cdot 4\)

  8. \((-6)(-9)\)

  9. \((-11)(5)\)

  10. \(8 \cdot (-6)\)

  1. \(7 \cdot 9 = 63\)

  2. \((-5)(8) = -40\)

  3. \(12 \cdot (-4) = -48\)

  4. \(6(11) = 66\)

  5. \((-8) \cdot (-3) = 24\)

  6. \(9(-7) = -63\)

  7. \(15 \cdot 4 = 60\)

  8. \((-6)(-9) = 54\)

  9. \((-11)(5) = -55\)

  10. \(8 \cdot (-6) = -48\)

Exercise 1.3 Compute.

  1. \(96 \div 8\)

  2. \(54 \div 6\)

  3. \(81 \div 9\)

  4. \(64 \div 8\)

  5. \(108 \div 12\)

  6. \(75 \div 5\)

  7. \(62 \div 7\)

  8. \(73 \div 9\)

  9. \(85 \div 11\)

  10. \(59 \div 8\)

  1. \(96 \div 8 = 12\)

  2. \(54 \div 6 = 9\)

  3. \(81 \div 9 = 9\)

  4. \(64 \div 8 = 8\)

  5. \(108 \div 12 = 9\)

  6. \(75 \div 5 = 15\)

  7. \(62 \div 7 = 8\) R \(6\)

  8. \(73 \div 9 = 8\) R \(1\)

  9. \(85 \div 11 = 7\) R \(8\)

  10. \(59 \div 8 = 7\) R \(3\)