Integers: Detailed Principles of Addition and Subtraction

Fundamentals of Adding and Subtracting Integers

The manipulation of integers involves specific rules for combining positive and negative values. Adding a negative integer is mathematically equivalent to regular subtraction. For example, the expression 5+(−1)5 + (-1) is processed as 5−15 - 1, resulting in 44. Conversely, subtracting a negative integer is mathematically equivalent to regular addition. For instance, the expression 5−(−1)5 - (-1) is processed as 5+15 + 1, resulting in 66.

Number Line Operations for Addition

When performing addition on a number line, the direction of movement is determined by the sign of the integer being added. Adding a positive integer requires counting forward (moving to the right). For example, to solve 1+31 + 3, one starts at 11 and moves 33 steps to the right to arrive at 44.

When adding a negative integer, one must count backward (moving to the left). In the instance of 1+(−3)1 + (-3), the operation involves moving −3-3 to the right, which effectively means moving 33 steps to the left from the starting point of 11, landing on −2-2. Further applications include:

  • 3+(−2)=13 + (-2) = 1: Start at 33 and move 22 steps to the left.
  • 2+(−4)=−22 + (-4) = -2: Start at 22 and move 44 steps to the left.
  • −1+(−4)=−5-1 + (-4) = -5: Start at −1-1 and move 44 steps to the left.
  • −5+(−1)=−6-5 + (-1) = -6: Start at −5-5 and move 11 step to the left.
  • −5+4=−1-5 + 4 = -1: Start at −5-5 and move 44 steps to the right.
  • 3+(−7)=−43 + (-7) = -4: Start at 33 and move 77 steps to the left.
  • −2+(−3)=−5-2 + (-3) = -5: Start at −2-2 and move 33 steps to the left.
  • 5+(−1)=45 + (-1) = 4: Start at 55 and move 11 step to the left.
  • −2+(−1)=−3-2 + (-1) = -3: Start at −2-2 and move 11 step to the left.
  • −4+1=−3-4 + 1 = -3: Start at −4-4 and move 11 step to the right.
  • 5+(−2)=35 + (-2) = 3: Start at 55 and move 22 steps to the left.
  • 7+(−5)=27 + (-5) = 2: Start at 77 and move 55 steps to the left.

Number Line Operations for Subtraction

Subtraction on a number line also follows directional rules based on the sign of the number. When subtracting a positive integer, one counts backward (moves to the left). For example, in the expression 3−53 - 5, one starts at 33 and moves 55 steps to the left to arrive at −2-2. Another instance is −6−4-6 - 4, where one starts at −6-6 and moves 44 units to the left, resulting in −10-10.

When subtracting a negative integer, one counts forward (moves to the right). For instance, 1−(−2)1 - (-2) is calculated by starting at 11 and moving 22 steps to the right, resulting in 33. Additional examples performed on number lines include:

  • −2−3=−5-2 - 3 = -5: Starting at −2-2 and moving 33 steps to the left.
  • −4−(−1)=−3-4 - (-1) = -3: This is processed as −4+1-4 + 1, moving 11 step to the right from −4-4.
  • −3+2=−1-3 + 2 = -1: Starting at −3-3 and moving 22 units to the right.
  • −5−(−4)=−1-5 - (-4) = -1: This is equivalent to −5+4-5 + 4, moving 44 steps to the right from −5-5.
  • 4−(−2)=64 - (-2) = 6: This simplifies to 4+24 + 2, moving 22 steps to the right from 44.
  • 8−(−2)=108 - (-2) = 10: This translates to 8+28 + 2, moving 22 units to the right from 88.

Principles of Like and Unlike Signs

A critical shortcut for integer arithmetic involves identifying pairs of signs. Two like signs yield addition. This applies to both positive pairs (+,++, +) and negative pairs (−,−-, -). For example, −2−(−1)-2 - (-1) becomes −2+1-2 + 1, and 3−(−2)3 - (-2) becomes 3+23 + 2. Two unlike signs yield subtraction (+,−+, - or −,+-, +). For instance, 2+(−4)2 + (-4) becomes 2−42 - 4, and 5+(−5)5 + (-5) becomes 5−55 - 5.

Practical applications of these sign rules include the following computations:

  • −2−(−1)=−2+1=−1-2 - (-1) = -2 + 1 = -1
  • 2+(−4)=2−4=−22 + (-4) = 2 - 4 = -2
  • 3−(−2)=3+2=53 - (-2) = 3 + 2 = 5
  • −5+2=−3-5 + 2 = -3
  • −2+6=4-2 + 6 = 4
  • −2−(−6)=−2+6=4-2 - (-6) = -2 + 6 = 4
  • 5+(−5)=5−5=05 + (-5) = 5 - 5 = 0
  • 5−(−5)=5+5=105 - (-5) = 5 + 5 = 10
  • −1+4=3-1 + 4 = 3
  • −1−4=−5-1 - 4 = -5
  • 4−(−2)=4+2=64 - (-2) = 4 + 2 = 6
  • −7+5=−2-7 + 5 = -2
  • −6−(−3)=−6+3=−3-6 - (-3) = -6 + 3 = -3
  • −9+3=−6-9 + 3 = -6

Multi-Step Integer Expressions

Complex expressions requiring multiple operations should be solved step-by-step from left to right, maintaining the signage rules throughout the process.

  • −2+(−3)−1-2 + (-3) - 1: Calculate −2+(−3)=−5-2 + (-3) = -5, then perform −5−1=−6-5 - 1 = -6.
  • 5+2+(−3)5 + 2 + (-3): Calculate 5+2=75 + 2 = 7, then perform 7−3=47 - 3 = 4.
  • 10−(−6)10 - (-6): Simplify to 10+6=1610 + 6 = 16.
  • 3−(−4)+(−1)3 - (-4) + (-1): Calculate 3−(−4)=73 - (-4) = 7, then perform 7−1=67 - 1 = 6.
  • −7+3+(−5)-7 + 3 + (-5): Calculate −7+3=−4-7 + 3 = -4, then perform −4−5=−9-4 - 5 = -9.
  • 6−4−(−5)6 - 4 - (-5): Calculate 6−4=26 - 4 = 2, then perform 2+5=72 + 5 = 7.
  • −1+6+8-1 + 6 + 8: Calculate −1+6=5-1 + 6 = 5, then perform 5+8=135 + 8 = 13.
  • −3+(−4)−(−5)-3 + (-4) - (-5): Calculate −3−4=−7-3 - 4 = -7, then perform −7+5=−2-7 + 5 = -2.
  • 1+6−(−8)1 + 6 - (-8): Calculate 1+6=71 + 6 = 7, then perform 7+8=157 + 8 = 15.
  • −10+(−6)−4-10 + (-6) - 4: Calculate −10−6=−16-10 - 6 = -16, then perform −16−4=−20-16 - 4 = -20.
  • −7−(−8)−(−9)-7 - (-8) - (-9): Calculate −7+8=1-7 + 8 = 1, then perform 1+9=101 + 9 = 10.
  • −3−(−2)+6-3 - (-2) + 6: Calculate −3+2=−1-3 + 2 = -1, then perform −1+6=5-1 + 6 = 5.
  • −2+(−3)+(−4)-2 + (-3) + (-4): Calculate −2−3=−5-2 - 3 = -5, then perform −5−4=−9-5 - 4 = -9.
  • 2−(−2)+(−5)2 - (-2) + (-5): Calculate 2+2=42 + 2 = 4, then perform 4−5=−14 - 5 = -1.
  • 8+(−1)−(−6)8 + (-1) - (-6): Calculate 8−1=78 - 1 = 7, then perform 7+6=137 + 6 = 13.
  • 9−(−4)−(−1)9 - (-4) - (-1): Calculate 9+4=139 + 4 = 13, then perform 13+1=1413 + 1 = 14.
  • −15−4−(−2)-15 - 4 - (-2): Calculate −15−4=−19-15 - 4 = -19, then perform −19+2=−17-19 + 2 = -17.
  • 8−12+(−3)8 - 12 + (-3): Calculate 8−12=−48 - 12 = -4, then perform −4−3=−7-4 - 3 = -7.
  • 4−(−8)+(−3)4 - (-8) + (-3): Calculate 4+8=124 + 8 = 12, then perform 12−3=912 - 3 = 9.

Equivalent Integer Expressions

Many integer expressions represent identical numerical values through different arrangements of signs and operations. Recognizing these equivalencies facilitates faster mental math.

  • The expression −3+(−4)-3 + (-4) is equivalent to −3−4-3 - 4 and −4+(−3)-4 + (-3).
  • The expression −3−(−4)-3 - (-4) is equivalent to −3+4-3 + 4.
  • The expression 1+(−2)1 + (-2) is equivalent to 1−21 - 2, −2+1-2 + 1, and (−2)+1(-2) + 1.
  • The expression 2+12 + 1 is equivalent to 1+21 + 2.
  • The expression −4+3-4 + 3 is equivalent to 3−43 - 4.
  • The expression −4−3-4 - 3 is equivalent to −4+(−3)-4 + (-3), −3−4-3 - 4, and −3+(−4)-3 + (-4).
  • The expression −1+2-1 + 2 is equivalent to 2−12 - 1 and 2+(−1)2 + (-1).
  • The expression −2−(−1)-2 - (-1) is equivalent to −2+1-2 + 1.
  • The expression (−1)−2(-1) - 2 is equivalent to −1+(−2)-1 + (-2).
  • The expression (−2)−(+1)(-2) - (+1) is equivalent to −2−1-2 - 1 and −2+(−1)-2 + (-1).