Subtracting Integers & Distance - Seventh Grade Mathematics

Subtracting signed numbers reveals that subtraction is fundamentally the addition of an additive inverse, linking numerical differences directly to physical distances on the number line.

Subtraction as Adding the Opposite

In elementary arithmetic, subtraction is often taught as taking away a physical quantity from a larger collection. However, when working with negative numbers, taking away a negative quantity can feel abstract. In seventh grade mathematics, we formalize subtraction through its rigorous algebraic definition: subtracting an integer is completely equivalent to adding its opposite.

               THE FUNDAMENTAL SUBTRACTION PRINCIPLE
                       a - b = a + (-b)
              Subtracting b is identical to adding -b.

Consider the expression positive five minus positive eight. Using the subtraction principle, rewrite this as positive five plus the opposite of positive eight, which is negative eight. Now apply the addition rules for integers with different signs: subtract five from eight to get three, and keep the negative sign from negative eight. The result is negative three.

Example 1:  7 - 12      -->  7 + (-12)     = -5
Example 2:  -4 - 6      -->  -4 + (-6)     = -10
Example 3:  3 - (-5)    -->  3 + (+5)      = 8
Example 4:  -9 - (-11)  -->  -9 + (+11)    = 2

Notice what happens in Example 3 when you subtract a negative number: three minus negative five becomes three plus positive five, yielding eight. Removing negative debt from an account increases the overall balance, just as removing negative charges in physics increases net electrical potential.

Modeling Subtraction on the Number Line

On the number line, subtraction can be visualized as finding the directed distance from the subtrahend (the number being subtracted) to the minuend (the starting number). This interpretation answers the question: what must be added to b to reach a?

               FINDING THE DIFFERENCE: 3 - (-4) = 7
       What vector moves from -4 to +3?  Answer: +7 units
  <----+----+----+----+----+----+----+----+----+----+---->
      -5   -4   -3   -2   -1    0    1    2    3    4
           [==================================>]
                         +7 units

To find three minus negative four, locate negative four on the number line and draw a directed vector to positive three. The vector points to the right, indicating a positive value, and spans a total distance of seven units. Therefore, three minus negative four equals positive seven.

Distance on the Number Line

A vital application of integer subtraction is determining the non-directional geometric distance between two locations on a coordinate line. Distance between two points can never be negative. If two points on a number line have coordinates a and b, the distance between them is given by the absolute value of their difference.

               DISTANCE FORMULA ON A NUMBER LINE
                       Distance = |a - b| = |b - a|

Because the absolute value of a quantity equals the absolute value of its opposite, the order in which you subtract the coordinates does not matter. The absolute value bars guarantee that the resulting distance is strictly non-negative.

Distance between -6 and +4:
Method A: |(-6) - 4| = |-6 + (-4)| = |-10| = 10 units
Method B: |4 - (-6)| = |4 + 6|     = |10|  = 10 units

Both methods yield ten units, perfectly matching the physical count of steps between negative six and positive four on the line.

Multi-Step Temperature and Elevation Problems

Real-world problems involving sudden shifts in temperature or steep changes in elevation require careful application of integer subtraction. When finding a total change, always subtract the initial value from the final value: change equals final value minus initial value.

Change in Value Formula:
  Change = Final State - Initial State
  If Change > 0, the quantity increased.
  If Change < 0, the quantity decreased.

Suppose a winter weather front causes the temperature in a mountain valley to drop from an initial measurement of four degrees Celsius at noon to a final measurement of negative nine degrees Celsius at midnight. The temperature change is negative nine minus positive four, which simplifies to negative nine plus negative four, equaling negative thirteen degrees Celsius. The negative sign confirms a thirteen-degree drop in temperature.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Rewrite the subtraction expression negative fifteen minus negative twenty-two as an equivalent addition expression and find the resulting value.

Exercise 2. At dawn, the surface temperature on a polar glacier was negative thirty-two degrees Fahrenheit. By mid-afternoon, the temperature rose to negative fourteen degrees Fahrenheit. Calculate the total change in temperature using the formula change equals final state minus initial state.

Exercise 3. Find the geometric distance between point P at coordinate negative seventeen and point Q at coordinate positive eight on a horizontal number line using the absolute value distance formula.

Exercise 4. A deep-sea research diver operates at negative eighty-five feet relative to sea level. The diver's support vessel rests at positive twelve feet above the water line. Determine the vertical distance separating the diver and the vessel.

Solutions

Solution to Exercise 1. To subtract negative twenty-two, add its opposite, which is positive twenty-two. The equivalent addition expression is negative fifteen plus twenty-two. Since the addends have opposite signs, subtract fifteen from twenty-two to get seven. Because positive twenty-two has the larger absolute value, the result is positive seven.

Solution to Exercise 2. Using the formula change equals final state minus initial state, the initial temperature is negative thirty-two and the final temperature is negative fourteen. The change is negative fourteen minus negative thirty-two. Rewrite this subtraction as adding the opposite: negative fourteen plus positive thirty-two. Subtracting fourteen from thirty-two yields positive eighteen. The temperature changed by positive eighteen degrees Fahrenheit, indicating an eighteen-degree increase.

Solution to Exercise 3. The geometric distance is given by the absolute value of the difference between the two coordinates: vertical bar negative seventeen minus eight vertical bar. Rewrite the subtraction as addition of the opposite: vertical bar negative seventeen plus negative eight vertical bar. The sum inside the absolute value is negative twenty-five. Evaluating the absolute value of negative twenty-five yields positive twenty-five units.

Solution to Exercise 4. The vertical distance is the absolute value of the difference between the vessel's elevation and the diver's elevation: vertical bar twelve minus negative eighty-five vertical bar. Rewrite the subtraction as addition: vertical bar twelve plus positive eighty-five vertical bar, which equals vertical bar ninety-seven vertical bar. The vertical distance separating the diver and the support vessel is ninety-seven feet.