## Using Zero when Subtracting

Posted by mark schwartz on April 12, 2016

While teaching pre-service teachers, we explored a bunch of subtraction algorithms. One which I presented was: *if you add or subtract the same amount from both numbers, the result is the same.*

For example, 73 ─ 28 = 75 ─30; 101 ─ 33 = 98 ─ 30; 11 ─ 8 = 13 ─ 10.

The amount added or subtracted is done to make one of the numbers easier to add or subtract.

Why does this work? Generalizing, let the problem be A ─ B. Subtracting (or adding) the same amount to both numbers gives (A ─ C) ─ (B ─ C). Simplifying this expression, the result is:

A ─ C ─ B + C . So, this procedure works because a zero ( ̶ C + C = 0) has been added; in my view, it simplifies subtraction.

One student asked if the same thing works for addition. We explored this too. It does work and it becomes more apparent that a zero is being added. *With addition, add an amount to one number but subtract that amount from the other.*

* *For example: 17 + 8 = 20 + 5; 113 + 78 = 111 + 80.

Again, generalizing the expression A + B by adding a zero gives A + B + C ─ C, then commuting and associating, you can write (A ─ C) + (B + C) or (A + C) + (B ─ C).

Another thing. Adding a zero enables operations with signed numbers and demonstrates the basis for the “rules”. Consider the problem + 2 ─ ( ─ 1). If this problem is done by placing positive and negative chips in a pile, the problem would begin by placing 2 positive chips in the pile. The problem now is asking the student to take one negative out of the pile but there aren’t any there. However, if a zero is added to the pile in the form of one negative chip and one positive chip, the operation can now be done, leaving 3 positive chips in the pile.

There are other “zero” ideas which came from some of my old textbooks and we talked about these too.

Mr. Smith, in his 1911 *Elements of Algebra *(pg. 33), demonstrated another way that zero can be used in subtraction. His example was 3 ̶ 5 = 3 ̶ 2 ̶ 3 = 3 ̶ 3 ̶ 2 = ̶ 2. He explains this by noting that taking away 5 is the same as taking away 2 then taking away 3.

Then, Mr. Ficklin, in his 1881 *Elements of Algebra *(pg. 30), demonstrated something similar. His example was 7 ̶ 10 = 10 ̶ 3 ̶ 10 = 10 ̶ 10 ̶ 3 = ̶ 3. He substituted an equivalent statement for 7, which allowed for a “zero” (10 ̶ 10).

We had a rather extended and robust exploration of subtraction, most notably about alternatives to “borrowing”. The concept of using a zero is similar in some sense to the Common Core subtraction algorithm, but the ”zero”examples shown here indicate that there is a wider spectrum of subtraction methods. Perhaps some of these should be considered for inclusion in the curriculum because they are valid and may be “discovered” by an imaginative student, who wouldn’t be wrong.

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