We often see bright kids in 5th or 6th grade getting the deer-in-the-headlights look when it comes to dealing with equations with multiple negative or minus signs. Often it is good to stop and think about the different roles the humble minus sign take on in integer arithmetic. In fact, at this level, there are three meanings of the minus sign we should be aware of, and be able to tell which role is applicable for each part of the math we’re trying to solve.
1. As “Take-away” or Difference
This is the first use of the minus sign we all learn in kindergarten or first grade – i.e. to denote the difference or how far a value is from another.
As a difference operator, the minus sign takes in two arguments (10 and 7) and produces one output (3). In this sense, the use of a dash symbol is most obvious, as we would often define source-destination with a dash in between.
2. As a Label for Negative Numbers
The second time we see the minus sign is when we learn about negative numbers or numbers that are on the left side of the zero on the number line.
Here, we are not viewing the negative numbers as “reflection” of the positive numbers yet. They are just numbers that extends the positive number line we were familiar with and which are progressing leftwards. Here the minus signs are understood as labels to be used as part of the symbol, e.g. “-3”.
3. As a “Flip” Operation
Soon after we learn about negative numbers, we would learn about the third meaning of the minus sign – the “flip” over zero. For a positive quantity or expression, the minus sign flips it across zero on the number line.
This time, as a flip operator, the minus sign takes in only one argument (10-7) and produces one result -(10-7). We can concatenate or nest the operation multiple times, each time just flipping the quantity over zero.
In summary, the minus sign has multiple roles to play and often takes on different meanings in the same equation. It helps to keep these roles in our mind when mixing them in a mathematical expression.