### What is a negative number?

Negative numbers can be a difficult concept to understand. In fact, the widespread acceptance of negative numbers is a relatively recent event in the history of mathematics. There are many ways to understand and explain negative numbers; and some of these interpretations are listed below.

1. The opposite of a positive number
2. A number that is less than zero
3. A number that is to the left of zero on a number line
4. A value on a scale that extends beyond zero
5. The amount of a loss or absence
6. A directed quantity
7. A comparison between two quantities
8. The result of subtracting a larger number from a smaller number
9. An equivalence class of ordered pairs of natural numbers

I will discuss my thoughts about these interpretations below. My purpose is not to tell the reader how to think about negative numbers, but to encourage the reader to think deeply about the matter and reach his or her own conclusions. My remarks are intended for other math teachers, so they might be too advanced for beginning learners.

Interpretation 1: The opposite of a positive number.

We know that the opposite of hot is cold, and the opposite of love is hate, but the opposite of a number might be an unfamiliar concept. Two numbers are opposites if their sum is zero. For example, the opposite of 6 is –6, because 6 + (–6) = 0. The same addition also shows that the opposite of –6 is 6.

This interpretation can be modeled using black chips and red chips, where black chips are used to show positive numbers and red chips are used to show negative numbers. We adopt a rule that we can add or remove chips, as long as we add or remove equal numbers of black chips and red chips. For example, a pile containing 4 black chips and 7 red chips represents the number –3, because we may remove 4 black chips and 4 red chips from the pile, leaving 3 red chips.

This interpretation is unsatisfying on its own, because it does not show how negative numbers are used in real life.

Interpretations 2, 3, and 4: A number that is less than zero.

This is self-explanatory, although it is not obvious how a number could be less than zero. Indeed, the concept makes no sense for many kinds of quantities. You can’t have less than zero enemies, eat less than zero servings of vegetables, or walk less than zero miles. A successful explanation of negative numbers has to explain why some quantities can be negative and some quantities cannot.

We can show the natural numbers on a number line, and we interpret “less than” and “greater than” to mean “to the left of” and “to the right of”, respectively. The number line continues to the right without end. Negative numbers arise when we extend the line to the left of zero.

A thermometer is a real-life example of a number line. The zero point is 0º on the Fahrenheit or Celsius scale. On a cold winter day, the temperature can go below zero on either scale. Students should be able to come up with other examples of scales that extend below zero. For example, elevations are defined in reference to sea level, and they can be negative. The lowest point in North America is the Badwater Basin in Death Valley, California. Its elevation is –86 meters, which means that it is 86 meters below sea level.

What these scales have in common is that each has a reference point which is designated as zero, and the quantity can be greater or less than this reference point. In the Celsius scale, the reference point is the freezing temperature of water. When measuring elevation, the reference point is sea level. When the reference point represents an absolute minimum value (such as absolute zero temperature) then negative numbers have no meaning.

Interpretation 5: Loss or absence.

Negative numbers often indicate a loss or decrease in a quantity. A decrease in a quantity may be thought of as a negative change or negative increase. We often see negative numbers in financial news stories when a company loses money or a market index declines in value. An American football team may gain or lose yards on a play, and this can be represented by a positive or negative number.

Negative numbers can also indicate the absence of something, such as a debt or deficit. In accounting, credits and debits are represented by positive and negative numbers respectively.

Interpretation 6: A directed quantity.

Negative numbers are useful for representing quantities that have two opposing directions. The directions can be literal (e.g. up or down, East or West) or metaphorical (e.g. profit or loss, winning or losing). In physics, one usually assumes that up is positive and down is negative. A falling object near Earth’s surface undergoes an acceleration of –9.8 meters per second per second; the acceleration is negative because the object is being pulled downwards by gravity. Latitude and longitude are measured with respect to the equator and the prime meridian. A location that is south of the equator has a negative latitude, and a location that is west of the prime meridian has a negative longitude.

Interpretations 7 and 8:  A comparison between two quantities.

Negative numbers usually arise when we are comparing two measurements of the same type. This is seen to be a generalization of the earlier interpretations. We may be comparing a quantity with an earlier value of the same quantity, which leads to an increase or decrease. Or we may be comparing the quantity to some reference point, such as the prime meridian or the freezing point of water.

The operation of subtraction expresses the difference between two quantities. We may usually think of subtraction as “taking away”, but we also use subtraction to answer the question “how many more?”  The subtraction fact 100 – 86 = 14 tells us that if we take 86 away from 100 then 14 remain, but it also tells us that 100 is 14 more than 86. This allows us to make sense of subtracting a larger number from a smaller number.  We can’t take 5 from 2, but we can say that 2 is 3 less than 5, and this is expressed by the subtraction fact 2 – 5 = –3.

Interpretation 9: An equivalence class of ordered pairs of natural numbers.

In formal mathematics, we construct the integers by defining an equivalence relation on ordered pairs of natural numbers. This approach is much too abstract for beginning learners, but it is a valuable perspective for math teachers.

The set of natural numbers is {0, 1, 2, 3, 4, …}. (Some people exclude 0.) Each integer is represented by an infinite number of different ordered pairs of natural numbers. Two ordered pairs (a,b) and (c,d) represent the same integer if and only if a+d = b+c. For example, the following ordered pairs all represent the same integer:

(0,3), (1,4), (2,5), (3,6), (4,7), …

The idea can be seen more clearly if we change our notation. If we write the ordered pair (a,b) as (a – b) instead, then we have

(0 – 3) = (1 – 4) = (2 – 5) = (3 – 6) = (4 – 7) = …

The standard notation for this integer is –3. What you should take away from this is that it is possible to define an integer as the result of subtracting two natural numbers.