Positive numbers are commonly defined as numbers greater than zero; the numbers to the right of zero on the number line. Zero is not a positive number. The opposite, or additive inverse, of a positive number is a negative number. Negative numbers are always preceded by a negative sign (–), while positive numbers are only preceded by a positive sign (+) when it is required to avoid confusion. Thus 15 and +15 are the same positive number.
Positive numbers are used to identify quantities, such as the length of a line, the area of a circle, or the volume of a glass jar. They are used to identify the magnitude of physical quantities, as well. For example, positive numbers are used to indicate the amount of electric power it takes to light a light bulb, the magnitude of the force required to launch a space shuttle, the speed required to reach a destination in a fixed time, the amount of pressure required pump water uphill, and so on.
Very often physical quantities also have a direction associated with them (represented by one-dimensional vectors). Positive numbers are used in conjunction with these quantities to indicate the direction. We may arbitrarily choose a certain direction as being positive and call the velocity, for instance, positive in that direction. Then a negative velocity corresponds to a velocity in the opposite direction. For instance, if north is chosen as the positive direction, a car traveling due north at a speed of 50 mph (80 km/h) has a velocity of 50 mph (80 km/h), and a car traveling due south at 50 mph has avelocity of←–50 mph (–80 km/h). In other instances, we may say a car has positive velocity when traveling in drive and negative velocity when traveling in reverse.
Force is also a directed quantity. Gravity exerts a force down on all massive bodies. To launch a space shuttle requires a force larger than that of gravity, and oppositely directed. If we choose down as positive, then the force of gravity is positive, and the force required for launch will be negative. There must be a net negative force on the shuttle, which really means a positive force larger than gravity applied in the negative direction.
This discussion gives meaning to positive as being greater than zero, or, in a geometric sense, as having a particular direction or location relative to zero. A more fundamental definition of positive numbers is based on the definition of positive integers or natural numbers such as the ones given by the German mathematician F. L. G. Frege or the Italian Giuseppe Peano. Frege based his ideas on the notion of one-toone correspondence from set theory. One-to-one correspondence means that each element of the first set can be matched with one element from the second set, and vice versa, with no elements from either set being left out or used more than once. Pick a set with a given number of elements, say the toes on a human foot. Then, form the collection of all sets with the same number of elements in one-to-one correspondence with the initial set, in this case the collection of every conceivable set with five elements. Finally, define the cardinal number 5 as consisting of this collection. Peano defined the natural numbers in terms of 1 and the successors of 1, essentially the same method as counting. Using either the Frege or Peano definitions produces a set of natural numbers that are essentially the same as the positive integers. Ratios of these are the positive rational numbers, from which positive real numbers can be derived. In this case, there is no need to consider “greater than 0” as a criterion at all—but this concept can then be derived.
Number line— A number line is a line whose points are associated with the real numbers, an arbitrary point being chosen to coincide with zero.
Rectangular coordinate system— A two-dimensional rectangular coordinate system consists of a plane in which the points are associated with ordered pairs of real numbers located relative to two perpendicular real number lines. The intersection of these lines coincides with the point (0,0), or origin.
Note that complex numbers are not considered positive or negative. Real numbers, however, are always positive, negative, or zero.
Boyer, Carl B. A History of Mathematics. 2nd ed. Revised by Uta C. Merzbach. New York: John Wiley and Sons, 1991.
Lay, David C. Linear Algebra and Its Applications. 3rd ed. Redding, MA: Addison-Wesley Publishing, 2002.
Pascoe, L.C. Teach Yourself Mathematics. Lincolnwood, Ill: NTC Publishing Group, 1992.
Mathematics League. “Positive and Negative Numbers” <http://www.mathleague.com/help/posandneg/posandneg.htm> [October 9, 2006).
J. R. Maddocks