What is a Families of Numbers

Families of Numbers

Numbers can be classified into families of numbers that share certain properties. There are many ways of putting numbers into classes in this way. In fact, just as there is an infinity of numbers, there is an infinite variety of ways in which they can be subdivided and distinguished from one another. For example the natural numbers, whole numbers with which we count objects in the real world, are just such a family, as are the integers—whole numbers including those less than zero. The rational numbers form another family, and help to define an even larger family, the irrational numbers. The families of algebraic and transcendental numbers are defined by other behaviors while the members of all these different families are members of the real numbers, defined in opposition to the imaginary numbers.

Saying that a number is a member of a certain family is a shorthand way of describing its various properties, and therefore clarifying what sort of mathematical questions we can usefully ask about it. Often, families arise from the creation of functions that describe how to construct a sequence of numbers. Alternatively, we can construct a function or rule to describe families that we recognize intuitively.

For instance, we instinctively recognize even numbers, but what are they? Mathematically, we could define them as all natural numbers of the form 2 × n where n is itself a natural number. Similarly, odd numbers are natural numbers of the form 2n + 1, while prime numbers are numbers greater than 1, whose only divisors are 1 and themselves.

Other families arise naturally in mathematics—for example in the Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, . . .), each number is the sum of the previous two. This pattern arises naturally in both biology and mathematics. Fibonacci numbers are also closely connected to the golden ratio.

Other examples include the multiplication tables, which are formed by multiplying the positive integers by a particular number, and the squares, where each number is the product of a natural number with itself: n times n, or n², or n squared.