How to Understand Number Sets

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A set is any group of objects. In mathematics, sets help group numbers that may or may not have common properties. Learning about some of the standard number sets with shared properties will help you understand their behavior.

Set Notation

The numbers in a ​number set​ are expressed as a comma-separated list enclosed by brackets. For example:

\{1, 2, 3\}

An individual object in a set is called an ​element​ of the set. In mathematics, it is represented by the element symbol. The expression below states that a is an element of the set A.

a ∈ A

This examples states that the number 3 is an element of set A.

A = \{3,9,14\}, 3 ∈ A

A set that has no members is referred to as the empty set or null set. It has its own set notation:

Ø = \{ \}

Integer Numbers Set

The set of ​whole numbers​ is defined as all positive numbers, plus zero. The ​integer numbers​ set includes the whole numbers, plus the negative versions of the positive numbers. It has its own set notation:

ℤ = \{...-3,-2,-1,0,1,2,3,...\}

Rational Numbers Set

Numbers that can be defined as fractions make up the ​rational numbers​ set. Any number that can be defined as ​a​ / ​b​, where ​b​ is nonzero, is a rational number. Zero is not an element of this set, but the other members of the integer numbers set are since they can be defined by the fraction ​a​ / 1. The rational number set has the following notation:

ℚ = \{x | x=\frac{a}{b}, a,b∈ℤ, b≠0\}

This notation states that a rational number is an element x such that x can be represented as a / b, where a and b are members of the integers number set and b is not equal to zero. Numbers that cannot be expressed in this form are known as irrational numbers.

A rational number can be expressed in decimal form by dividing the numerator by the denominator. For example, the fraction 1/5 is 0.2 in decimal form. Rational numbers have a fixed number of digits to the right of the decimal point, while ​irrational numbers​ have a non-repeating pattern of digits.

Real Numbers Set

When you combine all the rational and irrational numbers into a single set, you have the ​real numbers​ set. The set of real numbers can be represented as points on a number line that has 0 at its center, positive numbers to the right, and negative numbers to the left.

ℝ = \{x | -∞ < x <∞\}

The notation for the real numbers set indicates that it contains all the points on the number line, stretching to infinity in both the positive and negative directions.

What Is the Value of Z in Statistics?

A ​z​-score is a common measure of standard deviation used in statistics that allows you to calculate the probability of a certain value appearing within a normal distribution. There is no correlation between the ​Z​ number set and the ​z​-score concept.


About the Author

Catie Watson has a degree in Computer Science and spent 30 years working as a software engineer for Disney, Unisys and Siemens. She writes about science and technology online and in print publications and was a contributor to the textbook series “Computers, Internet and Society.”