key term - Number Systems

Definition

Number systems are the various ways of representing and manipulating numerical quantities. They provide a framework for understanding and working with different types of numbers, such as integers, rational numbers, and real numbers. Number systems are fundamental to the study of algebra, including topics like integer exponents and scientific notation.

5 Must Know Facts For Your Next Test

1. The four basic number systems are natural numbers, integers, rational numbers, and real numbers, each with its own unique properties and applications.
2. Integers are used to represent whole numbers, both positive and negative, and are denoted by the symbol $\mathbb{Z}$.
3. Rational numbers are numbers that can be expressed as a ratio of two integers, and are denoted by the symbol $\mathbb{Q}$.
4. Real numbers include both rational and irrational numbers, and are denoted by the symbol $\mathbb{R}$.
5. Understanding the properties and relationships between different number systems is essential for working with integer exponents and scientific notation.

Review Questions

• Explain how the different number systems (integers, rational numbers, and real numbers) are related and how they are used in the context of integer exponents and scientific notation.
  • The different number systems are hierarchical, with each system containing the previous one. Integers ($\mathbb{Z}$) are a subset of rational numbers ($\mathbb{Q}$), which are in turn a subset of real numbers ($\mathbb{R}$). This relationship is important when working with integer exponents and scientific notation, as the properties of these number systems determine how operations like multiplication and division can be performed. For example, integer exponents are used to represent very large or very small numbers in scientific notation, which is a way of expressing numbers in a standardized format using powers of 10.
• Describe the role of the real number system in the context of integer exponents and scientific notation, and explain how irrational numbers, such as $\pi$ and $\sqrt{2}$, are represented in this system.
  • The real number system ($\mathbb{R}$) is the most comprehensive number system, encompassing both rational numbers and irrational numbers. In the context of integer exponents and scientific notation, the real number system is essential because it allows for the representation of very large and very small numbers, including those that cannot be expressed as a ratio of two integers. Irrational numbers, such as $\pi$ and $\sqrt{2}$, are part of the real number system and cannot be represented as a simple fraction. These numbers are essential in various mathematical and scientific applications, and understanding their properties is crucial for working with integer exponents and scientific notation.
• Analyze how the properties of the integer number system, such as closure under addition and multiplication, impact the use of integer exponents and the manipulation of numbers in scientific notation.
  • The properties of the integer number system, such as closure under addition and multiplication, are fundamental to the use of integer exponents and the manipulation of numbers in scientific notation. The fact that the set of integers is closed under addition and multiplication means that the result of adding or multiplying two integers will always be another integer. This property allows for the consistent application of integer exponents, where the exponent represents the number of times a base is multiplied by itself. Additionally, the properties of integers ensure that operations with numbers in scientific notation, which rely on the use of integer exponents, can be performed reliably and predictably. Understanding these underlying number system properties is crucial for mastering the concepts of integer exponents and scientific notation.

Related terms

Integers:

The set of whole numbers, including positive and negative numbers, as well as zero.

Rational Numbers:

The set of numbers that can be expressed as a ratio of two integers, such as fractions and terminating or repeating decimals.

Real Numbers:

The set of all numbers, including both rational and irrational numbers, that can be represented on the number line.