How are integers and rational numbers are related
Integers and rational numbers are related as rational numbers are a subset of integers.
Integers are the set of whole numbers and their negative counterpart, including zero (..., -3, -2, -1, 0, 1, 2, 3, ...). They do not include fractions or decimals.
Rational numbers, on the other hand, can be expressed as a fraction of two integers, where the numerator and denominator are both integers and the denominator is not zero. Rational numbers include integers since any integer can be expressed as a fraction with a denominator of 1 (e.g., 3 can be expressed as 3/1). However, rational numbers also include fractions and decimals that cannot be simplified to integers (e.g., 2/3, 0.25).
In summary, every integer is a rational number, but not every rational number is an integer.
Integers and rational numbers are related in the following way:
1. Definition: Integers are whole numbers (positive, negative, or zero) without fractional or decimal parts. Rational numbers, on the other hand, are numbers that can be expressed as a ratio of two integers, where the denominator is not zero.
2. Representation: Every integer is a rational number. For example, the integer 5 can be expressed as 5/1, where the numerator (5) and denominator (1) are both integers.
3. Set inclusion: The set of integers is a subset of the set of rational numbers. This means that every integer is a rational number, but not every rational number is an integer. For instance, 7/3 is a rational number but not an integer because it has a fractional part.
4. Connection through decimals: Every integer can be expressed as a rational number in decimal form. For example, the integer 3 can be written as 3/1 or as the decimal 3.0. Similarly, the integer -2 can be written as -2/1 or as the decimal -2.0.
5. Operations: Both integers and rational numbers follow similar rules for operations such as addition, subtraction, multiplication, and division. These operations can be performed on both types of numbers using the same mathematical rules.
In summary, integers are a subset of rational numbers, as all integers can be expressed as rational numbers. Rational numbers, however, include both integers and numbers with fractional parts.