Rational Numbers Definitions and Examples
Introduction
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. The set of all rational numbers, often referred to as “the rationals”, is usually denoted by a boldface Q (or blackboard bold ?). The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base b ? 2. A real number that is not rational is called irrational. Irrational numbers include ?2 (square root of 2, an algebraic number), ? (pi, a transcendental number), and Euler’s constant e.
What are Rational Numbers?
Rational numbers are any number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0. All whole numbers and integers are rational numbers, as they can be expressed as fractions with a denominator of 1. For example:
1/1 = 1
2/1 = 2
-3/1 = -3
All decimal numbers can also be expressed as rational numbers. For example:
0.5 = 1/2
-0.75 = -3/4
2.25 = 9/4
Some irrational numbers, such as ? (pi), can also be expressed as rational numbers by using an infinite decimal expansion. For example:
? = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679…
= 355/113
Types of Rational Numbers
Rational numbers can be classified in different ways. One way is by how they are represented, which include fractions, decimals, and percentages. Another way is by their properties, such as being positive or negative, and whether or not they are prime numbers.
Fractions are the most common type of rational number. They are created when one number is divided by another number, resulting in a quotient and a remainder. The quotient is the integer part of the fraction, while the remainder is the decimal part. For example, if someone were to ask you to divide 7 by 3, the answer would be 2 with a remainder of 1 (2.333…). Therefore, the fraction 7/3 would be written as 2 1/3.
Decimals are another form of rational number that can be easily converted to fractions. A decimal is simply a number with a decimal point somewhere within it. For example, 0.5, 1.25, and 2.75 are all decimals. All decimals can be converted to fractions by moving the decimal point over until there is only one digit to the left of it, and counting how many digits there are total including zeroes on both sides of the original decimal point. So 0.5 would become 5/10 because there is one digit to the left of the original decimal point (0), and two digits total including zeroes (0 and 5). Decimals can also be
How to Identify Rational Numbers?
Rational numbers are any number that can be expressed as a fraction, where both the numerator and denominator are integers. The set of rational numbers includes all integers, since every integer can be written as a fraction with a denominator of 1.
To identify whether a given number is rational, you can check to see if it can be expressed as a fraction. If it can, then it is rational. If not, then it is notrational.
For example, the number 3 can be expressed as 3/1, which is a fraction with an integer numerator and an integer denominator. Therefore, 3 is a rational number. On the other hand, the number pi (3.14159…) cannot be expressed as a fraction with integers for both the numerator and denominator. Therefore, pi is notrational.
Rational Numbers in Decimal Form
Rational numbers in decimal form always have a finite or repeating decimal expansion. For example, 3/4 = 0.75, 8/5 = 1.6, 11/13 = 0.84615384615…
To convert a rational number to decimal form, divide the numerator by the denominator. The answer will either be a terminating decimal (one that eventually ends) or a repeating decimal (one where the digits repeat in a pattern).
When writing fractions as decimals, we can use place value to our advantage. Place value goes from right to left starting with the ones column, then the tens column, and so on. Because of this we can line up the fraction bars underneath each other and simply divide straight across:
For example:
1/4 = ?
3/8 = ?
1 | 4 3 | 8
– — —
4 | 4 8 | 8 So 1/4 = 0.25 and 3/8 = 0.375
List of Rational Numbers
A rational number is a number that can be expressed as a fraction, p/q, where p and q are integers and q ? 0. In other words, a rational number is a number that can be represented as a ratio of two integers.
The set of all rational numbers is denoted by Q. The following are some examples of rational numbers:
-3/5
2/3
7/4
-10/7
As you can see, a rational number can be positive or negative, and it can be an integer or a non-integer.
Adding and Subtracting Rational Numbers
When adding and subtracting rational numbers, it is important to remember that they are just like any other number. The only difference is that they can be written as a fraction. In order to add or subtract rational numbers, you need to first find the LCD. The LCD is the Lowest Common Denominator. This is the lowest number that both fractions can be divided by. Once you have the LCD, you can change each fraction so that they both have the same denominator. Once they have the same denominator, you can add or subtract them like regular numbers.
Multiplying and Dividing Rational Numbers
Rational numbers are any number that can be expressed as a fraction, and this includes all integers. In mathematics, a fraction is defined as a division of two integers, where the dividend is called the numerator and the divisor is called the denominator. For example, 3/4 would be read as “three-fourths.”
Integers are rational numbers, but not all rational numbers are integers. Any number that cannot be expressed as a fraction is an irrational number. For example, ?2 (the square root of 2) is an irrational number.
When multiplying and dividing rational numbers, we need to consider both the numerator and denominator. First, let’s look at an example of multiplying two rational numbers:
3/4 x 1/2 = 3/8
In this example, we multiply the numerators (3 and 1) together to get 3, and then we multiply the denominators (4 and 2) together to get 8. The answer is 3/8. Now let’s look at an example of dividing two rational numbers:
1/2 ÷ 1/4 = 2/1 or 2 (answer will be in lowest terms)
In this example, we divide the numerators (1 and 1) to get 1, and then we divide the denominators (2 and 4) to get 2. The answer is 2/1 or 2 (answer will be in lowest terms).
Rational vs Irrational Numbers
Rational numbers are defined as those numbers that can be expressed as a ratio of two integers. A rational number can be written as a fraction, in which case it is called a common fraction. For example, the number 3 can be written as 3/1, which is a common fraction.
Irrational numbers are defined as those numbers that cannot be expressed as a ratio of two integers. An irrational number cannot be written as a fraction, in which case it is called an improper fraction. For example, the number ? (pi) is an irrational number because it cannot be expressed as a ratio of two integers.
Conclusion
In conclusion, rational numbers are any numbers that can be expressed as a fraction or ratio. They are an important part of mathematics and are used in many different applications. Understanding how to work with rational numbers is essential for anyone studying mathematics or using it in their everyday life.
