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Examples of rational numbers

After identifying what are rational numbers in mathematics, there are several illustrative examples of the image of rational numbers, and these examples are:

The correct numbers

Integers are the numbers that include positive numbers, negative numbers, and zero, and all integers are considered rational numbers, where the numerator is the whole number, and the denominator is equal to one integer. Examples of this are:

• The number 10 is a rational number: where it can be written as a fraction 1/10, and the denominator is equal to one
• The number -4 is a rational number: Where it can be written as a fraction -4/1, and the denominator is equal to one
• The number 0 is a rational number: where it can be written as 0/1, and the denominator is integer one

Fractions and mixed numbers

The fraction is the one that can be written in the form of a numerator and denominator, so that both the numerator and denominator belong to the group of integers, and the value of the denominator is not equal to zero. They are rational numbers, and they can be translated with symbols. b to the set of integers, b not equal to zero, and the mixed numbers follow the same definition of fractions, and they are also rational numbers. Examples of this are:

• The fraction 2/4 is considered a rational number, since the numerator and denominator belong to the set of integers, and the denominator is not equal to zero.
• The mixed number 5 and 3/2 is considered a rational number, because it can be converted into a fraction of the form a/b, and its denominator is not equal to zero.

Note: Some fractions are not considered relative, as shown in the examples:

• 20/0 is an irrational number, so even though 20.0 belong to the set of integers, the denominator is zero, which will result in an undefined value.
• The fraction π/9 is an irrational number, and although the denominator is an integer and not equal to zero, pi is not a rational number.

decimals

Decimal fractions are relative fractions that can be written as a fractional form consisting of a numerator and denominator if they are finite or periodic fractions. Examples of this are:

• The decimal fraction 1.2 is considered a rational number, because it can be expressed as 1.2/1, and when both the numerator and denominator are multiplied by the number 10/10, the number 12/10 is a rational number, since the numbers 12 and 10 are integers, and the number 10 is not equal to zero .
• The recurring decimal… 5.555 is a rational number, because it can be written as the mixed number 5 and 1/5, and this mixed number can be converted to 10/5 which is considered a rational number.

An irrational number is called a number

The most famous examples of irrational numbers

Irrational numbers are infinite fractions, or a fraction with a denominator of zero. The most famous examples of irrational numbers are the following:[2]

• The number π: It is considered an irrational number, as its value is 3.1415926535897932384626433832795, an infinite decimal fraction.
• Niberian number E: It is considered an irrational number, as its value is 2.7182818284590452353602874713527, which is an infinite decimal.
• Some square and cube roots: where its value is equal to infinite decimals, such as the square root of 3.

Mathematics book solution second intermediate F1 first semester 1443

properties of rational numbers

The following is a set of general properties of rational numbers, including:

• When multiplying two rational numbers, the result is the product of the numerator of both numbers, and the product of the denominator of both numbers.
• When you add two rational numbers that have the same denominator, the result is the sum of the numerators of the two numbers, and the denominator remains the same.
• When subtracting two rational numbers that have the same denominator, the result is the product of subtracting the numerators of the two numbers, and the denominator remains the same.
• When multiplying, adding or subtracting two rational numbers that have the same denominator, the result is a rational number, and it cannot be otherwise.
• When dividing the numerator and denominator of a rational number by any integer whose value is not zero, the result is also a rational number.
• When the numerator and denominator of a rational number are multiplied by any integer whose value is not zero, the result is also a rational number.
• The square root is always a rational number, the value of which is the number inside the root.
• The product of irrational roots may sometimes result in an irrational number.
• If the numerator and denominator of a rational number have only 1, then it is called the standard form of a rational number.
• The process of adding or subtracting irrational numbers cannot lead to obtaining rational numbers, unless the two numbers have opposite signs and cancel each other out.

Various questions about rational numbers

Various questions help in the correct understanding of the definition of rational numbers, including the following:

• The first question: Do the following fractions represent rational numbers?
  • The fraction 8/5: It is considered a rational number, since both the numerator and denominator belong to the set of integers, and the denominator is not equal to zero.
  • The fraction 4/0 is not considered a rational number, because the denominator is zero, so it is an undefined number.
  • -8: It is considered a rational number, as it can be written as 1/8-
  • 0: is a rational number, which can be written as 0/1
• second question: Do the following decimals represent rational numbers?
  • 2.58585858558: a rational number, because it is a periodic fraction in which the two numbers 5.8 are repeated with the same frequency.
  • 1.4789: a rational number, because it is a terminating decimal.
• The third question: Are the following values ​​considered relative fractions, or are they not?
  • 2/4: a rational number, since both the numerator and denominator belong to the set of integers, and the denominator is not equal to zero.
  • 1 and 3/4: a rational number, because it is equal to the fraction 7/4, which is considered a rational number, since the numerator and denominator represent two integers, and the denominator is not equal to zero.
  • 7895/200: a rational number, since both the numerator and denominator belong to the set of integers, and the denominator is not equal to zero.

Compare rational numbers

Comparing rational numbers is a process that clarifies the relationship of numbers to each other through the comparison signs of the equal sign, the greater than sign, and the less than sign, so that it determines if a value is equal to, greater than, or less than the other value. :[3]

• The equal sign (=): It is used to indicate that two values ​​are equal, for example 3/4 = 3/4
• Less than (<) sign: It is used to indicate that the first value is less than the second value, for example 1/4 < 1/2
• Signal greater than (>): It is used to indicate that the first value is greater than the second value, for example 1/5 > 1/7

Comparing positive rational numbers

The positive rational number is the number in which the sign of the numerator and denominator is positive. The process of comparing rational numbers is carried out through the following steps:

• For example, compare the number 6/3 with the number 4/5:
• Unifying the denominators of rational numbers, by determining the least common multiple
• The least common multiple of 6/3 and 4/5 is 15
• We multiply the numerator and denominator of 6/3 by 5 to get 30/15
• We multiply the numerator and denominator of 4/5 by 3 to get 12/15
• We find that 30/15 = (5 x 3) / (5 x 6) = 6/3
• We find that 12/15 = (3 x 5 / (3 x 4) = 4/5
• The comparison becomes between 30/15 and 12/15.
• We compare the numerator of each number after unifying the denominators and the number with the largest numerator is the largest number, and since the numerator is an integer, it is compared in the same way as comparing whole numbers.
• We find that 30 is greater than 12
• So 30/15 is greater than 12/15, so 6/3 > 4/5

Compare negative rational numbers

A negative rational number is the number in which the sign of the numerator and denominator is negative, and the process of comparing negative rational numbers is done by the following:

• Unifying the denominators of rational numbers, by determining the least common multiple.
• We compare the numerators of both rational numbers in the same way as comparing negative numbers.
• The larger negative number is the smaller number.
• When comparing negative and positive numbers, a negative number is always smaller than a positive number, no matter its value.

Examples of comparing rational numbers

Illustrative examples help in the process of correct understanding of how to compare positive and negative rational numbers. Examples include the following:

• First example: Compare the numbers 4/8 and 7/8?
  • The number 7/8 > 4/8, since the denominators are the same, and the number 7 is greater than the number 4.
• Second example: Compare the numbers 2/3 and 4/9?
  • We find that the denominators are not equal, so we unify the denominators by finding the LCM = 9
  • We multiply the numerator and denominator of the fraction 2/3 by 3 to get 6/9
  • Compare the numbers 6/9 and 4/9
  • The number 4/9 <6/9, since the denominators are the same, and the number 4 is less than the number 6.
  • Number 4/9 <2/3
• Third example: Compare the numbers 2/4, 1/2?
  • We find that the denominators are not equal, so we unify the denominators by finding the LCM = 4
  • We multiply the numerator and denominator of the fraction 1/2 by 2 to get 2/4
  • Compare the numbers 2/4 and 2/4
  • The number 2/4 = 1/2, after unifying the denominators.

Arranging Rational Numbers

Arranging the numbers in descending order means placing the numbers according to their values ​​and their sequence from largest to smallest, and arranging the numbers in ascending order means placing numbers according to their values ​​and their sequence from smallest to largest, and the process of arranging the relative numbers is detailed as follows:

Arrange the rational numbers in descending order

The rational numbers are arranged in descending order from largest to smallest by following the following steps:

• Unifying the denominators between all rational numbers, by finding the least common multiple.
• Compare all numbers in the numerator with the same integer comparison mechanism.
• Positive numbers are always greater than negative numbers, no matter what their value.

Arranging rational numbers in ascending order

The rational numbers are arranged in ascending order from smallest to largest by following the following steps:

• Unifying the denominators between all rational numbers, by finding the least common multiple.
• Compare all numbers in the numerator with the same integer comparison mechanism.
• Negative numbers are always smaller than positive numbers, no matter what their value.

Examples of ordering rational numbers

Illustrative examples help to understand the mechanism of arranging rational numbers in ascending or descending order as follows:

• first example: Arrange the following rational numbers in descending order 1/3, 4/9, 5/3?
  • We combine the denominators of fractions by finding LCM = 9
  • 3/9 ، 4/9 ، 15/9
  • 15/9 > 4/9 > 3/9
  • 5/3 > 4/9 > 1/3
• Second example: Arrange the following rational numbers in ascending order – 8/9, 4/9, 1/9?
  • The denominators are the same, so the numerators of each of the denominators are compared?
  • A negative number is always smaller than a positive number.
  • 8/9- < 1/9 < 4/9

Is zero a rational number?

Yes, zero is a rational number. The rational numbers are the numbers that can be written in the form a / b, so that a and b belong to the group of integers, and b is not equal to zero, the number zero may not be in the denominator, but it can be in the numerator, whatever its denominator.[4]

Here we have come to the end of our article What are rational numbers in mathematics?, where we shed light on rational numbers, their properties, examples of them, and how to compare them.