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Negative Rational Numbers
Rational numbers are defined as the numbers that can be expressed in the form of p/q where q ≠ 0. When either the numerator or the denominator of p/q has a negative sign, the rational number is known as a negative rational number. We will be studying more about negative rational numbers in this article.
|1.||Negative Rational Numbers Definition|
|2.||Negative Rational Numbers on Number Line|
|3.||Standard Form of Negative Rational Numbers|
|4.||FAQs on Negative Rational Numbers|
Negative Rational Numbers Definition
The rational numbers that have a 'minus' sign are known as negative rational numbers. Given a rational number p/q, if either of 'p' or 'q' is negative, then we get - p/q which is a negative rational number. Therefore, when a 'minus' sign is included with a rational number, it gives us a negative rational number. For example: - 4/5, - 7/6, etc.
Negative Rational Numbers on Number Line
Representation of negative rational numbers on a number line is very similar to the representation of negative integers or negative fractions on a number line. The left-hand side of 0 on a number line represents the negative region and the right-hand side of 0 represents the positive region. Let us look into the steps to represent negative rational numbers on a number line as shown below.
Let us plot - 2/3 on the number line.
Step I: Draw a number line by marking 0 as the reference.
Step II: Identify the rational numbers between which the rational number lies and mark them. - 2/3 lies between 0 and -1.
Step III: Mark the number of divisions between the rational numbers marked in step II equivalent to the denominator of the given rational number. Here, - 2/3 has 3 as the denominator. Thus, we will make 3 divisions.
Step IV: Starting from 0, move towards the left by the number of steps equivalent to the numerator of the given rational number. For - 2/3, we will be moving 2 units towards the left.
Thus, we have represented - 2/3 on the number line as highlighted above.
Standard Form of Negative Rational Numbers
When the numerator and denominator of a negative rational number have a greatest common factor of 1 and the denominator is a positive number, then it is said to be in standard form. In other words, the standard form of a negative rational number, say - p/q is expressed in the simplest form in which a denominator is always a positive number. Thus, we can conclude that for a standard form of a negative rational number, a numerator is always a negative number.
If the denominator is a negative number, we multiply - 1 to both the numerator and denominator to change the denominator to a positive number. Let's look into some examples to understand this.
Example: Express 8/(-16) in standard form.
The denominator of the given rational number 8/(-16) is -16 which is a negative number. Hence, we multiply - 1 to both numerator and denominator.
[8 × (-1)] / [-16 × (-1)] = - 8/16
Also, we see that - 8/16 is not simplified. The GCF of 8 and 16 is 2. Thus, on simplification, - 8/16 = - 1/2. Hence, the standard form of the given negative rational number is - 1/2.
Check these articles related to the concept of negative rational numbers.
Negative Rational Numbers Examples
Example 1: Express the given negative rational number -2/4 in standard form.
Solution: The given negative rational number -2/4 has a negative numerator and positive denominator which follows the rules of the standard form. But, -2/4 is not in the simplified form. GCF of 2 and 4 is 2. Thus, on simplifying -2/4 we get -1/2. Hence, the standard form of the given rational number -2/4 is -1/2.
Example 2: Represent the negative rational numbers -1/5, -2/5, and -3/5 on the number line.
Solution: We know that, -1/5, -2/5, and -3/5 lies between the rational numbers 0 and -1.
Since the denominator is 5, we will make 5 divisions between 0 and -1 and mark the rational numbers by moving towards the left equivalent to the numerators of -1/5, -2/5, and -3/5.
Thus, we have represented -1/5, -2/5, and -3/5 on the number line.
Practice Questions on Negative Rational Numbers
FAQs on Negative Rational Numbers
What are Negative Rational Numbers?
Negative rational numbers are defined as rational numbers with a "minus" sign. For a given rational number x/y if either of 'x' or 'y' is a negative number, we get a negative rational number. For example: - 4/3, - 6/7, etc.
What are Positive and Negative Rational Numbers?
A positive rational number is defined as a number which is expressed as a/b where b ≠ 0 and both 'a' and 'b' are either positive or negative numbers. For example: 2/3, (-5)/(-7) = 5/7, etc. A negative rational number is very similar to a positive rational number except for the fact that either of 'a' or 'b' is negative. For example: - 1/2, - 3/7, etc.
What is the Sum of Two Negative Rational Numbers?
The sum of two negative rational numbers is always a negative rational number.
For example: - 3/5 + (- 4)/5
= - 3/5 - 4/5
= (- 3 - 4) / 5 = - 7/5
What is the Product of Two Negative Rational Numbers?
The product of two negative rational numbers is always a positive rational number.
For example: (- 2/3) × (- 4/5)
= [(- 2) × (- 4)] / (3 × 5) = 8/15
What are Non-Negative Rational Numbers?
Non-negative rational numbers are defined as a number x/y where y ≠ 0 and both 'x' and 'y' are either positive or negative numbers. In other words, any rational number that is positive is known as a non-negative rational number. For example: 1/5, (-7)/(-9) = 7/9, 0, etc.
How to Divide Negative Rational Numbers?
The division of rational numbers is very similar to division of fractions. The division of two negative rational numbers gives a positive result. When two rational numbers are divided, the second rational number is reciprocated and the operation changes to multiplication.
For example: - 3/5 ÷ (- 6/7)
= - 3/5 × 7/(-6)
= - 3/5 × (- 7/6)
= [- 3 × (- 7)] / (5 × 6]
= 21/30 = 7/10
Why is the Product of Two Negative Rational Numbers Positive?
The product of two negative rational numbers is positive because according to the rules of multiplying negative rational numbers, when two negative numbers are multiplied, it gives a positive result.
For example: (- 1/4) × (- 3/5)
= [(- 1) × (- 3)] / (4 × 5) = 3/20