Additive Identity of Rational Numbers

Rational numbers are numbers in the form of p/q, where p and q can be any integer and q ≠ 0. Natural numbers, whole numbers, integers, fractions, and decimals (terminating and recurring decimals) are all considered rational numbers. 

‘Rational’ is derived from ‘ratio’. Therefore, the idea of fractions, which represent ratios, is closely tied to rational numbers.  In simple words, a number is called rational if it can be written as a fraction where both the numerator and the denominator are integers (the denominator is not zero). 

The additive identity property states that adding 0 to a number does not change its value: a + 0 = a. For example, take the rational number ⅖, and its additive inverse is -⅖. According to the property:

 2/5 + (-2/5) = 0

Thus, the property is verified.

Two real numbers are said to be additive inverses of each other if their sum equals zero. In general, for any real number R, we have:

R + (-R) = 0

Here, R and -R are additive inverses of each other. Since rational numbers belong to the set of real numbers, this rule applies to them as well.

For example, if you take the rational number ⅚, its additive inverse is -⅚. 
So: 5/6 + (-5/6) = 0. 

Here, 5/6 is the additive inverse of -5/6, and vice versa. 

The additive identity property states that when you add zero to any number, the value of the number stays the same.

• Zero is the Additive Identity: For any real number a, adding 0 does not change the value of a: a + 0 = a and 0 + a = a.

• Applies to All Numbers: This property holds for all types of numbers, including natural numbers, whole numbers, integers, rational numbers, and real numbers.

• Additive Inverse Relation: Every number a has an additive inverse, which is -a. When you add a number and its additive inverse, the result is always zero.                 
• Neutral Element in Addition: Zero is called the neutral element in addition because it does not affect the sum.

• Associative and Commutative: The additive identity property follows the associative and commutative properties of addition:
  
  Associative: (a + b) + 0 = a + (b + 0)
  
  Commutative: a + 0 = 0 + a

In a mathematical system, the additive identity is an element that, when added to a number, does not change its value. For the set of rational numbers, this identity is 0. The additive identity property states that adding 0 to any number results in the same number:

a + 0 = 0 + a

For the additive inverse, a number and its inverse must sum to zero. For example, the additive inverse of -5/7 is 5/7, since:

(-5/7) + (5/7) = 0

For the multiplicative inverse, multiplying a number by its reciprocal results in 1. For example, the multiplicative inverse of 4/9 is 9/4, since:

(4/9) × (9/4) = 1

The additive identity of rational numbers (0) has practical applications in various real-life situations. Here are a few examples:

• Banking and Finance: When checking your bank balance, adding $0 to your account does not change the total amount. If you have $250 in your account and deposit $0, your balance remains $250.

• Temperature Measurement: If the temperature is 15°C, and there is no change (0°C increase or decrease), the temperature remains 15°C.

• Distance and Travel: If you start at 5 km and don’t move (add 0 km), your position stays the same at 5 km.