Additive Identity of Rational Numbers

We know that rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. This set includes natural numbers, whole numbers, integers, fractions and decimals that terminate or repeat. In short, anything that can be written as a ratio of two integers are rational numbers. 
 

In mathematics, an identity element for an operation is a special number that, when used in that operation with any number from the set, leaves the other number unchanged. In addition, the identity element is 0. This is because adding 0 to any number does not change its value. This rule applies not only to whole numbers or integers, but also to rational numbers. 
 

For any rational number p/q, where q ≠ 0, 
p/q + 0 = p/q and 0 + p/q = p/q. 

This shows that adding 0 does not change the value of the rational number.
 

The additive identity property states that adding a number to its additive inverse gives 0. For example, take the rational number 2/5, and its additive inverse is -2/5.

• According to the property:
  
  \(\frac{2}{5} + \frac{-2}{5} = 0\)
  
  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 \(\frac{5}{6}\), its additive inverse is \(\frac{-5}{6}\). 

So: \(\frac{5}{6} + \frac{-5}{6} = 0\). 

Here, \(\frac{5}{6}\) is the additive inverse of \(\frac{-5}{6}\), and vice versa. 

The additive identity is a number that when you add to any number, the value of the number stays the same.

• Zero is the Additive Identity: For any real number, 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)
  
  The sum remains the same regardless of how numbers are grouped.
  
  Commutative: a + 0 = 0 + a

In a mathematical system, the additive identity is an element that does not change a number's value when added. For 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 \(\frac{-5}{7}\) is \(\frac{5}{7}\), since:
  
  \(\frac{-5}{7} + \frac{5}{7} = 0\)

• For the multiplicative inverse, multiplying a number by its reciprocal results in 1. For example, the multiplicative inverse of \(\frac{4}{9}\) is \(\frac{9}{4}\), since:
  
  \(\frac{4}{9} × \frac{9}{4} = 1\)

The Additive Identity and the additive property of a number are easy to calculate. Here are a few tips and tricks for students to remember the additive identity and the additive inverse, along with practical guidance for parents and teachers. 

• To remember additive identity, think it as something that does not change the identity of number.
   
• To find the additive inverse of a number, just take its negative.
   
• Memorize, that for any number, 0 is the additive identity.
   
• Represent the numbers on number line. To find the additive inverse, the units it takes to return to 0 is the additive inverse. 
   
• Remember, any number plus zero is the same as adding 0 to any number.
   
• Use real-life examples, like showing that adding $0 to any amount of money does not change its value. 
   
• Parents and teachers can help students by highlighting patterns such as 5 + 0 = 5, -3 + 0 = -3 or \(\frac{7}{3} + 0 = \frac{7}{3}\), to show consistency across all rational numbers. 
   
• Use visual aids, such as counters or colored blocks, to show that adding zero means adding nothing, so the value stays the same. 
   
• Teach students that they can remove any terms like ‘+0’ or ‘0+’ in an equation to make simplification easier. 
   
• Explain to students that a number and its additive inverse always bring you back to 0 on the number line, just like walking four steps and then walking back four steps returns you to the start.

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 changes by 0°C, it 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.

• Sports: If a cricketer has 52 runs in 6 overs and misses the next two balls. The new runs after 6 over and 2 balls will be the same, that is 52 runs.

• Shopping: When shopping, if you did not buy any items, then the money you will be left with will be the same.