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Chapter 1: Problem 21

What is the identity element for addition?

Short Answer

Expert verified
The identity element for addition is 0, as adding 0 to any number 'a' does not change its value: \(a + 0 = a\) and \(0 + a = a\).

Step by step solution

01

Identify the Property of the Identity Element for Addition

The identity element for addition is denoted as 'i'. When we add the identity element to any number, let's say 'a', the result should be the same number 'a'. Mathematically, this can be written as: \(a + i = a\) or \(i + a = a\)
02

Determine the Identity Element for Addition

Now we need to find the value of 'i' that satisfies this equation for any number 'a'. Let's consider the addition of the number 'a' and its additive inverse '-a': \(a + (-a) = 0\) We know that when we add a number and its additive inverse, we get the neutral element for addition (i.e. 0). Therefore, when we add '-a' (the additive inverse of 'a') to 'a', we should get the identity element, 'i': \(i = a + (-a)\) However, since 'a' can be any number, it is clear that the value of 'i' that satisfies this equation for any 'a' is 0.
03

Conclude the Identity Element for Addition

In conclusion, the identity element for addition is 0: \(a + 0 = a\) or \(0 + a = a\) Adding 0 to any number 'a' does not change its value, hence 0 is the identity element for addition.

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