In the converse of a statement we switch the hypothesis with the conclusion.
Converse of the Statement: If a^2 is odd, then a is an odd number. Are They True? Both the statement and its converse are true.
Practice makes perfect
We want to write the converse of the given statement. In this statement we have a hypothesis and a conclusion. A hypothesis tells us about the condition. When this condition is satisfied, the conclusion is true.
Statement
If {\color{#0000FF}{\underbrace{a\text{ is an odd number}}_\text{hypothesis}}}, then a^2is odd_(conclusion).
Now, in the converse of the statement we switch the hypothesis with the conclusion. This means that the previous hypothesis becomes a new conclusion and the previous conclusion becomes new hypothesis.
Converse of the Statement
If a^2is odd_(new hypothesis), then {\color{#0000FF}{\underbrace{a\text{ is an odd number}}_\text{new conclusion}}}.
Let's think if our statement and its converse are true or false. To do so, let's take a look at the couple examples of odd numbers and even numbers and their squares.
Number a
Odd or Even?
Squared Number a^2
Odd or Even?
1
Odd
1^2=1
Odd
2
Even
2^2=4
Even
3
Odd
3^2=9
Odd
4
Even
4^2=16
Even
We can see that when the number is odd its square is also odd. On the other hand, even numbers have even squares. This means that both our statement and its converse are true.
Extra
More Formal Proof
Let's try to prove that the statement and its converse are true for any odd number. Let 2n+ 1 represent an odd number where n is any natural number. Notice that 2n is even because any number multiplied by 2 is even.
2n+ 1
After each even number we have an odd number. This means that 2n+ 1 is odd because it is one more than an even number 2n. Now let's find the square of our number.
We end with an odd number because 4n(n+1) is even and by adding 1 to it we get an odd number. This means that any odd number has an odd square. We can check if the converse of this statement is true using a similar method.
