Note that division by a fraction can be rewritten as a÷ 1a=a* a.
Conjecture: The quotient of a number and its reciprocals equals the square of the number.
Practice makes perfect
Note that the reciprocal of a number is when the numerator and denominator switch places. Note that any number a can be rewritten as a fraction with a denominator of 1.
Number:& a/1 [0.6em]
Reciprocal:& 1/a
Let's attempt to find a pattern by calculating the quotient of a few numbers and their reciprocal.
2÷ 1/2&=4 &&8÷ 1/8=64 [0.8em]
5÷ 1/5&=25 &&9÷ 1/9=81
Examining the products, we see that all of them equal the number squared. With this information we can write a conjecture.
Conjecture: &The quotient of a number
&and its reciprocals equals the
&square of the number
Let's test this a few times with some other numbers.
3÷ 1/3&=9 &&8÷ 6÷ 1/6=36 [0.8em]
4÷ 1/4&=16 &&9÷ 7÷ 1/7=49
It would appear that our conjecture is true.
