Without thinking too much about it, most of us can use negative numbers for things like recording temperatures below zero, but this was not always true. In fact, there was quite an opposition to the introduction of negative numbers. William Fredn, in his book, The Principles of Algebra (1796), refused to accept negative numbers. He said,
...But numbers are divided into two sorts, positive and negative; and an attempt is made to explain the nature of negative numbers by allusion to book-debts and other arts...You may put a mark before one, which it will obey: it submits to being taken away from another number greater than itself, but to attempt to take it away from a number less than itself is ridiculus...
In the early work solving equations, negative roots were not even considered. Hindu mathematicians in the sixth century, and before that, Chinese mathematicians figured out all the rules for operations with negative numbers. Yet the Arabs did not include any of this in their work three centuries later. French mathematician Blaise Pascal (1623-1662) did not see the need for negative numbers. Augustus de Morgan (1806-1871), an English mathematician, thought numbers less than zero were unimaginable.
In the sixteenth century the negative numbers gained some acceptance. Cardan published the solution to cubic equations. In order for all three distinct roots to be found we must use complex numbers, and thus the square roots of negative numbers. Only in the nineteenth century were negative numbers accepted as part of the number system.