*three-ness*. Manipulation of number – with no connection to physical objects – was a great intellectual leap.

## Beyond counting

*can*be solutions of equations – just as valid as their positive namesakes. Likewise the history of zero is just as fraught with controversy and confusion. Zero initially served as a placeholder in the representation of number. For example, it is the zeros which tell you about the size of the numbers 15 and 105 and 1005. But zero as a number in its own right took a long time to gain acceptance. Just like negative values, the solutions to some equations can be zero.

*and*all the values between them) along with zero can be represented on a numberline stretching infinitely in both directions.

## Impossible square-roots

## Imaginary numbers

*imaginary*(as opposed to the useful, real numbers). But his name for them stuck. The square-root of minus one – whatever it was – gained its own symbol. It was denoted in equations by the letter \(i\), which made arithmetic with them less cumbersome. No doubt it shielded nervous mathematicians from having to think too much about how different \(\sqrt{-1}\) was from the familiar, real numbers.

*complex numbers*.