*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.