# Definition:Negative/Number/Historical Note

## Historical Note on Negative Number

The idea of a **negative number** was the cause of considerable philosophical difficulty.

Negative numbers made no sense to the ancient Greeks. A number expressed a magnitude and that was all.

Diophantus of Alexandria recognised that certain equations yielded both a positive root and a negative root, but rejected the negative root as nonsensical. If an equation had no positive root, for example $x + 10 = 5$, then it was not a proper equation.

The early Hindu mathematicians recognised the existence of negative roots, but were still wary of them. As Bhaskara II Acharya put it:

*The second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots.*

The Chinese were using negative numbers by the $12$th century as a matter of course. However, they still did not recognise negative roots.

The general acceptance of negative numbers in calculations seems in fact to have started with merchants and accountants. The symbols $+$ and $-$ originated in $15$th century German warehouses for indicating whether a container was over or underweight.

Michael Stifel referred to negative numbers as **absurd** or **fictitious**.

It was Gerolamo Cardano who was one of the first to accept negative numbers, and he even went as far as considering their square roots in his *Ars Magna*.

However, as late as the end of the $18$th century, William Frend, together with Francis Maseres, published between them a number of works, most notably *The Principles of Algebra* in $1796$, which rejected the concept of negative numbers as invalid.

## Sources

- 1939: E.G. Phillips:
*A Course of Analysis*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.1$ Introduction - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $-1$ and $i$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $-1$ and $i$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers: Negative numbers