Niraj Yadav

Throughout history, the concept of negative numbers has baffled, confused, and even outraged many thinkers. Their journey into mainstream mathematics is marked by scepticism and incredulity, as captured by various historical quotes. Here, we explore some of these fascinating perspectives.

“Above all, he [the student] must reject the definition still sometimes given of the quantity -a, that it is less than nothing. It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing; above all, that the notion should have outlived the belief in judicial astrology and the existence of witches, either of which is ten thousand times more possible.”

–Augustus De Morgan, 1898

In his book On the Study and Difficulties of Mathematics, De Morgan expresses astonishment at the acceptance of negative numbers. Comparing the belief in negative quantities to the belief in astrology and witches, he underscores the perceived absurdity of the concept.

“Numbers that are less than nothing? Inconceivable! The next thing you’ll tell me is that there’s a witch outside with my horoscope.”

–Augustus De Morgan, as interpreted by Jessica

“How can you have a negative number of monkeys?”

–Bhascara I, 7th century

“How can you buy a negative amount of cloth from a merchant?”

–Chuquet, 1400s

Chuquet, a French mathematician from the 15th century, also struggled with the idea. His practical example of buying cloth illustrates the confusion and impracticality of negative quantities in everyday transactions.

“You may put a mark before 1, which it will obey: it submits to be taken away from another number greater than itself, but to attempt to take it away from the number less than itself is ridiculous. Yet this is attempted by algebraists who talk of numbers less than nothing ….”

–William Frend, The Principles of Algebra, 1796

“I know people who cannot understand that when you subtract four from zero, what is left is zero.”

–Blaise Pascal, Pensées, 17th century

Blaise Pascal acknowledged the confusion surrounding negative numbers.

Brahmagupta and Operations with Negative Numbers

While many historical figures struggled with negative numbers, Indian mathematician Brahmagupta made significant strides in this area. In 628, in his book Brahmasphutasiddhanta (The Opening of the Universe), Brahmagupta wrote about the rules for doing arithmetic with zero and negative numbers. These rules are clearly stated in the six verses of the eighteenth chapter, numbered 18.30 to 18.35:

• A negative quantity subtracted from zero gives a positive quantity.
• A positive quantity subtracted from zero gives a negative quantity.
• The product or quotient of two positive quantities is a positive quantity.
• The product or quotient of two negative quantities is a positive quantity.
• The product or quotient of a negative quantity and a positive quantity is a negative quantity.

Brahmagupta’s work was pioneering, providing a structured approach to understanding and working with negative numbers long before they were accepted in the Western world.

Reference:

https://sci.sdsu.edu/crmse/projectz/documents/bishop-tsu-witches-astrology-and-negatives.pdf

https://www.indiachapter.in/index.php?/user/article/2/36/67