# Origins of the Equals Signs

Some authors use different equals signs for different purposes. For the most part, they are "$=$", "$\equiv$", "$:=$", and "$:\equiv$'. I read that "$=$" dates back to 1557, and is of mathematical origin.

What about the others? I have read that "$:=$" appears in some programming languages from the $70$'s. But, does it occur earlier? What about $\equiv$ and $:\equiv$, what are their origins?

• I use $\equiv$ to mean "equivalent" which is not exactly the same thing as equal. It has been a long time since I have done any programming but as I remember $a:=b$ defines $a$ as equal to $b$ where "$a=b$" is a logical proposition, returning a true or false result. Nov 9, 2017 at 22:32
• the sign $\equiv$ is a logical symbol, sometimes used in mathematics to denote congruence. It means that the chain of symbols to both sides represent the same thing. The double point $:$ in both notations means "definition". This is to differentiate from an equation or a consequential identity. Nov 9, 2017 at 22:51
• The use of $\equiv$ for congruence is introduced by Gauss in 1801 (an online reference, another reference is paragraph 408 of Florian Cajor's book A history of mathematical notations). Nov 9, 2017 at 22:52
• This question may be worth moving to or asking instead on history of science and mathematics stackexchange. In any case, another possibly useful reference would be Gentzen's paper where he tried to standardize various logical symbols. He mentions that he opted not to use $\equiv$ in the same way as Russel since it has other uses elsewhere in mathematical literature. Nov 9, 2017 at 22:59
• @DougM my memory is that $a = b$ defines $a$ but $a==b$ is the logical proposition. Nov 9, 2017 at 23:08

The material here is extracted from Florian Cajori's book A history of mathematical notations.
If there are any mistakes, it is most likely caused by my own misunderstanding.

The symbol $\equiv$ has been used by various branches of mathematics. It has been used for

1. arithmetic as congruence - first introduced by Gauss in $1801$.

Ref: an online reference and paragraph $408$ of Cajori's book.

2. geometry as geometric congruence - first introduced by Riemann.

It appears in G.F.B.Riemann's Elliptische Funklionen (Leipzig 1899), p 1, 6.

Ref: paragraph $374$ of Cajori's book.

3. logic - can be dated back at least to $1910$.

It appears as 'definitional identity' in E.H.Moore, Introduction to a Form of General Analysis (1910) p.18. In first volume of Whitehead and Russell's Principia Mathematica (1910, p5-38), it has been used as bi-conditional (i.e $p \equiv q$ stands for $p$ implies $q$ and $q$ implies $p$) instead.

Ref: paragraphs $694, 695$ of Cajori's book.

Gottlob Ferge may have used $\equiv$ before in $1879$ (he switched to use $=$ later in his publication in $1893$), the reference I have is not clear what happens.

Ref: paragraph $687$ of Cajori's book.

• +1, but partly correct. W&R's Principia (see page 7 and 12) has $\equiv$ for bi-conditional and $=$ followed by Df. on the right-end od the formula for definitional identity. Nov 10, 2017 at 13:19
• @MauroALLEGRANZA fixed, thanks for pointing that out. Nov 10, 2017 at 13:30