Tuesday, 4 October 2022

# 7 Absolute value/difference,… from old math term notes

 Absolute Value The word absolute is from a variant of absolve and has a meaning related to free from restriction or condition. The first use of "absolute value" in English seems to have been to apply to real values. Jeff Miller's website on the Earliest Known Uses of Some of the Words of Mathematics says," Absolute value is found in English in 1850 in The elements of analytical geometry; comprehending the doctrine of the conic sections, and the general theory of curves and surfaces of the second order by John Radford Young (1799-1885): "we have AF the positive value of x equal to BA - BF, and for the negative value, BF must exceed BA, that is, F must be on the other side of A, as at F', hence making AF' equal to the absolute value of the negative root of the equation" [University of Michigan Digital Library]." [See the page here] In 1876 Karl Weierstrass applied the term to magnitude of complex numbers. From Miller's site again we find "Absolute value was coined in German as absoluten Betrag by Karl Weierstrass (1815-1897), who wrote:

Ich bezeichne den absoluten Betrag einer complex Groesse x mit |x|. [I denote the absolute value of complex number x by |x|.]"

In "The Words of Mathematics", Steven Schwartzman suggests that the use of the word for real values only became common in the middle of the 20th century. This may be true, but the use for signed numbers also appears in 1889 by Wentworth according to Miller; "In 1889, Elements of Algebra by G. A. Wentworth has: 'Every algebraic number, as +4 or -4, consists of a sign + or - and the absolute value of the number; in this case 4.' " (above). In the 1893 edition of the same book he uses the term again, as shown below, without any symbol.

The revision of Hall and Knight's Algebra, for Colleges and Schools {"Revised and Enlarged for the use of American Schools"} by F. L. Sevenoak in 1905 also uses the term without a sign. By 1934, the word is still used without symbol in Walter W. Hart's Progressive First Algebra,(pg 78), but in the 1939 edition of College Algebra by Rosenbach and Whitman, the symbol is used as shown below

The symbol for absolute value is usually a pair of vertical lines containing the number, as created by Weierstrass in 1876 (see above). |3| is read as "The absolute value of three". The absolute value of a real number is its distance from zero, so |3| = |-3| = 3. In words that says that the absolute value of three is equal to the absolute value of -3 , and that both have a value of three.

For complex numbers the absolute value is also called magnitude or length of the complex number. Complex numbers are sometimes drawn as a vector using an Argand Diagram, and the length of the vector Z=a+bi is |a+bi|. Stated another way, the value of |a+bi|= 

A symbol for the Absolute Difference of two numbers, or the absolute value of the difference was created by Oughtred around 1630. Miller writes, "The tilde was introduced for this purpose by William Oughtred (1574-1660) in the Clavis Mathematicae (Key to Mathematics), composed about 1628 and published in London in 1631, according to Smith, who shows a reversed tilde (Smith 1958, page 394)." This seems no longer to be common in basic maths classes in England today (current coments anyone?). After posting a request for information to the Historia Matematica discussion group about the use of the tilde to indicate absolute difference in England I received the following update from Herbert Prinz:

"In modern English texts on navigation, nautical astronomy or its history, the tilde,~, is frequently used to express the function | a - b |, where |x| stands for absolute value. E. g. Cotter, The Complete Nautical Astronomer, 1969. I am not sure when this practice started. In older texts on the same subject, say, Moore, The Practical Navigator, 1800, one does not find the tilde used in this way. For one, because instructions were given mostly verbally without the use of any symbols at all. And second, the distinction from '-' was unnecessary, as it was always understood, if not explicitly stated, that one must subtract the smaller.  While talking about symbols, I should add that shortly after Robert Recorde began using the equal sign, =, now common, Viete used the same symbol for the absolute difference between two numbers.   


In England the absolute value is often referred to as the modulus function, and the two bars that make up the symbol are sometimes called "modulus signs" according to a note posted by Vicky Neale on the Ask NRich math site. The term modulus is used both in America and England to represent the magnitude or length of a complex number. The term is also used in a number of other specialty ways in mathematics, the best known being the "congruence modulus". The modulus of a congruence, often shortened to "mod" is the base value with which the congruence is computed. We say A is Congruent to B modulus C, if A divided by C and B divided by C have the same remainder. C is called the modulus of congruence. It would be written A≡B [mod C]

Modulus comes almost unchanged from the Latin from the diminutive of modus (measure or amount), modulus for a small measure. Vicky also pointed out that at one time the term was used for, "A unit of payment used at Trinity College.... Fellows received some number of moduli". Ms Neale also said she was unfamiliar with the use of the ~ for absolute difference.

It was Gauss, Disquisitiones arithmeticae in 1801, who introduced the term modulus of congruence, and the abreviation, "mod". Cajori credits Jean Argand for the first use of modulus for the length of a vector in 1814. I am not sure when the British public schools started to use the term for the absolute value of a number, and would love to know if someone has old books with these terms (or others for the same idea).

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