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Kenneth E. Iverson
Kenneth E. Iverson (1920–2004)

Iverson Notation

The q language descends from the mathematical notation devised at Harvard by Ken Iverson for his work on input-output matrices with economist Vassily Leontieff – for which Leontieff received the Nobel Prize.

Fred Brooks and Ken Iverson used the notation to teach the world’s first programming course, for which they wrote the textbooks Automatic Data Processing and A Programming Language.


Iverson went from Harvard to IBM, where his notation was implemented as a programming language and dubbed APL, after the title of his book.1 Before PCs and spreadsheets, IBM managers throughout the world used APL for personal computing.

APL has inspired major developments in computer science, such as functional programming, and NumPy. In 1979 Iverson received the Turing Award. His Turing lecture “Notation as a Tool of Thought” is a classic.

A+ and J

Iverson was mentor to his friend’s son, Arthur Whitney. They both worked for some years at I.P. Sharp Associates (IPSA) in Toronto.

For IPSA Australia, Whitney rewrote the 500Kb APL interpreter as an 80Kb binary for a minicomputer: SHARP APL/HP.

Later, in New York, he designed and implemented an ASCII-only subset of APL, dubbed A (later, A+) that for twenty years served as the platform for Morgan Stanley’s trading applications.

Ken Iverson and Arthur Whitney
Ken Iverson & Arthur Whitney at APL91

In 1990 Iverson stopped working on APL and founded Jsoftware to distribute his ASCII-only J programming language. Whitney wrote the first draft of the J interpreter.

kdb+ and q

Later iterations of A+ became the k programming language. Whitney and Janet Lustgarten founded Kx Systems to distribute it as kdb+.

Q was written as an embedded domain-specific language – a wrapper – to make k more accessible, with SQL-like queries and English-like keywords replacing some of k’s dense rank overloads.

After Kx Systems, Whitney and Lustgarten founded Shakti to release a further iteration of k.



  1. A proposal to call the notation Iverson’s Better Math was rejected as facetious.