**Li Zhi** was a Chinese mathematician who described methods for solving equations.

- Usually Chinese names have a number of different spellings, each trying in a different way to match the pronunciation of the original.
- However this is not the reason that Li Zhi is also known as Li Yeh.
- Rather it is because he was known as a young man as Li Zhi, but since this was the same name as the third T'ang emperor, he later changed his name to Li Yeh.
- Li Zhi took the civil service examinations in Lo-yang, a city in the northwestern Honan province in 1230.
- After successfully completing his examinations, Li Zhi was appointed registrar of the district of Kaoling but the advancing Mongol armies prevented him taking up the appointment.
- Li Zhi only escaped being massacred himself thanks to the intervention of one of the Jurchen officials who had gone over to the side of the Mongols.
- For over 15 years Li Zhi lived in poverty as a hermit in the Shansi province.
- Three years after completing his masterpiece, Li Zhi's financial position improved somewhat and he returned to Hopeh province where he made is home near Feng Lung mountain in the Yuan municipality.
- He lived here until 1257 when Kublai, a grandson of the Mongol leader Genghis Khan who was leading further Mongol advances, sent for Li Zhi to ask his advice on governing the state and on civil service examinations.
- Knowing that Li Zhi was a great expert on scientific matters, Kublai also asked him to explain the reasons for earthquakes.
- Li Zhi continued to work on mathematics and completed another important text Yi gu yan duan (New steps in computation) in 1259.
- politely declined with the plea of ill health and old age.
- Kublai Khan made another attempt to get Li Zhi's services in 1264 when he set up his Academy.
- This time Li Zhi felt that he had no option but to join the Academy, but after a short while he resigned, again saying that he was too ill and too old for the task.
- We now look at some of the very remarkable contributions which Li Zhi made to mathematics.
- This was a notation for an equation and, although the work of Li Zhi is the earliest source of the method, it must have been invented before his time.
- Li Zhi placed the coefficients in an array as in the following example.
- We have given the example using numerals which are natural with the language that we write this archive but, of course, Li Zhi would have used Chinese characters.
- But he does not limit his reflections to equations of degree two or three; for him, the fact that polynomial equations of arbitrarily high degree are involved is of little importance.
- In other words, like many other algebraists, Chinese or not, he demonstrates algebra by using it ...
- To solve the above equation Li Zhi would bring the leading coefficient to -1 and then give the solution; in this case 20.
- Li Zhi seems happy with equations of any degree and, although methods to solve equations do not appear explicitly, one has to assume that he used methods similar to those Ruffini and Horner discovered over 600 years later.
- Chapter 1 contains three sections, the first giving the names of the constituents, the second section lists all the values of the lengths of the segments, so in essence contains all the answers to the problems, while the third section comprises of 692 formulae for areas of triangles and lengths of segments.
- If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
- Li Zhi's New steps in computation although written much later, is a more elementary work.
- The New steps in computation is based on an earlier book which it is said was written by Chaing Chou of P'ing-yang (although nothing else is known of the author, nor is there any knowledge of the date of this earlier work).
- Li Zhi's book contains 64 problems, of which he says that 21 are from the earlier text.
- This older geometric style method of solving equations was used by Chinese mathematicians before Li Zhi and so the New steps in computation gives historians a unique opportunity to see the new coefficient array method beside the older method of sections.

Born 1192, Ta-hsing (now Beijing), China. Died 1279, Hopeh province, China.

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Ancient Chinese, Chinese, Origin China, Puzzles And Problems

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive