Problem solver: Mathematics maestro Terry Tao. Credit:Hugh Hamilton "He is arguably the world's best mathematician," says Joseph Rudnick, the dean of Physical Sciences at the University of California, Los Angeles (UCLA), where Tao, now 39, has been a mathematics professor since he was 24. "Other mathematicians speak of him in tones of awe." His talents, says Rudnick, are "other-worldly". Tao has pages of awards, fellowships, prizes and medals to his name - most notably, the Fields Medal, the maths world's equivalent of the Nobel Prize, which he received in 2006 when he was 31 "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory". Around about then, people started to describe him as "the Mozart of math". Last year, Facebook founder Mark Zuckerberg and Russian tech tycoon Yuri Milner recognised Tao's transformational contributions when they announced he was one of five recipients of their inaugural 2015 Breakthrough Prize in Mathematics. Before the announcement, Tao tried to convince Milner not to give him the $US3 million prize. He thought the money should be dispersed between more people. "I didn't feel I was the most qualified for this prize," he told The New York Times. (So far, Tao has put some of his prize money towards a travel fellowship for graduate students in developing countries, and a student fellowship for gifted American high-school students.) Tao's modesty is almost as remarkable as his mind. It's the way he was raised, the Chinese philosophy, says his father, Billy Tao. "Even if you are very successful, you tend to say, 'No, I'm just pretty average,' " he says.

The sum of us: Terry Tao with his wife, Laura, and children William, 11 and Madeleine, 3, in their Los Angeles home. Credit:Hugh Hamilton "This is a world-class genius but there is not one iota of conceit," says Miraca Gross, Emeritus Professor of Gifted Education at the University of NSW. Gross met Tao when he was about three. "It was more like talking with an intelligent six- or seven-year-old." The little boy with shining eyes and glossy black hair told her he liked everything to do with numbers. He would become one of 60 children with an IQ in excess of 160 in Gross's longitudinal study of gifted children. And, when he grew up, he would become something of a celebrity: after he won the Fields Medal, Holden asked Tao to star in an ad campaign featuring Australian heroes. He declined the offer. "It just felt kind of awkward." In December, he was a somewhat bemused guest on the American satirical television talk show, The Colbert Report. Looking atypically preppy in a watermelon-pink, V-neck sweater and blue-checked shirt, he took questions about prime numbers (a number that can only be divided evenly by 1 and itself) and twin primes - pairs of primes that are two apart. "Is there a cousin prime?" host Stephen Colbert asked. "Oh, wow," said Tao. "Okay, all right, cousin primes are pairs of primes that are four apart, like seven and 11." And then, almost daringly: "And then there are sexy primes - pairs of primes that are six apart." Colbert was ready: "Are cousin primes ever sexy but you're afraid to say it 'cause it sounds creepy?" Tao's uproarious laughter suggested he might not previously have ventured into such territory.

He has groupies, maths geeks who fill lecture halls when he speaks. "They're expecting some sort of lecture full of, I don't know, some transcendence of wisdom," says Tao, "but they're meant for serious students of mathematics, so I don't know what some of these students get out of it." Afterwards they loiter for autographs; he finds the attention discomfiting but deploys a puckish sense of humour and a high-pitched, eyes-closed giggle to tackle the issue. "Yeah, okay, usually not female," he says, spinning the toy pony in his hands. Terry Tao's website and celebrated personal blog offer some insight into his world. The blog is foreign territory for all but the few; a mathematical menagerie inhabited by the likes of gradient shrinking solitons, probability spaces, random matrices, oscillatory integrals, sieve coefficients, parity problems and prime gaps. Meanwhile, his website is a defence mechanism for a man in demand: multiple pages of information to deflect every possible inquiry or invitation. Dozens of past lecture trips are listed but these days, he writes, "I am currently declining essentially all further invitations to discretionary travel or speaking engagements." He's also refusing invitations to join editorial boards, to publish monographs and books, requests for career advice and appeals to collaborate. "I get a lot of emails," he says. Nevertheless, Tao's work has always been marked by collaborations with his peers. "It means that other people get the benefit of working with him," says Rudnick. Professor Dimitri Shlyakhtenko, chair of the mathematics department at UCLA, adds that the ease with which Tao forms "extremely deep" collaborations "has a tremendous scientific effect. There have been many instances where his participation in projects has led to significant breakthroughs, even on the fringes of mathematics." Shlyakhtenko cites Tao's work with Stanford University mathematics and statistics professor Emmanuel Candès. During a casual conversation with Tao in 2004, Candès mentioned issues he had encountered in his work with radiologists to make magnetic resonance imaging (MRI) more efficient. The next day, with customary speed, Tao had sent his thoughts about the issue to Candès. The mathematicians' subsequent paper launched the field of compressed sensing, a mathematical algorithm that can, in certain cases, obtain high-resolution data from low-resolution samples of information.

"It kind of set off a whole area of applied mathematics," says Shlyakhtenko. Compressed sensing has led to faster MRI scanning methods, which can be life-saving. There are potential applications for the military and in intelligence gathering, as well as in the search for other galaxies. As a pure mathematician, Tao revels in the thrill of the chase rather than the application of a theory. And he lets himself get distracted, meander, looking for beauty and artistry in numbers. "A lot of the math that I do, it's not sort of premeditated. I talk online or with a colleague and I get interested and I just follow where it leads," he says. "We work on these problems because they're intrinsically interesting." Tao's soft, melodic voice is silent for a bit and his eyes close as he considers the question of what element of his work has brought him the most satisfaction. Finally, he speaks. The compressed sensing breakthrough "still impresses me the most", he says. "Most of what I do does not save lives." Dimitri Shlyakhtenko says that what sets Tao apart from other mathematicians is his ability to absorb an argument, extract its key idea, and rework it into a form that is crystal clear. "After Terry has looked into some area of mathematics it often becomes more accessible to other mathematicians." Others believe that Tao's genius lies in the number of areas in which he works. The nature of pure mathematics is such that, at a certain level, most practitioners confine themselves to one area and may have only a rudimentary understanding of other mathematical topics.

Broadcaster and mathematical interpreter Adam Spencer, who has a first-class honours degree in pure maths from the University of Sydney and last year self-published his Big Book of Numbers, is excitable on the subject. "The great mathematicians, the once-in-a-generation mathematicians, take things from different areas and realise, 'Hold it, that problem you're trying to work out in group theory, I think that's essentially the same question that these guys in algebraic topology have been doing.' " Tao, says Spencer, is that once- in-a-generation guy. "Terry Tao's like a piano player who can play classical and jazz and is really f...ing good on the saxophone, and, if you gave him an AFL footy and half an hour, he could probably handball." Terry Tao recalls the day his aunt found him rolling around her living room floor in Melbourne with his eyes closed. He was about 23. He was trying to visualise a "mathematical transform". "I was pretending I was the thing being transformed; it did work actually, I got some intuition from doing that." His aunt is likely still puzzled. "Sometimes to understand something you just use whatever tools you have available." Detergent can be a mathematical tool, and it's a part of Tao's earliest memory. "I was about three, and my grandmother was washing the windows at our house," he recalls. "I wanted her to put the detergent on the windows in the shape of numbers." Tao's father, Billy, remembers an earlier day, the day it became clear that his oldest son was unusual. The Taos were visiting friends. "After a while, the parents said, 'Oh, the children are very quiet,' " says Billy Tao. When they investigated, they found Terry showing older children how to read and count. He was little more than two years old. In the years ahead, when people would ask the father why his son was so smart, he'd tell them, "Oh, because I make him eat fried rice."

Billy Tao is an energetic paediatric allergist. His wife, Grace, is a first-class honours graduate in mathematics and physics. They met at Hong Kong University and migrated to Australia in 1972. Billy was the talker, Grace the quiet one. It was a dazzling conjunction of genetic material. Nigel, their youngest, had an IQ of 180 when he was tested as a child. He won International Mathematical Olympiad medals and would go on to become a software engineer at Google. Billy and Grace's middle son, Trevor, diagnosed with autism when he was 2 1/2 years old, is an international master in chess and a talented pianist and composer with a double degree in maths and music and a PhD in mathematics. And then there was Terry. Grace and Billie were gobsmacked by his progress and decided he should start school early. He was 3 1/2 when they put him in a tiny uniform and sent him off to primary school. But their first experiment in the "radically accelerated" schooling that would eventually become a hallmark of his education was not a success. The teacher complained that Terry distracted the class. "Other children always wanted to play with him. He became like a little toy," says Billy. Nor was Terry socially ready for the situation. The Taos took him out of school and put him in a local kindergarten. "I have to look after Terry's environment a bit," Billy decided. The family embarked on an adventure in bespoke education, developing a regime of staggered classes for their oldest son. Grace, who'd taught high-school maths and physics in Hong Kong and Adelaide, did not return to paid employment after Terry was born. Instead, she became the superintendent of her special- needs household and its scheduling demands.

Terry, who mastered primary-school maths while he was still in kindergarten, started school when he was five. By the time he was six, he had taught himself BASIC computer language and had written programs on maths problems. In early 1983, the Adelaide Advertiser published one of the first articles on the child prodigy: "Tiny Terence, 7, is high school whiz" read the headline. Tao still attended some primary school classes but was taking high-school maths and science subjects. "They accelerated me, not because they wanted their kid to be the first or whatever, but just because they thought [staying in grades to match his age] wasn't the best environment for me," says Tao, who started full-time high school when he was eight. By nine, he was shuttling between high-school classes and maths lectures and tutorials at Flinders University. "I didn't have very good self-awareness," says Tao. "I'm sure I was very annoying. It's a good thing I don't have a video of myself from that era." But there is a video: an ABC report in 2006, after he was awarded the Fields Medal, showed file footage of a little figure trudging intently through the university campus, wearing a backpack and carrying a can of soft-drink. He looked to be about 12. Tao says that it was only when he was 17 and started postgraduate studies at Princeton that he started to get a sense of "what it was like to grow up more normally among peers". The Tao family was tight: Miraca Gross recalls Terry as tender and nurturing towards his siblings; he says they thought in almost exactly the same way: "For many years, with Nigel in particular, we could complete each other's sentences." The boys invented their own games: explorations of imaginary countries on huge, hand-drawn maps, or curious Scrabble derivations using the game combined with chess pieces, mahjong tiles and chequers. "Most of them were pretty rubbish," says Nigel Tao, who remembers peering over his oldest brother's shoulder as he worked on the family's Commodore computer. Terry Tao's high-school friend, Julian Cochran, says there was a "special atmosphere" in the household. "There was a kind of great joy assumed in inventing things, discovering things," says the Adelaide composer in an email. The boys immersed themselves in an early on-screen world, creating computer games, solving programming problems, cracking up with laughter.

Billy Tao, always a voluble presence, would sometimes burst in to show off something he was working on. "He certainly loves to talk," says his son. "He's always taken an interest in all kinds of things - politics or science or whatever." Both father and son acknowledge that, these days, there's infrequent contact. "It doesn't mean we don't get on with each other, it's just that we appreciate that each other is very busy and if we don't write it usually means we're doing well," says Billy, 69. "If I want to know what's happening to him, I just look up his website." Says Nigel Tao, "We're not big on small talk." Little Madeleine is snuggling up to her father on the couch in the living room of the family home in the manicured Westwood neighbourhood, a short walk from Terry Tao's UCLA office. He's talking domestic matters, showing off a notebook covered in teddy-bear wrapping paper. It's a handwritten cookbook filled with recipes for dishes such as beef noodles in black bean sauce, honey prawns and Tao's favourite, creamed corn chicken. Grace Tao gave her son and his wife, Laura, the book after they were married. "Because I'm a terrible cook," says Laura, who was a student in one of Tao's classes at UCLA in 2000 and went on to work at the solar system research and development group JPL (Jet Propulsion Laboratory) after graduating as an electrical engineer. Laura Tao doesn't remember who instigated the coffee date they went on some time after she started slipping homework under her professor's office door. She liked how relaxed and stable Tao seemed. That hasn't changed. "He's never stressed out," says Laura. "I see him tired but he never raises his voice."

A shared life would seem incompatible with a genius mathematician's singular focus. "When you're concentrating hard, hours can fly by and it's just you and a math problem," says Tao. Such focus is almost impossible these days, even though, in an echo of his mother's path, Laura has given up paid employment to manage the household, shield Tao from mundane matters and buy the polo shirts he wears. "There's a reason why we stick to polo shirts: the other kinds he doesn't button up right." UCLA also attempts to keep him from being besieged. "The institution fully appreciates Terry as one of the great figures in mathematics, in fact, one of the great figures in scholarship of our time," says Joseph Rudnick, the dean. "The math department does its best to guard Terry's time for him." But there's not much the department can do to improve his sleep. "What's difficult now is Maddie sleeps with us and so if she doesn't sleep, we don't sleep," says Tao. The extraordinary and the ordinary can co-exist: the world's smartest man is responsible for the morning school run. "I have to drop William off at eight o'clock." His son is 11. He's learning piano, guitar and clarinet. One day he was scouted by a talent agency in a mall - "We like his face", they said - and he has appeared in commercials for Ford and Disney. "He's different from me in some ways," says Tao, who likes to watch Doctor Who with his son. "He likes math, he's good at it, but he likes free-form; he likes things that I was never into, like writing. I was never very good at school with ... humanities ... anything which was more a matter of opinion." Write about "home", a year 8 English teacher once told him. Tao was stumped. "It was so vague. I didn't know what to write about." He liked anything that was black and white; he needed a right answer and a wrong answer. It was why he liked Latin. "Latin has very precise rules," says Tao, who started to translate The Hitchhiker's Guide to the Galaxy into Latin when at high school.

Things could have been different for Terry Tao. He might have used his brain with evil intent. "Okay, I don't think it would count as evil, but a lot of my PhDs, they go into the finance industry, Wall Street, and typically they earn ridiculous salaries. In fact, I don't even know exactly how much they earn. It's probably good for my health not to know." He is now absent-mindedly stroking the family's fluffy tortoiseshell cat, which has jumped onto his lap. Tao himself was once head-hunted by a hedge fund. "But I don't know, these things never sort of really interested me." He's done some consultancy work for the US intelligence bureau, the National Security Agency. "It's not as glamorous as it sounds. You spend a year going through security clearance and then you work on some problems which you don't know where they came from, they don't tell you that much," he says, and then corrects himself. "No, it's interesting work; it's kind of fun actually..." Tao's colleagues and the autograph hunters and geeks like Adam Spencer will be relieved that Tao chose to devote his beautiful mind to academia and to the inquisition of vintage conjectures such as those relating to prime numbers. "There's probably no thing in mathematics that is more fundamental to understanding numbers than prime numbers," says Spencer, who points out their application in data encryption. In 2004, working with English mathematician Ben Green, Tao famously proved the theory that there are arbitrarily long arithmetic progressions of prime numbers. His initial response was terror. "There's a history of this problem; people previously tried to solve it, or thought they solved it and it didn't work, so we didn't believe we'd solved it for a long time." The Green-Tao theorem was an earthquake in the mathematical world. Eventually though, for Tao, it all seemed so natural. "The funny thing is that the moment you actually work something out, it doesn't seem so difficult anymore; once you see it the right way, it's hard to actually go back and remember what it was like when you didn't understand it."

Madeleine has brought in a balloon. She dressed as Ariel from The Little Mermaid for the party. "She's a little princess," says Tao, like every father ever. "Do you like math?" he asks her. "How far can you count to?" "One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14." Madeleine stops. "Okay, 14's pretty good," says Tao. "She's been stuck on 15 for a while." It's hard to tell, he says, whether she has inherited his mighty maths genes. Madeleine is bouncing the balloon into the air now, skipping around the room. "You don't let them touch the ground," she explains. Follow Stephanie Wood on Facebook.

PRIME MOVERS

First, a reminder of primary-school mathematics: a prime number is a whole number greater than one that can be divided evenly only by one or itself. So think 2, 3, 5, 7, 11, 13, 17, 19 ... Broadcaster Adam Spencer gets excited about prime numbers. "They are the building blocks of all numbers," he says. "Every number either is a prime or has a unique decomposition into primes." So think about a non-prime number like 4 and its factors: 2 x 2. Or 6 (2 x 3), or 8 (2 x 2 x 2), or 9 (3 x 3), or 10 (2 x 5) or 16 (2 x 2 x 2 x 2). "But if I tell you a given number that's really massive, you can't tell me the next prime number. And if I give you any number and ask, 'Is it prime or not?', it's actually really hard to work out; there's no simple test for whether a number's prime. There's probably no thing in mathematics that is more fundamental to understanding numbers, that teases us more, than prime numbers." One area in which prime numbers are absolutely critical is internet security. The multiplication of large prime numbers is fundamental in encrypting data. "The fact that you can send me your credit card details over the phone and no one else can hack in and take that is all due to prime numbers," says Spencer. "If I came up tomorrow with a test, a simple test, for whether a number's prime, internet security would collapse. Primes are awful sexy like that. They just sort of sit there laughing at us."