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Here at Live Science, we love numbers. And on Pi Day — March 14, or 3/14 — we love to celebrate the world’s most famous irrational number, pi, whose first 10 digits are 3.141592653. 

As the ratio of a circle’s circumference to its diameter, pi is not just irrational, meaning it can’t be written as a simple fraction. It is also transcendental, meaning it’s not the root, or solution, to any polynomial equation, such as x+2X^2+3 = 0.

Pi may be one of the best-known numbers, but for people who are paid to think about numbers all day long, the circle constant can be a bit of a bore. We asked several mathematicians to tell us their favorite non-pi numbers. Here are some of their answers.Tau

People often eat pie on pi day. (Image credit: Shutterstock)

You know what’s cooler than one pie? … Two pies. In other words, two times pi, or the number “tau,” which is roughly 6.28.

“Using tau makes every formula clearer and more logical than using pi,” said John Baez (opens in new tab) , a mathematician at the University of California, Riverside. “Our focus on pi rather than 2pi is a historical accident.”

Tau is what shows up in the most important formulas, he said.

While pi relates a circle’s circumference to its diameter, tau relates a circle’s circumference to its radius — and many mathematicians argue that this relationship is much more important. Tau also makes seemingly unrelated equations nicely symmetrical, such as the one for a circle’s area and an equation describing kinetic and elastic energy.

But tau will not be forgotten on Pi Day! As per tradition, the Massachusetts Institute of Technology will send out decisions at 6:28 p.m. today. A few months from now, on June 28, it will be Tau Day.Natural log base

The natural log is expressed as the symbol “e.” (Image credit: Shutterstock)

The base of natural logarithms — written as “e” for its namesake, the 18th-century Swiss mathematician Leonhard Euler — may not be as famous as pi, but it also has its own holiday. So, while 3.14 is celebrated on March 14, natural log base — the irrational number beginning with 2.718 — is lionized on Feb. 7.

The base of natural logarithms is most often used in equations (opens in new tab) involving logarithms, exponential growth and complex numbers.

“[It] has the wonderful definition as being the one number for which the exponential function y = e^x has a slope equal to its value at every point,” Keith Devlin (opens in new tab) , emeritus professor and former director of the Stanford University Mathematics Outreach Project in the Graduate School of Education, told Live Science. In other words, if the value of a function is, say, 7.5 at a certain point, then its slope, or derivative, at that point is also 7.5. And, “like pi, it comes up all the time in mathematics, physics and engineering,” Devlin said.Imaginary number i

The imaginary number i. (Image credit: Shutterstock)

Take the “p” out of “pi,” and what do you get? That’s right, the number i. No, that’s not really how it works, but i is a pretty cool number. It’s the square root of -1, which means it’s a rule breaker, as you’re not supposed to take the square root of a negative number.

“Yet, if we break that rule, we get to invent the imaginary numbers, and so the complex numbers, which are both beautiful and useful,” Eugenia Cheng (opens in new tab) , a mathematician at the School of the Art Institute of Chicago, told Live Science in an email. (Complex numbers can be expressed as the sum of both real and imaginary parts.)

The imaginary number i is an exceptionally weird number because -1 has two square roots: i and -i, Cheng said. “But we can’t tell which one is which!” Mathematicians have to just pick one square root and call it i and the other -i.

“It’s weird and wonderful,” Cheng said.i to the power of i

The number i raised to the power of i is actually a real number roughly equal to 0.207. (Image credit: Shutterstock)

Believe it or not, there are ways to make i even weirder. For example, you can raise i to the power of i — in other words, take the square root of -1 raised to the square root of -1 power.

“At a glance, this looks like the most imaginary number possible — an imaginary number raised to an imaginary power,” David Richeson, a professor of mathematics at Dickinson College in Pennsylvania and author of the book “Tales of Impossibility: The 2,000-Year Quest to Solve the Mathematical Problems of Antiquity (opens in new tab) ” (Princeton University Press, 2019), told Live Science. “But, in fact, as Leonhard Euler wrote in a 1746 letter, it is a real number!”

Finding the value of i to the i power involves rearranging Euler’s identity, a formula relating the irrational number e, the imaginary number i, and the sine and cosine of a given angle. When you solve the formula for a 90-degree angle (which can be expressed as pi over 2), you can simplify the equation to show that i to the power of i equals e raised to the power of negative pi over 2.

It sounds confusing (here’s the full calculation (opens in new tab) , if you dare to read it), but the result equals roughly 0.207 — a very real number. At least, in the case of a 90-degree angle.

“As Euler pointed out, i to the i power does not have a single value,” Richeson said, but rather takes on “infinitely many” values depending on the angle you’re solving for. (Because of this, it’s unlikely we’ll ever celebrate an “i to the power of i day.”)Belphegor’s prime number

Belphégor’s prime number, named after a demon, is a palindromic prime with a 666 hiding in the middle. (Image credit: Louis Le Breton/Dictionnaire Infernal)

Belphegor’s prime number is a palindromic prime number with a 666 hiding between 13 zeros and a 1 on each side. The ominous number can be abbreviated as 1 0(13) 666 0(13) 1, where the (13) denotes the number of zeros between the 1 and 666.

Although he didn’t “discover” the number, scientist and author Cliff Pickover (opens in new tab) made the sinister-looking number famous when he named it after Belphegor (or Beelphegor), one of the seven demon princes of hell in the Bible.

The number apparently even has its own devilish symbol, which looks like an upside-down symbol for pi. According to Pickover’s website, the symbol is derived from a glyph in the mysterious Voynich manuscript, an early-15th-century compilation of illustrations and text that no one seems to understand.2^{aleph_0}

There are many types of infinities, and 2^{aleph_0} is a number that describes the size of a particular infinite set.

Harvard mathematician W. Hugh Woodin (opens in new tab) has devoted many years of research to infinite numbers. It’s no surprise, then, that his favorite number is an infinite one: 2^{aleph_0}, or 2 raised to the power of aleph-naught, also called aleph-null. Aleph numbers are used to describe the sizes of infinite sets, where a set is any collection of distinct objects in mathematics. (So, for example, the numbers 2, 4 and 6 can form a set of size 3.)

As for why Woodin chose the number, he said, “Realizing that 2^{aleph_0} is not aleph_0 (i.e., Cantor’s theorem (opens in new tab) ) is the realization that there are different sizes of infinite. So that makes the conception of 2^{aleph_0} rather special.”

In other words, there’s always something bigger: Infinite cardinal numbers are infinite, so there is no such thing as the “largest cardinal number.”Apéry’s constant

Apéry’s constant is an irrational number that begins with 1.2020569 and continues infinitely and shows up in physics equations describing magnetism and the electron. (Image credit: Ian Cuming/Getty Images)

Harvard mathematician Oliver Knill (opens in new tab) told Live Science his favorite number is the Apéry’s constant (zeta(3)), “because there is still some mystery associated with it.” In 1979, French mathematician Roger Apéry proved that a value that would come to be known as Apéry’s constant is an irrational number. (It begins with 1.2020569 and continues infinitely.) The constant is also written as zeta(3), where zeta(3) is the Riemann zeta function when you plug in the number 3.

One of the biggest outstanding problems in math, the Riemann hypothesis, makes a prediction about when the Riemann zeta function equals zero and, if proven, would allow mathematicians to better predict how the prime numbers are distributed.

Riemann’s Zeta function. (Image credit: Furfur)

Of the Riemann hypothesis, renowned 20th-century mathematician David Hilbert once said (opens in new tab) , “If I were to awaken after having slept for a thousand years, my first question would be, ‘Has the Riemann hypothesis been proven?'”

So what’s so cool about this constant? It turns out that Apéry’s constant shows up in fascinating places in physics, including in equations governing the electron’s magnetism and orientation to its angular momentum.The number 1

The number 1 has many interesting properties, such as being the only number that is neither prime nor able to be factored into two numbers. (Image credit: Shutterstock)

Ed Letzter (opens in new tab) , a mathematician at Temple University in Philadelphia (and father of former Live Science staff writer Rafi Letzter), had a practical answer:

“I suppose this is a boring answer, but I’d have to choose 1 as my favorite, both as a number and in its different roles in so many different more abstract contexts,” he told Live Science.

One is the only number by which all other numbers divide into integers. It’s the only number divisible by exactly one positive integer (itself, 1). It’s the only positive integer that’s neither prime nor composite.

In both math and engineering, values are often represented as between 0 and 1: “100%” is just a fancy way of saying 1. It’s whole and complete.

And, of course, throughout the sciences, 1 is used to represent basic units. A single proton is said to have a charge of +1. In binary logic, 1 means yes. It’s the atomic number of the lightest element, and it’s the dimension of a straight line.Euler’s identity

Leonhard Euler was a Swiss mathematician, and his identity ties together three fascinating numbers: pi, natural log and i. (Image credit: Jakob Emanuel Handmann/Wikimedia Commons)

Euler’s identity, which is actually an equation, is a real mathematical jewel, at least as described by the late physicist Richard Feynman. It has also been compared to a Shakespearean sonnet.

In a nutshell, Euler’s identity ties together a number of mathematical constants: pi, natural log e and the imaginary unit i.

“[It] connects these three constants with the additive identity 0 and the multiplicative identity of elementary arithmetic: e^{i*Pi} + 1 = 0,” Devlin said.The number 0

Zero may have many useful properties, but it was a surprisingly late concept to emerge. (Image credit: Fernando Trabanco Fotografía/Getty)

If we’re already talking about how awesome 1 is, then why not throw in the even weirder and cooler number 0? For most of written human history, the concept of zero wasn’t all that important. Clay tablets from ancient Babylonian times didn’t always distinguish between numbers like 216 and 2106, according to the University of St. Andrews in Scotland (opens in new tab) . 

The ancient Greeks began to develop the idea of using zero as an empty place indicator to distinguish numbers of different magnitudes, but it wasn’t until roughly the seventh century that Indian mathematicians, like Brahmagupta, began describing the modern idea of zero. Brahmagupta wrote that any number multiplied by zero is zero, but he struggled with division, saying that a number, n, divided by zero just comes out as n/0, rather than the modern answer, which is that the result is undefined. (The Maya had also independently derived the concept of zero by A.D. 665.) 

Zero is extremely useful, but it is a very tricky concept for many people to wrap their heads around. We have examples such as 1 horse or 3 chickens in our day-to-day lives, but using a number to represent nothing is a larger conceptual leap. “Zero is in the mind but not in the sensory world,” the late Robert Kaplan, a Harvard math professor, told Vox (Kaplan died in 2021). Still, without 0 (and 1), we wouldn’t be able to represent all of the digital binary code that makes our contemporary world run. (Data on computers is represented by strings of 0s and 1s.)The square root of 2

This painting by Raphael depicts philosophers from the school of Athens, including Plato, Aristotle and Pythagoras, who was credited with discovering the square root of 2. (Image credit: mikroman6/Getty)

Perhaps the most dangerous number ever conceived, the square root of 2 supposedly led to the first mathematical murder in history. The Greek mathematician Hippasus of Metapontum is credited with discovering it in the fifth century B.C., according to the University of Cambridge (opens in new tab) . While working on a separate problem, Hippasus is said to have stumbled on the fact that an isosceles right triangle whose two base sides are 1 unit in length will have a hypotenuse that is √2, which is an irrational number. 

According to legend, Hippasus’ contemporaries, members of the quasi-religious order known as the Pythagoreans, threw him into the sea after hearing about his great discovery. That’s because the Pythagoreans believed that “all is number” and the universe only contained whole numbers and their ratios. Irrational numbers like √2 (and pi), which can’t be expressed as a ratio of whole numbers and go on forever after the decimal place, were seen as an abomination. 

These days, we’re a little calmer about √2, often calling it Pythagoras’ constant. It starts off as 1.4142135623 … (and, of course, goes on forever). ) Pythagoras’ constant has all sorts of uses. Besides proving the existence of irrational numbers, it is used by the International Organization for Standardization (ISO) to define the A paper size. The 216 definition (opens in new tab) of the A paper states that the sheet’s length divided by its width should be 1.4142. This means that a piece of A1 paper divided in half by width will yield two A2 pieces of paper. Divide an A2 in half again, and it will produce two A3 pieces of paper, and so on.Slice of pi

NASA uses a truncated estimate of pi that only includes 15 places after the decimal point. That’s all the agency needs to get everywhere they want to go in space. (Image credit: NASA/JPL)

Sometimes, the number cooler than pi is…a truncated version of pi. At least, that’s the case for NASA and scientists at the Jet Propulsion Laboratory (JPL) in California. For interplanetary navigation, JPL uses the number 3.141592653589793, according to JPL’s Chief Engineer for Mission Operations and Science, Marc Rayman (opens in new tab) . At that level of accuracy, Rayman said, NASA can get everything it builds wherever it needs to go.

To see why, it’s helpful to do some number-crunching. The most distant spacecraft from Earth is the Voyager 1 spacecraft, which is more than 14.6 billion miles (23.5 billion kilometers) distant from us. At this distance, you can calculate a circumference of a circle that is roughly 94 billion miles (more than 150 billion km) around, yet adding extra decimal places to pi only shaves off a half-inch (1.2 centimeters) of error from the calculation, Rayman said.

Even if scientists wanted to calculate the radius of a circle the size of the known universe to the accuracy of a width of a hydrogen atom, it would only need 37 digits after the decimal point to reach that accuracy, Rayman said.

So what’s cooler than pi? A slice of pi.

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Starmer survives rebellion as watered-down welfare cuts pass key vote

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Starmer survives rebellion as watered-down welfare cuts pass key vote

Sir Keir Starmer’s controversial welfare bill has passed its first hurdle in the Commons despite a sizeable rebellion from his MPs.

The prime minister’s watered-down Universal Credit and Personal Independent Payment Bill, aimed at saving £5.5bn, was backed by a majority of 75 on Tuesday evening.

A total of 49 Labour MPs voted against the bill – the largest rebellion since 47 MPs voted against Tony Blair’s Lone Parent benefit in 1997, according to Professor Phil Cowley from Queen Mary University.

Politics latest: Chancellor left in ‘impossible situation’ after PM survives welfare rebellion

After multiple concessions made due to threats of a Labour rebellion, many MPs questioned what they were voting for as the bill had been severely stripped down.

They ended up voting for only one part of the plan: a cut to Universal Credit (UC) sickness benefits for new claimants from £97 a week to £50 from 2026/7.

The Institute for Fiscal Studies (IFS) said the bill voted through “is not expected to deliver any savings over the next four years” because the savings from reducing the Universal Credit health element for new claimants will be roughly offset by the cost of increasing the UC standard allowance.

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Just 90 minutes before voting started on Tuesday evening, disabilities minister Stephen Timms announced the last of a series of concessions made as dozens of Labour MPs spoke of their fears for disabled and sick people if the bill was made law.

How did your MP vote on Labour’s welfare bill?

In a major U-turn, he said changes in eligibility for the personal independence payment (PIP), the main disability payment to help pay for extra costs incurred, would not take place until a review he is carrying out into the benefit is published in autumn 2026.

An amendment brought by Labour MP Rachael Maskell, which aimed to prevent the bill progressing to the next stage, was defeated but 44 Labour MPs voted for it.

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Welfare bill blows ‘black hole’ in chancellor’s accounts

A Number 10 source told Sky News’ political editor Beth Rigby: “Change isn’t easy, we’ve always known that, we’re determined to deliver on the mandate the country gave us, to make Britain work for hardworking people.

“We accept the will of the house, and want to take colleagues with us, our destination – a social security system that supports the most vulnerable, and enables people to thrive – remains.”

But the Conservative shadow chancellor Mel Stride called the vote “farcical” and said the government “ended up in this terrible situation” because they “rushed it”.

He warned the markets “will have noticed that when it comes to taking tougher decisions about controlling and spending, this government has been found wanting”.

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‘Absolutely lessons to learn’ after welfare vote

Work and Pensions Secretary Liz Kendall said: “I wish we’d got to this point in a different way. And there are absolutely lessons to learn.

“But I think it’s really important we pass this bill at the second reading, it put some really important reforms to the welfare system – tackling work disincentives, making sure that people with severe conditions would no longer be assessed and alongside our investment in employment support this will help people get back to work, because that’s the brighter future for them.”

She made further concessions on Monday in the hope the rebels’ fears would be allayed, but many were concerned the PIP eligibility was going to be changed at the same time the review was published, meaning its findings would not be taken into account.

Her changes were:

• Current PIP claimants, and any up to November 2026, would have the same eligibility criteria as they do now, instead of the stricter measure proposed

• A consultation into PIP to be “co-produced” with disabled people and published in autumn 2026

• For existing and future Universal Credit (UC) claimants, the combined value of the standard UC allowance and the health top-up will rise “at least in line with inflation” every year for the rest of this parliament

• The UC health top-up, for people with limited ability to work due to a disability or long-term sickness, will get a £300m boost next year – doubling the current amount – then rising to £800m the year after and £1bn in 2028/29.

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Women’s Euros: Extreme heat warnings in place as tournament kicks off

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Women's Euros: Extreme heat warnings in place as tournament kicks off

The Women’s Euros begin in Switzerland today – with extreme heat warnings in place.

Security measures have had to be relaxed by UEFA for the opening matches so fans can bring in water bottles.

Temperatures could be about 30C (86F) when the Swiss hosts open their campaign against Norway in Basel this evening.

Players have already seen the impact of heatwaves this summer at the men’s Club World Cup in the US.

Players take a drink during a training session of Spain soccer team at the Euro 2025, in Lausanne, Switzerland Tuesday, July 1, 2025 Pic: AP
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The Spain squad pauses for refreshments during a training session. Pic: AP

It is raising new concerns in the global players’ union about whether the stars of the sport are being protected in hot and humid conditions.

FIFPRO has asked FIFA to allow cooling breaks every 15 minutes rather than just in the 30th minute of each half.

There’s also a request for half-time to be extended from 15 to 20 minutes to help lower the core temperature of players.

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FIFPRO’s medical director, Dr Vincent Gouttebarge, said: “There are some very challenging weather conditions that we anticipated a couple of weeks ago already, that was already communicated to FIFA.

“And I think the past few weeks were confirmation of all worries that the heat conditions will play a negative role for the performance and the health of the players.”

Football has seemed focused on players and fans baking in the Middle East – but scorching summers in Europe and the US are becoming increasingly problematic for sport.

Chloe Kelly celebrates with Beth Mead, right, after scoring her side's sixth goal at Wembley Stadium, in London, Friday, May 30, 2025. AP
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England are the tournament’s defending champions. Pic: AP

While climate change is a factor, the issue is not new and at the 1994 World Cup, players were steaming as temperatures rose in the US.

There is now more awareness of the need for mitigation measures among players and their international union.

FIFPRO feels football officials weren’t responsive when it asked for kick-off times to be moved from the fierce afternoon heat in the US for the first 32-team Club World Cup.

FIFA has to balance the needs of fans and broadcasters with welfare, with no desire to load all the matches in the same evening time slots.

Electric storms have also seen six games stopped, including a two-hour pause during a Chelsea game at the weekend.

This is the dress rehearsal for the World Cup next summer, which is mostly in the US.

Players are also feeling the heat at the Club World Cup in the US. Pic: AP
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Players are also feeling the heat at the Club World Cup. Pic: AP

The use of more indoor, air conditioned stadiums should help.

There is no prospect of moving the World Cup to winter, as Qatar had to do in 2022.

And looking further ahead to this time in 2030, there will be World Cup matches in Spain, Portugal and Morocco. The temperatures this week have been hitting 40C (104F) in some host cities.

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Wildfires erupt in Italy and France amid heatwave

FIFA said in a statement to Sky News: “Heat conditions are a serious topic that affect football globally.

“At the FCWC some significant and progressive measures are being taken to protect the players from the heat. For instance, cooling breaks were implemented in 31 out of 54 matches so far.

“Discussions on how to deal with heat conditions need to take place collectively and FIFA stands ready to facilitate this dialogue, including through the Task Force on Player Welfare, and to receive constructive input from all stakeholders on how to further enhance heat management.

“In all of this, the protection of players must be at the centre.”

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SEC approves conversion of Grayscale’s large-cap crypto fund into ETF

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<div>SEC approves conversion of Grayscale's large-cap crypto fund into ETF</div>

<div>SEC approves conversion of Grayscale's large-cap crypto fund into ETF</div>

Grayscale has slowly narrowed the arbitrage opportunities on its crypto trusts by converting them into exchange-traded funds.

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