Pi Day: 12 numbers that are cooler than pi

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2 years ago

March 17, 2023

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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.

Entertainment

Pete Townshend’s Quadrophenia talked about modern masculinity before Gen Z was born

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1 hour ago

April 5, 2025

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Pete Townshend's Quadrophenia talked about modern masculinity before Gen Z was born

Despite The Who’s Quadrophenia being set over 60 years ago, Pete Townshend’s themes of identity, mental health, and modern masculinity are just as relevant today.

The album is having a renaissance as Pete Townshend’s Quadrophenia A Mod ballet is being brought to life via dance at Sadler’s Wells East, and Sky News has an exclusive first look.

As Townshend puts it, the album he wrote is “perfect” for the stage.

“My wife Rachel did the orchestration for me, and as soon as I heard it I said to her it would make a fabulous ballet and we never really let that go,” he tells Sky News.

“Heavy percussion, concussive sequences. They’re explosive moments. They’re also romantic movement moments.”

If you identify with the demographics of Millennial, Gen Y or Gen Z, you might not be familiar with The Who and Mod culture.

But in post-war Britain the Mods were a cultural phenomenon characterised by fashion, music, and of course, scooters. The young rebels were seen as a counter-culture to the establishment and The Who, with Roger Daltry’s lead vocals and Pete Townshend’s writing, were the soundtrack.

Quadrophenia the album is widely regarded as an essay on the British adolescent experience at the time, focusing on the life of fictional protagonist Jimmy – a young Mod struggling with his sanity, self-doubt, and alienation. 

Townshend sets the rock opera in 1965 but thinks its themes of identity, mental health, and modern masculinity are just as relevant today.

He says: “The phobias and the restrictions and the unwritten laws about how young men should behave. The ground that they broke, that we broke because I was a part of it.

“Men were letting go of [the] wartime-related, uniform-related stance that if I wear this kind of outfit it makes me look like a man.”

This struggle of modern masculinity and identity appears to be echoing today as manosphere influencers like Andrew Tate, incel culture, and Netflix’s Adolescence make headlines.

For dancer Paris Fitzpatrick, who takes on the lead role of Jimmy, the story resonates.

“I think there’s a connection massively and I think there may even be a little more revival in some way,” he tells Sky News.

“I love that myself. I love non-conforming to gender norms and typical masculinity; I think it’s great to challenge things.”

Despite the album being written before he was born, the dancer says he was familiar with the genre already.

“I actually did an art GCSE project about Mods and rockers and Quadrophenia,” he says.

“I think we’ll be able to bring it to new audiences and hopefully, maybe people will be inspired to to learn more about their music and the whole cultural movement of the early 60s.”

In 1979, the album was adapted into a film directed by Franc Roddam starring Ray Winstone and Sting but Townshend admits because the film missed key points he is “not a big fan”.

“What it turned out to be in the movie was a story about culture, about social scenario and less about really the specifics of mental illness and how that affects young people,” he adds, also complimenting Roddam’s writing for the film.

Perhaps a testament to Pete Townshend’s creativity, Quadrophenia started as an album, was successfully adapted to film and now it will hit the stage as a contemporary ballet.

It appears that over six decades later Mod culture is still cool and their issues still relatable.

Quadrophenia, a Mod Ballet will tour to Plymouth Theatre Royal from 28 May to 1 June 2025, Edinburgh Festival Theatre from 10 to 14 June 2025 and the Mayflower, Southampton from 18 to 21 June 2025 before having its official opening at Sadler’s Wells Theatre, London on 24 June running to 13 July 2025 and then visiting The Lowry, Salford from 15 to 19 July 2025.

Business

‘We will see closures’: The industries hit the hardest by national insurance hike

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1 hour ago

April 5, 2025

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'We will see closures': The industries hit the hardest by national insurance hike

The cost of having staff is going up this Sunday as the increase in employers’ national insurance kicks in.

Chancellor Rachel Reeves announced in the October budget employers will have to pay a 15% rate of national insurance contributions (NIC) on their employees from 6 April – up from 13.8%.

She also lowered the threshold at which employers pay NIC from £9,100 a year to £5,000 a year, meaning they start paying at an earlier point on staff salaries.

This is on top of the national minimum wage rising, the business relief rate for hospitality, retail and leisure reducing from 75% to 40% and the rising cost of ingredients and services.

Sky News spoke to people working in some of the industries that will be hardest hit by the rise in NIC: Nurseries, hospitality, retail, small businesses and care.

NURSERIES

Nearly all (96% of 728) nurseries surveyed by the National Day Nurseries Association (NDNA) said they will have no choice but to put up fees because of the NIC rise, leaving parents to pick up the shortfall.

The NDNA has warned nurseries could close due to the rise, with 14% saying their business is at risk, 69% reducing spending on resources and 39% considering offering fewer places with government-funded hours as 92% said they do not cover their costs.

Sarah has two children, with her youngest starting later this month, but they were just informed fees will now be £92 a day – compared with £59 at the same nursery when her eldest started five years ago.

“I’m not sure how we will afford this. Our salaries haven’t increased by 50% during this time,” she said.

“We’re stuck as there aren’t enough nursery spaces in our area, so we will have to struggle.”

Karen Richards, director of the Wolds Childcare group in Nottinghamshire, has started a petition to get the government to exempt private nurseries – the majority of providers – from the NIC changes as she said it is unfair nurseries in schools do not have to pay the NIC.

She told Sky News she will have to find about £183,000 next year to cover the increase across her five nurseries and reducing staff numbers is “not off the table” but it is more likely they will reduce the number of children they have.

Joeli Brearley, founder of the Pregnant Then Screwed campaign group, told Sky News: “Parents are already drowning in childcare costs, and now, thanks to the national insurance hike, nurseries are passing even more fees on to families who simply can’t afford it.

“It’s the same story every time – parents pay the price while the government looks the other way. How exactly are we meant to ‘boost the economy’ when we can’t even afford to go to work?”

Purnima Tanuku, executive chair of the NDNA, said staffing costs make up about 75% of nurseries’ costs and they will have to find £2,600 more per employee to pay for the NIC rise – £47,000 for an average nursery.

“The government says it wants to offer ‘cheaper childcare’ for parents on the one hand but then with the other expects nurseries to absorb the costs of National Insurance Contributions themselves,” she told Sky News.

“High-quality early education and care gives children the best start in life and enables parents to work. The government must invest in this vital infrastructure to make sure nurseries can continue to deliver this social and economic good.”

HOSPITALITY

The hospitality industry has warned of closures, price rises, lack of growth and shorter opening hours.

Dan Brod, co-owner of The Beckford Group, a small southwest England restaurant and country pub/hotel group, said the economic situation now is “much worse” than during COVID.

The group has put plans for two more projects on hold and Mr Brod said the only option is to put up prices, but with the rising supplier costs, wages, business rates and NIC hike they will “stay still” financially.

He told Sky News: “What we’re nervous about is we’re still in the cost of living crisis and even though our places are in very wealthy areas of the country, Wiltshire, Somerset and Bath, people are feeling the situation in their pockets, people are going out less.”

Mr Brod said they are not getting rid of any staff as their business strongly depends on the quality of their hospitality so they are having to make savings elsewhere.

“I’m still optimistic, I still feel that humans need hospitality but we’re not valued as an industry and the social benefit is never taken into account by government.”

Aktar Islam, owner/chef at two Michelin-starred Opheem in Birmingham, said the NIC rise will cost him up to £120,000 more in staff costs a year and to maintain the financial position he is in now they would have to make “another million pounds”.

He got emails from eight suppliers on Thursday saying they were raising their costs, and said he will have to raise prices but is concerned about the impact on diners.

The restaurateur hires four commis chefs to train each year but will not be able to this year, or the next few.

“It’s very short-sighted of the government, you’re not going to grow the economy by taxing hospitality out of existence, these sort of businesses are the lifeblood of our economy,” he said.

“They think if a hospitality business closes another will open but people know it’s tough, why would they want to do that? It’s not going to happen.”

The chef sent hundreds of his “at home” kits to fellow chefs this week for their staff as an acknowledgement of how much of a “s*** show” the situation is – “a little hug from us”.

RETAIL

Some of the UK’s biggest retailers, including Tesco, Boots, Marks & Spencer and Next, wrote to Rachel Reeves after the budget to say the NIC hike would lead to higher consumer prices, smaller pay rises, job cuts and store closures.

The British Retail Consortium (BRC), representing more than 200 major retailers and brands, said the costs are so significant neither small or large retailers will be able to absorb them.

Andrew Bailey, the governor of the Bank of England, told the Treasury committee in November that job losses due to the NIC changes were likely to be higher than the 50,000 forecast by the Office for Budget Responsibility (OBR).

Nick Stowe, chief executive of Monsoon and Accessorize, said retailers had the choice of protecting staff numbers or cancelling investment plans.

He said they were trying to protect staff numbers and would be increasing prices but they would likely have to halt plans to increase store numbers.

Helen Dickinson, head of the BRC, told Sky News the national living wage rise and NIC increase will cost businesses £5bn, adding more than 10% to the cost of hiring someone in an entry-level role.

A further tax on packaging coming in October means retailers will face £7bn in extra costs this year, she said.

“This huge cost burden will undoubtedly reduce investment in stores and jobs and is likely to lead to higher prices,” she added.

SMALL BUSINESSES

A massive 85% of 1,400 small business owners surveyed by the Federation of Small Businesses (FSB) in March reported rising costs compared with the same time last year, with 47% citing tax as the main barrier to growth – the highest level in more than a decade.

Just 8% of those businesses saw an increase in staff numbers over the last quarter, while 21% had to reduce their workforce.

Kate Rumsey, whose family has run Rumsey’s Chocolates in Wendover, Buckinghamshire and Thame, Oxfordshire, for 21 years, said the NIC rise, minimum wage increase and business relief rate reduction will push her staff costs up by 15 to 17% – £70,000 to £80,000 annually.

To offset those costs, she has had to reduce opening hours, including closing on Sundays and bank holidays in one shop for the first time ever, make one person redundant, not replace short-term staff and introduce a hiring freeze.

The soaring price of cocoa has added to her woes and she has had to increase prices by about 10% and will raise them further.

She told Sky News: “We’re very much taking more of a short-term view at the moment, it’s so seasonal in this business so I said to the team we’ll just get through Q1 then re-evaluate.

“I feel this is a bit about the survival of the fittest and many businesses won’t survive.”

Tina McKenzie, policy chair of the FSB, said the NIC rise “holds back growth” and has seen small business confidence drop to its lowest point since the first year of the pandemic.

With the “highest tax burden for 70 years”, she called on the chancellor to introduce a “raft of pro-small business measures” in the autumn budget so it can deliver on its pledge for growth.

She reminded employers they can claim the Employment Allowance, which has doubled after an FSB campaign to take the first £10,500 off an employer’s annual bill.

CARE

The care sector has been warning the government since the October that budget care homes will be forced to close due to the financial pressures the employers’ national insurance rise will place on them.

Care homes receive funding from councils as well as from private fees, but as local authorities feel the squeeze more and more their contributions are not keeping up with rising costs.

The industry has argued without it the NHS would be crippled.

Raj Sehgal, founding director of ArmsCare, a family-run group of six care homes in Norfolk, said the NIC increase means a £360,000 annual impact on the group’s £3.6m payroll.

In an attempt to offset those costs, the group is scrapping staff bonuses and freezing management salaries.

It is also considering reducing day hours, where there are more staff on, so the fewer numbers of night staff work longer hours and with no paid break.

Mr Sehgal said: “But what that does do unfortunately, is impact the quality you’re going to be able to provide, at a time when we need to be improving quality, but something has to give.

“The government just doesn’t seem to understand that the funding needs to be there. You cannot keep enforcing higher costs on businesses and not be able to fund those without actually finding the money from somewhere.”

He said the issue is exacerbated by the fact local authority funding, despite increasing to 5%, will not cover the 10% rise.

“It’s going to be a really, really tough ride. And we are going to see a number of providers close their doors,” he warned.

Nadra Ahmed, executive co-chair of the National Care Association, said those who receive, or are waiting to access, care as well as staff will feel the impact the hardest.

“As providers see further shortfalls in the commissioning of care services, they will start to limit what they can do to ensure their viability or, as a last resort exit the market,” she said.

“This is very short-sighted, with serious consequences, which alludes to the understanding of this government.”

Government decided to ‘wipe the slate clean’

A Treasury spokesperson told Sky News the government is “pro-business” but has “taken the difficult but necessary decisions to wipe the slate clean and properly fund our public services after years of declines”.

“Our budget choices have already delivered an NHS with falling waiting lists, a £3.7bn rescue package for social care, and vital protection for Britain’s small businesses,” they said.

“We’re making tough choices today to secure a better tomorrow through our Plan for Change. By investing in economic growth and early years education while capping corporation tax, we’re putting more money in working people’s pockets and giving every child the best start in life.”

Politics