what is mathematics hypothesis

A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false.

In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis .

In symbolic logic , a hypothesis is the first part of an implication (with the second part being known as the predicate ).

In general mathematical usage, "hypothesis" is roughly synonymous with " conjecture ."

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July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

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Riemann hypothesis

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Riemann hypothesis , in number theory , hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function , which is connected to the prime number theorem and has important implications for the distribution of prime numbers . Riemann included the hypothesis in a paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (“On the Number of Prime Numbers Less Than a Given Quantity”), published in the November 1859 edition of Monatsberichte der Berliner Akademie (“Monthly Review of the Berlin Academy”).

Riemann extended the study of the zeta function to include the complex numbers x + i y , where i = Square root of √ −1 , except for the line x = 1 in the complex plane. Riemann knew that the zeta function equals zero for all negative even integers −2, −4, −6,… (so-called trivial zeros) and that it has an infinite number of zeros in the critical strip of complex numbers that fall strictly between the lines x = 0 and x = 1. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1 / 2 . Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.

In 1914 English mathematician Godfrey Harold Hardy proved that an infinite number of solutions of ζ( s ) = 0 exist on the critical line x = 1 / 2 . Subsequently it was shown by various mathematicians that a large proportion of the solutions must lie on the critical line, though the frequent “proofs” that all the nontrivial solutions are on it have been flawed. Computers have also been used to test solutions, with the first 10 trillion nontrivial solutions shown to lie on the critical line.

A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography .

The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics . It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated Hilbert’s idea with a list of important problems for the 21st century; the Riemann hypothesis was number one. In 2000 it was designated a Millennium Problem , one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., U.S., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency ( DARPA ) listed it as one of the DARPA Mathematical Challenges, 23 mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Nineteen: Settle the Riemann Hypothesis. The Holy Grail of number theory.”

A statement that could be true, which might then be tested.

Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls.

Sometimes the hypothesis won't be tested, it is simply a good explanation (which could be wrong). Conjecture is a better word for this.

Example: you notice the temperature drops just as the sun rises. Your hypothesis is that the sun warms the air high above you, which rises up and then cooler air comes from the sides.

Note: when someone says "I have a theory" they should say "I have a hypothesis", because in mathematics a theory is actually well proven.

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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a solution have?

Many researchers have spent a lot of time on it, trying to prove it, but why is it important to solve the problem?

I've tried relating the situation to problems in my field. For instance, solving the $P \ vs. NP$ problem has important implications if $P = NP$ is shown, and important implications if $P \neq NP$ is shown. For instance, there would be implications regarding the robustness or security of cryptographic protocols and algorithms. However, it's hard to say WHY the Riemann Hypothesis is important.

Given that the Poincaré Conjecture has been resolved, perhaps a hint about what to expect if and when the Riemann Hypothesis is resolved could be obtained by seeing what a proof of the Poincaré Conjecture has led to.

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• 7 $\begingroup$ Define "practical use". And it is important because it is there and it is fun to know stuff, though this would probably not fit within the "practical use" tag... $\endgroup$ –  DonAntonio Commented May 28, 2013 at 9:51
• $\begingroup$ It is very likely to be a duplicate isn't it? $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 9:52
• $\begingroup$ @DominicMichaelis; I don't follow, a duplicate of what? $\endgroup$ –  Mythio Commented May 28, 2013 at 10:00
• 4 $\begingroup$ Well it is a very famous hypothesis so you won't be the first one to ask it, partial answers are for example here $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 10:14
• 3 $\begingroup$ immortality ... in the sense of Pythagoras ($a^2 + b^2 = c^2$) and Einstein ($E=mc^2$) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:43

5 Answers 5

Proving the Riemann Hypothesis will get you tenure, pretty much anywhere you want it.

• 16 $\begingroup$ Worth the downvotes :) $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:51
• 17 $\begingroup$ Not to mention the 1M on your bank account... $\endgroup$ –  Matemáticos Chibchas Commented Dec 29, 2013 at 3:44
• 7 $\begingroup$ I've already proven it, I just need to check my calculations... ;) $\endgroup$ –  King Squirrel Commented May 4, 2014 at 17:26
• 38 $\begingroup$ @KingSquirrel If only that damn margin were to be a bit wider... ;-) $\endgroup$ –  triple_sec Commented Aug 26, 2014 at 7:46
• 1 $\begingroup$ @KingSquirrel I have proved it too, just waiting for the prize to be increased to $10M.;) $\endgroup$ –  ishandutta2007 Commented Aug 12, 2022 at 8:55

The Millennium problems are not necessarily problems whose solution will lead to curing cancer. These are problems in mathematics and were chosen for their importance in mathematics rather for their potential in applications.

There are plenty of important open problems in mathematics, and the Clay Institute had to narrow it down to seven. Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis. There are many (many many) theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true.

In contrast, a solution to some of the other Millennium problems is (highly likely) not going to lead to anything dramatic. For instance, the $P$ vs. $NP$ problem. I personally doubt it is probable that $P=NP$ . The reason it's an important question is not because we don't (philosophically) already know the answer, but rather that we don't have a bloody clue how to prove it. It means that there are fundamental issues in computability (which is a hell of an important subject these days) that we just don't understand. Solving $P \ne NP$ will be important not for the result but for the techniques that will be used. (Of course, in the unlikely event that $P=NP$ , enormous consequences will follow. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.)

The Poincaré conjecture is an extremely basic problem about three-dimensional space. I think three-dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three-dimensional space, but I'm sure there are.

• 16 $\begingroup$ about Perelman's solution and the Poincaré conjecture : in theoretical physics, more precisely in general relativity, there are equations relating the distribution of mass in the universe to the curvature of space-time. Perelman's solution used curvature (via the Ricci flow) to get topological information, i.e. the shape of the compact simply connected manifold (homeomorphic to a sphere). these kind of techniques are related to determnining the shape of the universe (space-time) which is a 4-manifold, depending on the curvature (which is determined by mass repartition, which we can observe) $\endgroup$ –  Albert Commented May 28, 2013 at 13:22
• 2 $\begingroup$ @Glougloubarbaki - it would be great if you could expound your comment as an answer. $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:45
• 2 $\begingroup$ Is it really obvious that "enormous consequences will follow" from a proof of P = NP? $\endgroup$ –  bof Commented Nov 14, 2014 at 4:53
• $\begingroup$ @bof highly likely, yes. At the very least lots and lots of expert computer scientists will look at the proof in disbelief as it shatters something they really felt very confident about. And, unless the proof will be completely unusable computationally, some really hard to solve problems will be found polynomial solutions. $\endgroup$ –  Ittay Weiss Commented Nov 14, 2014 at 4:57
• 1 $\begingroup$ P=NP. I'd love an opportunity to demonstrate my NTM. Nonetheless, the argument that no-one believes P=NP is just patently false. P=NP via BPP. $\endgroup$ –  Jonathan Charlton Commented Jan 18, 2016 at 6:44

Explaining the true mathematics behind the Riemann Hypothesis requires more text that I'm allotted (took most of my undergraduate degree in mathematics to even touch the surface; required all of graduate school to fully appreciate the beauty).

In very simple terms, the Riemann Hypothesis is mostly about the distribution of prime numbers. The idea is that mathematicians have some very good approximations (emphasis on approximate) for the density of the primes (so you give me an integer, and I can use these approximate functions to tell you roughly how many primes are between 0 [really 2] and that integer). The reason we use these approximations is that no [known] function exists that efficiently and perfectly computes the number of primes less than a given integer (we're talking numbers with literally millions of zeros). Since we can't determine the exact values (again, I'm simplifying a lot of this) the problem mathematicians want to know is exactly HOW good are these approximations.

This is where the Riemann Hypothesis comes in to play. For well over a century, mathematicians have known that a special form of the polylogarithm function (again, more fun math if you're bored) is a really great approximation for the prime counting function (and it's way easier to compute). The Riemann Hypothesis, if true, would guarantee a far greater bound on the difference between this approximation and the real value. In other words, the importance of the Riemann Hypothesis is that it tells us a lot about how chaotic the primes numbers really are. That's an incredibly high-level explanation and the Riemann Hypothesis deals with literally hundreds of other concepts, but the main point is understanding the distribution of the primes.

The Riemann hypothesis is a conjecture about the Riemann zeta function $$\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s}$$ This is a function $\mathbb{C} \rightarrow \mathbb{C}$. With the definition I have provided the zeta function is only defined for $\Re(s)\gt1$. With some complex analysis you can proof that there is a continuous (actually holomorphic if you know what it means) extension of the function so that it is defined in whole $\mathbb{C}$. The Riemann zeta function has some trivial zero points like $-2,-4,-6.$ The hypothesis says that the other zero points lie on the critical line $\Re(s)=\dfrac12$. This hypothesis had many application in analysis and number theory. The first proof of the prime number theorem used this conjecture.

In order to give an anwer to your question a would like to refer to this website, where you can find tons of applications of the Riemann hypothesis.

• 38 $\begingroup$ This is not an answer to the question, but a (very) brief introduction to what the RH is. OP asked about what applications proving RH would have. $\endgroup$ –  Lord_Farin Commented May 28, 2013 at 9:58
• 2 $\begingroup$ the link was interesting (+1) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:46
• $\begingroup$ And we want the zeros of $\zeta(s)$ because $\frac{\zeta'(s)}{s\zeta(s)}$ is the Laplace transform of $\Psi(u) = \sum_{p^k < e^u} \log p$ a function showing us the distribution of primes numbers. If the RH is true, then $\frac{\Psi(u) -e^u}{e^{u/2}}= o(e^{-u/2})+\log 2\pi + \sum_\gamma \frac{e^{i \gamma u}}{1/2+i\gamma}$ where $\gamma$ are the imaginary parts of the non-trivial zeros $\endgroup$ –  reuns Commented Dec 23, 2016 at 7:33

The techniques used in the proofs of some of the most difficult theorems are used to prove so many other theorems. A proof of 1 of these theorems will give us access to an incredible amount of new techniques that will definitely make mathematics shorter,simpler and easier to understand.

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• Hypothesis | Definition & Meaning

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Explanation of Hypothesis

Contradiction, simple hypothesis, complex hypothesis, null hypothesis, alternative hypothesis, empirical hypothesis, statistical hypothesis, special example of hypothesis, solution part (a), solution part (b), hypothesis|definition & meaning.

A hypothesis is a claim or statement  that makes sense in the context of some information or data at hand but hasn’t been established as true or false through experimentation or proof.

In mathematics, any statement or equation that describes some relationship between certain variables can be termed as hypothesis if it is consistent with some initial supporting data or information, however, its yet   to be proven true or false by some definite and trustworthy experiment or mathematical law. 

Following example illustrates one such hypothesis to shed some light on this very fundamental concept which is often used in different areas of mathematics.

Figure 1: Example of Hypothesis

Here we have considered an example of a young startup company that manufactures state of the art batteries. The hypothesis or the claim of the company is that their batteries have a mean life of more than 1000 hours. Now its very easy to understand that they can prove their claim on some testing experiment in their lab.

However, the statement can only be proven if and only if at least one batch of their production batteries have actually been deployed in the real world for more than 1000 hours . After 1000 hours, data needs to be collected and it needs to be seen what is the probability of this statement being true .

The following paragraphs further explain this concept.

As explained with the help of an example earlier, a hypothesis in mathematics is an untested claim that is backed up by all the known data or some other discoveries or some weak experiments.

In any mathematical discovery, we first start by assuming something or some relationship . This supposed statement is called a supposition. A supposition, however, becomes a hypothesis when it is supported by all available data and a large number of contradictory findings.

The hypothesis is an important part of the scientific method that is widely known today for making new discoveries. The field of mathematics inherited this process. Following figure shows this cycle as a graphic:

Figure 2: Role of Hypothesis in the Scientific Method 

The above figure shows a simplified version of the scientific method. It shows that whenever a supposition is supported by some data, its termed as hypothesis. Once a hypothesis is proven by some well known and widely acceptable experiment or proof, its becomes a law. If the hypothesis is rejected by some contradictory results then the supposition is changed and the cycle continues.

Lets try to understand the scientific method and the hypothesis concept with the help of an example. Lets say that a teacher wanted to analyze the relationship between the students performance in a certain subject, lets call it A, based on whether or not they studied a minor course, lets call it B.

Now the teacher puts forth a supposition that the students taking the course B prior to course A must perform better in the latter due to the obvious linkages in the key concepts. Due to this linkage, this supposition can be termed as a hypothesis.

However to test the hypothesis, the teacher has to collect data from all of his/her students such that he/she knows which students have taken course B and which ones haven’t. Then at the end of the semester, the performance of the students must be measured and compared with their course B enrollments.

If the students that took course B prior to course A perform better, then the hypothesis concludes successful . Otherwise, the supposition may need revision.

The following figure explains this problem graphically.

Figure 3: Teacher and Course Example of Hypothesis

Important Terms Related to Hypothesis

To further elaborate the concept of hypothesis, we first need to understand a few key terms that are widely used in this area such as conjecture, contradiction and some special types of hypothesis (simple, complex, null, alternative, empirical, statistical). These terms are briefly explained below:

A conjecture is a term used to describe a mathematical assertion that has notbeenproved. While testing   may occasionally turn up millions of examples in favour of a conjecture, most experts in the area will typically only accept a proof . In mathematics, this term is synonymous to the term hypothesis.

In mathematics, a contradiction occurs if the results of an experiment or proof are against some hypothesis.  In other words, a contradiction discredits a hypothesis.

A simple hypothesis is such a type of hypothesis that claims there is a correlation between two variables. The first is known as a dependent variable while the second is known as an independent variable.

A complex hypothesis is such a type of hypothesis that claims there is a correlation between more than two variables.  Both the dependent and independent variables in this hypothesis may be more than one in numbers.

A null hypothesis, usually denoted by H0, is such a type of hypothesis that claims there is no statistical relationship and significance between two sets of observed data and measured occurrences for each set of defined, single observable variables. In short the variables are independent.

An alternative hypothesis, usually denoted by H1 or Ha, is such a type of hypothesis where the variables may be statistically influenced by some unknown factors or variables. In short the variables are dependent on some unknown phenomena .

An Empirical hypothesis is such a type of hypothesis that is built on top of some empirical data or experiment or formulation.

A statistical hypothesis is such a type of hypothesis that is built on top of some statistical data or experiment or formulation. It may be logical or illogical in nature.

According to the Riemann hypothesis, only negative even integers and complex numbers with real part 1/2 have zeros in the Riemann zeta function . It is regarded by many as the most significant open issue in pure mathematics.

Figure 4: Riemann Hypothesis

The Riemann hypothesis is the most well-known mathematical conjecture, and it has been the subject of innumerable proof efforts.

Numerical Examples

Identify the conclusions and hypothesis in the following given statements. Also state if the conclusion supports the hypothesis or not.

Part (a): If 30x = 30, then x = 1

Part (b): if 10x + 2 = 50, then x = 24

Hypothesis: 30x = 30

Conclusion: x = 10

Supports Hypothesis: Yes

Hypothesis: 10x + 2 = 50

Conclusion: x = 24

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What Is a Hypothesis? (Science)

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A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject.

In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.

In the study of logic, a hypothesis is an if-then proposition, typically written in the form, "If X , then Y ."

In common usage, a hypothesis is simply a proposed explanation or prediction, which may or may not be tested.

Writing a Hypothesis

Most scientific hypotheses are proposed in the if-then format because it's easy to design an experiment to see whether or not a cause and effect relationship exists between the independent variable and the dependent variable . The hypothesis is written as a prediction of the outcome of the experiment.

Null Hypothesis and Alternative Hypothesis

Statistically, it's easier to show there is no relationship between two variables than to support their connection. So, scientists often propose the null hypothesis . The null hypothesis assumes changing the independent variable will have no effect on the dependent variable.

In contrast, the alternative hypothesis suggests changing the independent variable will have an effect on the dependent variable. Designing an experiment to test this hypothesis can be trickier because there are many ways to state an alternative hypothesis.

For example, consider a possible relationship between getting a good night's sleep and getting good grades. The null hypothesis might be stated: "The number of hours of sleep students get is unrelated to their grades" or "There is no correlation between hours of sleep and grades."

An experiment to test this hypothesis might involve collecting data, recording average hours of sleep for each student and grades. If a student who gets eight hours of sleep generally does better than students who get four hours of sleep or 10 hours of sleep, the hypothesis might be rejected.

But the alternative hypothesis is harder to propose and test. The most general statement would be: "The amount of sleep students get affects their grades." The hypothesis might also be stated as "If you get more sleep, your grades will improve" or "Students who get nine hours of sleep have better grades than those who get more or less sleep."

In an experiment, you can collect the same data, but the statistical analysis is less likely to give you a high confidence limit.

Usually, a scientist starts out with the null hypothesis. From there, it may be possible to propose and test an alternative hypothesis, to narrow down the relationship between the variables.

Example of a Hypothesis

Examples of a hypothesis include:

• If you drop a rock and a feather, (then) they will fall at the same rate.
• Plants need sunlight in order to live. (if sunlight, then life)
• Eating sugar gives you energy. (if sugar, then energy)
• White, Jay D.  Research in Public Administration . Conn., 1998.
• Schick, Theodore, and Lewis Vaughn.  How to Think about Weird Things: Critical Thinking for a New Age . McGraw-Hill Higher Education, 2002.
• Scientific Method Flow Chart
• Six Steps of the Scientific Method
• What Are the Elements of a Good Hypothesis?
• What Are Examples of a Hypothesis?
• What Is a Testable Hypothesis?
• Null Hypothesis Examples
• Scientific Hypothesis Examples
• Scientific Variable
• Scientific Method Vocabulary Terms
• Understanding Simple vs Controlled Experiments
• What Is an Experimental Constant?
• What Is a Controlled Experiment?
• What Is the Difference Between a Control Variable and Control Group?
• DRY MIX Experiment Variables Acronym
• Random Error vs. Systematic Error
• The Role of a Controlled Variable in an Experiment

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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What is a hypothesis test?

The null hypothesis significance testing (nhst) framework, our simple scenario.

• Our null hypothesis is $H_0\colon \pi=\frac{1}{2}$.  This says that the proportion is what we believe it should be.
• Our alternative hypothesis is $H_1\colon \pi\ne\frac{1}{2}$.  This says that the proportion has changed.

Testing our hypotheses

• We can work out the critical region for $X$, that is, those extreme values of $X$ which would lead us to reject the null hypothesis at 5% significance.  (This can be done even before performing the experiment.)  The probability of $X$ taking a value in this critical region, assuming that the null hypothesis is true, should be 5%, or as close at we can get to 5% without going over it.  In symbols, we can say: $$\mathrm{P}(\text{$X$ in critical region} | \text{$H_0$ is true}) \le 0.05.$$ Then we reject the null hypothesis if $X$ lies in that region.
• We can work out the probability of $X$ taking the value it did or a more extreme value, assuming that the null hypothesis is true.  This is known as the p-value .  If the p-value is less than 0.05, then we will reject the null hypothesis at 5% significance. [ note 1 ]  In symbols, we can write $$\text{p-value} = \mathrm{P}(\text{$X$ taking this or a more extreme value} | \text{$H_0$ is true}).$$

Other types of scenario

• Does this drug/treatment/intervention/... have any effect?
• Which of these drugs/... is more effective, or are they equally effective?
• Is the mean height/mass/intelligence/test score/... of this population equal to some predicted value?
• Is the standard deviation of the height/mass/... equal to some predicted value?
• For two distinct groups of people, is their mean height/mass/... of each group the same?
• Does this group of people's heights/masses/... appear to be following the probability distribution we expect?
• Do these two populations' heights/masses/... appear to have the same distribution as each other?
• Do this population's heights and weights appear to be correlated?

Interpreting the results

The key question that hypothesis testing (nhst) answers, what a hypothesis test does not tell us, a non-significant result.

• It could be that the null hypothesis is true.  In this case, we would have to be unlucky to get a significant p-value, so most of the time, we will end up accepting the null hypothesis.  (If the null hypothesis is true, we would reject it with a probability of only 0.05.)  
• On the other hand, it could be that the alternative hypothesis is true, but we did not use a large enough sample to obtain a significant result (or we were just unlucky).  In such a case, we could say that our test was insensitive .  In this situation (the alternative hypothesis is true but we do not reject the null hypothesis), we say that we have made a Type II error .  The probability of this happening depends on the sample size and on how different the true $\pi$ is from $\frac{1}{2}$ (or whatever our null hypothesis says), as is explored in Powerful Hypothesis Testing .

A significant result

• It could be that the null hypothesis is true.  In this case, we reject the null hypothesis with a probability of $0.05=\frac{1}{20}$, that is, one time in 20 (at a significance level of 5%), so we were just unlucky.  
• On the other hand, the alternative hypothesis could indeed be true.  Either the sample was large enough to obtain a significant result, or the sample size wasn't that large, but we were just lucky.

Using this tree diagram, we can work out the probabilities of $H_0$ being true or $H_1$ being true given our experimental results.  To avoid the expressions becoming unwieldy, we will write $H_0$ for "$\text{$H_0$ true}$", $H_1$ for "$\text{$H_1$ true}$" and "$\text{p}^+$" for "observed p-value or more extreme".  Then we can write (conditional) probabilities on the branches of the tree diagram leading to our observed p-value: [ note 2 ]

The two routes which give our observed p-value (or more extreme) have the following probabilities: $$\begin{align*} \mathrm{P}(H_0\cap \text{p}^+) &= \mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) \\ \mathrm{P}(H_1\cap \text{p}^+) &= \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1) \end{align*}$$ (Recall that $\mathrm{P}(H_0\cap \text{p}^+)$ means "the probability of $H_0$ being true and the p-value being that observed or more extreme".) We can therefore work out the probability of the alternative hypothesis being true given the observed p-value, using conditional probability: $$\begin{align*} \mathrm{P}(H_1|\text{p}^+) &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(H_0\cap\text{p}^+)+\mathrm{P}(H_1\cap\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)}{\mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) + \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)} \end{align*}$$ Though this is a mouthful, it is a calculation which only involves the four probabilities on the above tree diagram.  (This is an example of Bayes' Theorem , discussed further in this resource .) However, we immediately hit a big difficulty if we try to calculate this for a given experiment.  We know $\mathrm{P}(\text{p}^+ | H_0)$: this is just the p-value itself.  (The p-value tells us the probability of obtaining a result at least this extreme given that the null hypothesis is true.)  But we don't know the probability of the null hypothesis being true or false (that is, $\mathrm{P}(H_0)$ and $\mathrm{P}(H_1)=1-\mathrm{P}(H_0)$), nor do we know the probability of the observed result if the alternative hypothesis is true ($P(\text{p}^+|H_1)$), as knowing that the proportion of greens is not $\frac{1}{2}$ does not tell us what it actually is.  (Similar issues apply to all the other contexts of hypothesis testing listed above.)  So we are quite stuck: in the null hypothesis significance testing model, it is impossible to give a numerical answer to our key question: "Given our results, what is the probability that the alternative hypothesis is true?"  This is because we don't know two of the three probabilities that we need in order to answer the question. An example might highlight the issue a little better.  Let us suppose that we are trying to work out whether a coin is biased (alternative hypothesis), or whether the probability of heads is exactly $\frac{1}{2}$ (null hypothesis).  We toss the coin 50 times and obtain a p-value of 0.02.  Do we now believe that the coin is biased?  Most people believe that coins are not biased, and so are much more likely to attribute this result to chance or poor coin-tossing technique than to the coin being biased. On the other hand, consider a case of a road planner who introduces a traffic-calming feature to reduce the number of fatalities along a certain stretch of road.  The null hypothesis is that there is no change in fatality rate, while the alternative hypothesis is that the fatality rate has decreased.  A hypothesis test is performed on data collected for 24 months before and 24 months after the feature is built.  Again, the p-value was 0.02.  Do we believe that the alternative hypothesis is true?  In this case, we are more likely to believe that the alternative hypothesis is true, because it makes a lot of sense that this feature will reduce the number of fatalities. Our "instinctive" responses to these results are tied up with assigning values to the unknown probabilities in the formula above.  For the coin, we would probably take $\mathrm{P}(H_0)$ to be close to 1, say $0.99$, as we think it is very unlikely that the coin is biased, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.1$: if the coin is biased, the bias is not likely to be very large, and so it is only a bit more likely that the result will be significant in this case.  Putting these figures into the formula above gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.01 \times 0.1}{0.99 \times 0.02 + 0.01 \times 0.1} \approx 0.05,$$ that is, we are still very doubtful that this coin is biased, even after performing the experiment.  Note that in this case, the probability of these results given that the null hypothesis is true is 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.05=0.95$, which is very different.  This shows how dramatically different the answers to the two questions can be. On the other hand, for the fatalities situation, we might assume quite the opposite: we are pretty confident that the traffic-calming feature will help, so we might take $\mathrm{P}(H_0)$ to be $0.4$, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.25$ (though the traffic-calming may help, the impact may be relatively small).  Putting these figures into the formula gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.6 \times 0.25}{0.4 \times 0.02 + 0.6 \times 0.25} \approx 0.95,$$ so we are now much more convinced that the traffic-calming feature is helping than we were before we had the data.  This time, the probability of these results given that the null hypothesis is true is still 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.95=0.05$, which is not that different. This approach may seem very disturbing, as we have to make assumptions about what we believe before we do the hypothesis test.  But as we have seen, we cannot answer our key question without making such assumptions.  

Other approaches and some warnings

• Because our test is two-tailed (in the alternative hypothesis, the true proportion could be less than $\frac{1}{2}$ or more than $\frac{1}{2}$), we must be careful when calculating the p-value: we calculate the probability of the observed outcome or more extreme occurring, and then double the answer to account for the other tail.  We could also compare the probability of the value or more extreme to 0.025 instead of 0.05, but that would not be called a p-value. Likewise, when we determine the critical region, we will have two parts: a tail with large values of $X$ and a tail with small values of $X$; we require that the probability of $X$ lying in the large-value tail is as close as possible to 0.025 without going over it, and the same for the probability of $X$ lying in the small-value tail.  
• There are complications here when working with two-tail tests as opposed to one-tail tests.  We will ignore this problem, as it does not significantly affect the overall discussion.  
• "Likelihood" is a technical term.  For a discrete test statistic $X$, the likelihood of $H_0$ given the data $X=x$ means $P(X=x|H_0)$, in other words, how likely would this data be if $H_0$ were true.  It is not the probability of $H_0$ being true given the data.

Further reading