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# Hypothesis|Definition & Meaning

## Definition

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.**

## Explanation of Hypothesis

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:**

### Conjecture

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.**

### Contradiction

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.**

### Simple 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.**

**Complex **Hypothesis

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.

**Null **Hypothesis

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.**

**Alternative **Hypothesis

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**.**

**Empirical **Hypothesis

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

**Statistical **Hypothesis

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.

### Special Example of Hypothesis

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

### Solution Part (a)

Hypothesis: 30x = 30

Conclusion: x = 10

Supports Hypothesis: Yes

### Solution Part (b)

Hypothesis: 10x + 2 = 50

Conclusion: x = 24

Supports Hypothesis: Yes

*All images/mathematical drawings were created with GeoGebra.*