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• How to write an essay introduction | 4 steps & examples

How to Write an Essay Introduction | 4 Steps & Examples

Published on February 4, 2019 by Shona McCombes . Revised on July 23, 2023.

A good introduction paragraph is an essential part of any academic essay . It sets up your argument and tells the reader what to expect.

The main goals of an introduction are to:

• Catch your reader’s attention.
• Give background on your topic.
• Present your thesis statement —the central point of your essay.

This introduction example is taken from our interactive essay example on the history of Braille.

The invention of Braille was a major turning point in the history of disability. The writing system of raised dots used by visually impaired people was developed by Louis Braille in nineteenth-century France. In a society that did not value disabled people in general, blindness was particularly stigmatized, and lack of access to reading and writing was a significant barrier to social participation. The idea of tactile reading was not entirely new, but existing methods based on sighted systems were difficult to learn and use. As the first writing system designed for blind people’s needs, Braille was a groundbreaking new accessibility tool. It not only provided practical benefits, but also helped change the cultural status of blindness. This essay begins by discussing the situation of blind people in nineteenth-century Europe. It then describes the invention of Braille and the gradual process of its acceptance within blind education. Subsequently, it explores the wide-ranging effects of this invention on blind people’s social and cultural lives.

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Table of contents

Step 1: hook your reader, step 2: give background information, step 3: present your thesis statement, step 4: map your essay’s structure, step 5: check and revise, more examples of essay introductions, other interesting articles, frequently asked questions about the essay introduction.

Your first sentence sets the tone for the whole essay, so spend some time on writing an effective hook.

Avoid long, dense sentences—start with something clear, concise and catchy that will spark your reader’s curiosity.

The hook should lead the reader into your essay, giving a sense of the topic you’re writing about and why it’s interesting. Avoid overly broad claims or plain statements of fact.

Examples: Writing a good hook

Take a look at these examples of weak hooks and learn how to improve them.

• Braille was an extremely important invention.
• The invention of Braille was a major turning point in the history of disability.

The first sentence is a dry fact; the second sentence is more interesting, making a bold claim about exactly  why the topic is important.

• The internet is defined as “a global computer network providing a variety of information and communication facilities.”
• The spread of the internet has had a world-changing effect, not least on the world of education.

Avoid using a dictionary definition as your hook, especially if it’s an obvious term that everyone knows. The improved example here is still broad, but it gives us a much clearer sense of what the essay will be about.

• Mary Shelley’s  Frankenstein is a famous book from the nineteenth century.
• Mary Shelley’s Frankenstein is often read as a crude cautionary tale about the dangers of scientific advancement.

Instead of just stating a fact that the reader already knows, the improved hook here tells us about the mainstream interpretation of the book, implying that this essay will offer a different interpretation.

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Next, give your reader the context they need to understand your topic and argument. Depending on the subject of your essay, this might include:

• Historical, geographical, or social context
• An outline of the debate you’re addressing
• A summary of relevant theories or research about the topic
• Definitions of key terms

The information here should be broad but clearly focused and relevant to your argument. Don’t give too much detail—you can mention points that you will return to later, but save your evidence and interpretation for the main body of the essay.

How much space you need for background depends on your topic and the scope of your essay. In our Braille example, we take a few sentences to introduce the topic and sketch the social context that the essay will address:

Now it’s time to narrow your focus and show exactly what you want to say about the topic. This is your thesis statement —a sentence or two that sums up your overall argument.

This is the most important part of your introduction. A  good thesis isn’t just a statement of fact, but a claim that requires evidence and explanation.

The goal is to clearly convey your own position in a debate or your central point about a topic.

Particularly in longer essays, it’s helpful to end the introduction by signposting what will be covered in each part. Keep it concise and give your reader a clear sense of the direction your argument will take.

As you research and write, your argument might change focus or direction as you learn more.

For this reason, it’s often a good idea to wait until later in the writing process before you write the introduction paragraph—it can even be the very last thing you write.

When you’ve finished writing the essay body and conclusion , you should return to the introduction and check that it matches the content of the essay.

It’s especially important to make sure your thesis statement accurately represents what you do in the essay. If your argument has gone in a different direction than planned, tweak your thesis statement to match what you actually say.

To polish your writing, you can use something like a paraphrasing tool .

You can use the checklist below to make sure your introduction does everything it’s supposed to.

Checklist: Essay introduction

My first sentence is engaging and relevant.

I have introduced the topic with necessary background information.

I have defined any important terms.

My thesis statement clearly presents my main point or argument.

Everything in the introduction is relevant to the main body of the essay.

You have a strong introduction - now make sure the rest of your essay is just as good.

• Argumentative
• Literary analysis

This introduction to an argumentative essay sets up the debate about the internet and education, and then clearly states the position the essay will argue for.

The spread of the internet has had a world-changing effect, not least on the world of education. The use of the internet in academic contexts is on the rise, and its role in learning is hotly debated. For many teachers who did not grow up with this technology, its effects seem alarming and potentially harmful. This concern, while understandable, is misguided. The negatives of internet use are outweighed by its critical benefits for students and educators—as a uniquely comprehensive and accessible information source; a means of exposure to and engagement with different perspectives; and a highly flexible learning environment.

This introduction to a short expository essay leads into the topic (the invention of the printing press) and states the main point the essay will explain (the effect of this invention on European society).

In many ways, the invention of the printing press marked the end of the Middle Ages. The medieval period in Europe is often remembered as a time of intellectual and political stagnation. Prior to the Renaissance, the average person had very limited access to books and was unlikely to be literate. The invention of the printing press in the 15th century allowed for much less restricted circulation of information in Europe, paving the way for the Reformation.

This introduction to a literary analysis essay , about Mary Shelley’s Frankenstein , starts by describing a simplistic popular view of the story, and then states how the author will give a more complex analysis of the text’s literary devices.

Mary Shelley’s Frankenstein is often read as a crude cautionary tale. Arguably the first science fiction novel, its plot can be read as a warning about the dangers of scientific advancement unrestrained by ethical considerations. In this reading, and in popular culture representations of the character as a “mad scientist”, Victor Frankenstein represents the callous, arrogant ambition of modern science. However, far from providing a stable image of the character, Shelley uses shifting narrative perspectives to gradually transform our impression of Frankenstein, portraying him in an increasingly negative light as the novel goes on. While he initially appears to be a naive but sympathetic idealist, after the creature’s narrative Frankenstein begins to resemble—even in his own telling—the thoughtlessly cruel figure the creature represents him as.

If you want to know more about AI tools , college essays , or fallacies make sure to check out some of our other articles with explanations and examples or go directly to our tools!

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Your essay introduction should include three main things, in this order:

• An opening hook to catch the reader’s attention.
• Relevant background information that the reader needs to know.
• A thesis statement that presents your main point or argument.

The length of each part depends on the length and complexity of your essay .

The “hook” is the first sentence of your essay introduction . It should lead the reader into your essay, giving a sense of why it’s interesting.

To write a good hook, avoid overly broad statements or long, dense sentences. Try to start with something clear, concise and catchy that will spark your reader’s curiosity.

A thesis statement is a sentence that sums up the central point of your paper or essay . Everything else you write should relate to this key idea.

The thesis statement is essential in any academic essay or research paper for two main reasons:

• It gives your writing direction and focus.
• It gives the reader a concise summary of your main point.

Without a clear thesis statement, an essay can end up rambling and unfocused, leaving your reader unsure of exactly what you want to say.

The structure of an essay is divided into an introduction that presents your topic and thesis statement , a body containing your in-depth analysis and arguments, and a conclusion wrapping up your ideas.

The structure of the body is flexible, but you should always spend some time thinking about how you can organize your essay to best serve your ideas.

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Mathematics

• Geometry Essays

Geometry Essays (Examples)

290+ documents containing “geometry” .

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Geometry proof geometry as a.

The student then places it on the playing field. The system allows a chosen playing card to be dragged by means of a mouse to the playing field and, if properly placed, to "stick" in place on the playing field. (Improperly placed cards "snap" back to their original file position.) After each card has been correctly placed, a line between properly placed cards is generated connecting proper statements and reasons to each other and the GIVEN or CONCLUSION displays the completed proof (Herbst, 2002). In working with geometric proofs, it is important for the student and teacher alike to approach this new and intimidating subject with an open mind. Even though students may have never experienced any type of logic or reasoning prior to the introduction of proofs, if presented correctly, this new way of approaching math can be both fun and enlightening. Teachers should keep this in mind when….

Discovering Geometry: A Guide for Parents. 2008, Key Curriculum Press. Retrieved October 19, 2009 at http://www.keymath.com/documents/dg4/GP/DG4_GP_02.pdf

Herbst, Patricio G. Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century, Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. 283-312,

Geometry of Design Elam Kimberly

The Golden section has a special relationship to the Fibonacci sequence. This is a mathematical sequence in which the first two numbers being 0 and 1, each subsequent number is a sum of the previous two numbers: 0, 1,1, 2 (1+1), 3 (2+1), 5 (3+2), etc. Like the Golden section, the Fibonacci numbers are used to understand the way trees branch, leaves occur, fruit ripens, etc. -- it is a set of numbers that explains nature's patterns. Subdividing shapes has no effect on their ratio or relationship to Fibonacci. Chapter 4 -- Root Rectangles- the idea of the root angel reduction allows the Golden section to become more vital in several aspects of modern life. Not only does this impact modern design of furniture, technology, and appliances; it has a far larger and more robust meaning as we begin to understand the roots of organic chemistry and the structure of living organisms. One….

In 2003, a study showed that based on psychometric data, the Golden mean appears in the chronological cycle of brain waves. This was empirically confirmed in 2008.

In 2010, the journal Science reported that the golden ratio is present at the atomic level.

For a designer, the importance of the ratio cannot be overstated; from the Volkswagen Beetle to the Gutenberg Bible, the Golden ratio is everywhere; natural and man-made.

Geometry Manipulative Elementary Geometry Manipulative

This will not only introduce elementary students to geometry, but also begin the complicated thinking associated with algebraic concepts. Using the formula to plug in the known degrees and then find the x is the beginning of much more abstract algebraic thinking. Handout Circles rule our lives and have rules of their own! Each circle measures to 350 degrees, and with this knowledge we can begin to find unknown angles! If a circle measures 360, that means that a half circle measures half -- 180 degrees. In a half circle, there are many different angle combinations. But, we know that they all equal out to 180 degrees. Knowing this, we can find the great unknown! Well, we know that the total of the two angles equals 180 degrees. Therefore, angle 1 = angle 2 = 180 degrees. Let's just plug the numbers into the equation. 63 + x = 180. The first step is to isolate the….

National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

Fractual Geometry

Fractal Geometry is a somewhat new branch of mathematics that was developed in 1980 by enoit . Mandelbrot, a research mathematician in I..M.'s Thomas Day Watson laboratory in New York. Mandelbrot was experimenting with the theories of Gaston Julia, a French mathematician when he discovered the fractal set was discovered. Julia dedicated his life to the study of the iteration of polynomials and rational functions. Around the 1920s, Julia published a paper on the iteration of a rational function, which brought him to fame. However, after his death, he was all but forgotten...until the 1970's when Mandelbrot, who was inspired by Julia's work, revived his work. y using computer graphics, Mandelbrot was able to show the first pictures of the most beautiful fractals known today. Mandelbrot, who is now Professor of Mathematics at Yale, made the discovery of fractal geometry by going against establishment and academic mathematics -- going beyond Einstein's theories to….

Bibliography

Mandelbrot, Benoit B. The Fractal Geometry of Nature W.H. Freeman and Company, 1977.

Crilly, R.A. Fractals and Chaos. Springer-Verlag, 1991.

Dictionary of Scientists, Oxford University Press, Market House Books Ltd., 1999

Theme of Love in Relation to Natural Sciences and Geometry in Metaphysical Poetry

Natural Sciences and Geometry in Metaphysical Poetry Love in metaphysical poetry: Donne and Marvell "Metaphysical texts, primarily characterized through the conflation of traditional form with seditious linguistic techniques such as satire, irony, wit, parody and rhetoric, generate a microcosmic emphasis in many of the texts" even while the authors ultimately address 'macro' concerns of religion and man's place in the universe (Uddin 45). In poems such as John Donne's "The Flea" and "A Valediction Forbidding Mourning" and Andrew Marvell's "The Definition of Love," subjects such as the poet's adoration for his beloved take on a much higher significance than the personal sphere within the context of the poem. Metaphysical poetry embodies what is often considered a paradox: it is, on one hand, intensely emotional, but it is also, on the other hand, quite explicit in its suggestion of universality. "Introspection, being 'a careful examination of one's own thoughts, impressions and feelings'….

Donne, John. " The Flea." Poetry Foundation.

 http://www.poetryfoundation.org/poem/175764  [16 Jan 2013]

Donne, John. "A Valediction Forbidding Mourning."

 http://www.luminarium.org/sevenlit/donne/mourning.php  [16 Jan 2013]

Measurement Geometry Representation in Math

MATH - Measurement, Geometry, Representation Part A A standard unit of measurement offers a point of reference by which items of weight, length, or capacity can be delineated. It is a quantifiable semantic that aids every individual comprehend the relation of the object with the measurement. For instance, volume can be expressed in metrics such as gallons, ounces, and pints. On the other hand, a non-standard unit of measurement is something that might fluctuate or change in terms of weight or length. The item that was measured is a door as this is an object that can be found in everyday life. Through the use of standard units of measurement, I found that the door is equivalent to 2 meters or 200 centimeters in length. This particular standard measure was used for the reason that it is possible to utilize it using items including tape measures. On the other hand, with respect….

Capturing Astronomical Geometry the Greek Way

Mathematics -- to the Moon & Back Once upon a time, Alexander, a young man from Athens fell in love with a local girl, Adrianna, whose beauty was far greater than any other young woman he had ever seen. Alexander was so smitten with Adrianna that he promised her the moon. Being an astute girl, Adrianna told Alexander that she wasn't at all sure that he could deliver the moon, but he could begin to convince her that he was intelligent and clever by measuring the distance from the earth to the moon. Alexander had long heard the stories about his Greek ancestors who were experts in mathematics and astronomy, so he sought out some wise elders to learn more. Alexander spent some time with two elders, one of whom told him he knew how to measure the size of the earth (which, Alexander mused, was bound to impress, Adrianna), and another….

Intersection Theory Bridging Concepts in Mathematics and Beyond

Introduction Intersection theory, a fundamental aspect of algebraic geometry, serves as a bridge between various mathematical disciplines, offering insights into the geometric properties of different mathematical spaces. This essay provides an overview of intersection theory, exploring its origins, fundamental principles, applications, and the influence it exerts on other mathematical fields. Origins and Historical Context Intersection theory originated from the need to understand and quantify the intersections of geometric shapes in various dimensions. Historically, mathematicians like Bernhard Riemann and André Weil contributed significantly to its development. Weil, in particular, was instrumental in laying the groundwork for modern intersection theory through his work on algebraic surfaces and their intersection numbers. His contributions, along with those of other 20th-century mathematicians, transformed intersection theory into a major branch of algebraic geometry. Fundamental Principles At its core, intersection theory is concerned with the study of how subspaces of a given space intersect with each other. It involves calculating intersection numbers….

Basic Facts

Teach Geometry Dear Parent, This letter is in response to your question: Why are students in elementary school learning geometry when they do not yet know the basic facts and should be spending their time working on them instead? There are two parts to the answer. The first is concerned with the learning of math facts. It is an ongoing process for students in the elementary grades. It begins with the development of number sense, which is a child's facility and flexibility in using and manipulating numbers (Chard, Baker, Clarke, Jungjohann, Davis, and Smolkowski, 2008, p. 12). Some students develop number sense in preschool or informally in familial settings before kindergarten; other children do not begin to develop number sense until their formal schooling begins, whether because of opportunity or because of developmental readiness. Developing number sense takes time. It does not happen quickly and it does not happen because a child….

Chard, D.J., Baker, S.K., Clarke, B., Jungjohann, K., Davis, K., and Smolkowski, K. (2008).

Preventing early mathematics difficulties: The feasibility of a rigorous kindergarten mathematics curriculum. Learning Disability Quarterly 31(1), pp. 11-20.

Common core standards adoption by state. (2012). ASCD. Retrieved from  http://www.ascd.org/common-core-state-standards/common-core-state-standards - adoption-map.aspx

Cooke, B.D., and Buccholz, D. (2005). Mathematical communication in the classroom: A teacher makes a difference. Early Childhood Education Journal 32(6), pp. 365-369).

Art and Mathematics Are Related

Note the distinct similarities. An examination of Escher's Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher's woodcut and the Poincare disk, the images showcased appear smaller as one's eye moves toward the edge of the circle. However, this is an illusion created by our traditional, Euclidean perceptions. Because of the way that distance is measured in a hyperbolic space, all of the objects shown in the circle are actually the same size. As we follow the backbones of the fish in Escher's representation, we can see, then, that the lines separating one fish from the next are actually all the same distance even though they appear to grow shorter. This is because, as already noted, the hyperbolic space stretches to infinity at its edges. There is no end. Therefore, the perception that the lines are getting smaller toward the edges is, in….

Works Cited

Corbitt, Mary Kay. "Geometry." World Book Multimedia Encyclopedia. World Book, Inc., 2003.

Dunham, Douglas. "A Tale Both Shocking and Hyperbolic." Math Horizons Apr. 2003: 22-26.

Ernst, Bruno. The Magic Mirror of M.C. Escher. NY: Barnes and Noble Books, 1994.

Granger, Tim. "Math Is Art." Teaching Children Mathematics 7.1 (Sept. 2000): 10.

Don't Have One

relearn several mathematical concepts and learn how to instruct other about them. It also became necessary to learn the different components of educating students on math based upon their current knowledge and abilities and how the teacher will evaluate the students to make that determination. Not only did I learn how to teach the subject, but I was also instructed on how to submit and fulfill standards. In short, this class taught me how to be an effective and efficient math teacher for students from kindergarten up to the eighth grade. This class had good moments, difficult moments, and has influenced both what concepts I will teach my students and how I will teach them when the time comes. It is hard determining which of the components learned in this class were the most important. Each mathematical concept will be necessary when entering the teaching profession. Certainly it was useful….

Works Cited:

Billstein, R., Libeskind, S., & Lott, J.W. (2010). A Problem Solving Approach to Mathematics

for Elementary School Teachers (10th ed.). Boston, MA: Addison-Wesley.

National and State Subject Matter Content Standards

National and State Subject Matter Content Standards for Math According to the California standards for high school students, the geometry curriculum contains six critical components: "to establish criteria for congruence of triangles based on rigid motions; establish criteria for similarity of triangles based on dilations and proportional reasoning; informally develop explanations of circumference, area, and volume formulas; apply the Pythagorean Theorem to the coordinate plan; prove basic geometric theorems; and extend work with probability" (Common Core Standards, California Department of Education: 69). The elucidated standards are often quite specific in terms of how students are asked to apply basic concepts such as measuring angles; understanding the different properties of parallel lines; and manipulating various polygons. Not only must the students prove theorems but they must also be able to construct such shapes using a variety of methods in a hands-on fashion (Common Core Standards, 2013, California Department of Education: 70). Deeper….

Common Core Standards. California Department of Education. ca.gov. [21 Oct 2013]  http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf 

Common Core Standards. Official Website. [21 Oct 2013]

 http://www.corestandards.org/Math/Content/HSG/introduction

Mountain Bike Rear Suspension Pds

The system also has to undergo thousands of cycles and vibrations and needs to be able to stand up to the same reliability standards as the rest of the components on the bike. Conventional and Proven ear Suspension Designs Given all of the previously mentioned considerations, the design itself is important in making sure the rider and the manufacturer are getting the most out of the system. The Fox acing Homepage (2011) has some excellent examples of both the strut style rear suspension as well as the shock with spring and strut combination system. The latter is typically reserved for use on higher-end advanced bikes since these systems are costlier and requires more maintenance. To be more specific, the Van C product represents the higher-end strut and spring combination while the Float design is a basic, oil dampened design for use on more entry-level designs. The Van C model is also an….

Bu, Yan; Tian Huang, Zhongxia Xiang, Xiaofan Wu and Chun Chen. (2010). "Optimal design of mountain bicycle based on biomechanics." Transactions of Tianjin University,

Volume 16, Number 1, 45-49.

DT Swiss Homepage. (2011). Accessed Jan. 5 at:

 http://www.dtswiss.com/Products/Suspension/DT-Swiss-Shocks.aspx .

Sales of Off-Highway Trucks Is

When fully loaded the weight distribution is 40% on the front axle and 60% on the rear axle. Given the likely adhesion conditions, the powertrain will drive all axles. Suspension geometry design and assessment Steering design Turning circle When the vehicle is cornering, each wheel must go through a turning circle. The outer turning circle, is to our main subject of interest. This calculation is never precise because when a vehicle is cornering the perpendiculars via the centres of all wheels never intersect at the curve centre point (Ackermann condition). Additionally, while the vehicle is moving, certain dynamic forces will always arise that will eventually affect the cornering manoeuvre (MAN,2000). The formula used. Vehicle Model T31, 19.314 FC Wheelbase lkt = 5000 mm Front axle Model V9-82L Tyres 315/80 . 22.5 Wheel 22.5 x 9.00 Track width s = 2058 mm Scrub radius r0 = 58 mm Inner steer angle ?i = 50.0° Outer steer angle ?a = 30°30' = 30.5° 1. Distance between….

Catapillar (2001). 769D-Off-Highway Truck. http://xml.catmms.com/servlet/ImageServlet?imageId=C199012

Ivanov V, Shyrokau B, Augsburg K, Vantsevich V (2010)System Fusion in Off-Road Vehicle Dynamics Control09/2010; in proceeding of: Joint 9th Asia-Pacific ISTVS Conference, at Sapporo, Japan

MAN (2009).vehicle calculations

Rafael, M, a. Lozano, J. Cervantes, V. Mucino, C.S. Lopez-Cajun (2009).A method for powertrain selection of heavy-duty vehicles with fuel savings. International Journal of Heavy Vehicle Systems

Strategic Scaffolding at Apple Strategic

raster graphics, wire-frame and 3D modeling performance, and refresh rates of their screens. What began to occur in the company's culture as a result of this focus on graphics performance and CPU acceleration was a bifurcation or splitting of product lines. At the high end Apple was gradually turning into a workstation company that could easily challenge Sun Microsystems or Silicon Graphics for supremacy of graphically-based calculations. At the low-end, the company was pursuing an aggressive strategy of dominating special-purpose laptops. This strategy was entirely predicated on the core metrics of price/performance on hardware defining a culture that put pricing above all else, paradoxically nearly driving the company out of business during this period. The focus on metrics that were meant to purely define the Apple competitive advantage made the company descend into pricing wars with competitors whose business models were much more attuned to pricing competition. The metrics the….

Berling, Robert J. (1993). The emerging approach to business strategy: Building a relationship advantage. Business Horizons, 36(4), 16. Accessed from: http://www.berlingassociates.com/features/horizon.pdf

Keidel, Robert . THE GEOMETRY OF STRATEGY New York: Routledge, May 2010

Sakakibara, Kiyonori, Lindholm, Chris, & Ainamo, Antti. (1995). Product development strategies in emerging markets: The case of personal digital assistants. Business Strategy Review, 6(4), 23. Accessed from  http://www.soc.utu.fi/laitokset/iasm/SakakibaraEtAlPDAs1995.pdf

What is intersection theory? How can we use it to help understand the experiences of those around us

Intersection theory, in abstract terms, is a complex mathematical framework used in algebraic geometry to understand the intricate "meetings" between shapes within a mathematical space. While its roots and applications stem from the world of shapes and equations, it can be viewed through a more metaphorical lens to gain insights into human experiences. Here's how:

Understanding Intersection:

• Imagine two overlapping circles:  Their intersection represents shared experiences, common ground, or areas of mutual understanding between two individuals. Intersection theory, in this metaphor, provides tools to quantify and analyze these overlaps, their complexities, and how they influence the overall experience.

How it helps:

Can you provide guidance on how to outline an essay focusing on Intersection Theory

Outline for an Essay on Intersection Theory I. Introduction Begin with a compelling hook or question that captures the reader's attention. Define intersection theory and explain its significance in algebraic geometry. State the thesis statement, which should articulate the main argument or purpose of the essay. II. Background and Historical Context Provide a brief overview of the historical development of intersection theory. Discuss the contributions of key mathematicians, such as Bézout, Euler, and Poincaré. Explain the role of intersection theory in resolving classical geometric problems. III. Fundamental Concepts Define the basic concepts of intersection theory, such as: Intersection number Cycle Homology and cohomology....

I need some suggestions for abstract for maths in physics project essay topics. Can you offer any?

1. The Fibonacci sequence and its applications in physics 2. Chaos theory and its implications for understanding complex systems in physics 3. The role of symmetry in modern physics 4. Fractal geometry and its applications in modeling natural phenomena 5. The use of wave equations in describing physical processes 6. The concept of infinity in calculus and its significance for physics 7. The mathematical foundations of quantum mechanics 8. Differential equations and their role in modeling physical systems 9. The geometry of spacetime in general relativity 10. The role of group theory in understanding the fundamental forces of nature 11. The applications of calculus in solving problems in classical mechanics 12.....

Is there anything in the news related to abstract of maths in physics that would make a good essay subject?

One interesting topic that has been in the news recently is the use of abstract mathematical concepts in physics, specifically in the field of quantum physics. There has been a lot of research and debate on the role of abstract mathematical structures in understanding and describing the behavior of particles at the quantum level. For example, the use of complex numbers, matrices, and abstract algebraic structures has been essential in developing the mathematical framework of quantum mechanics. One recent development in this area is the use of category theory, a branch of mathematics that studies abstract structures and relationships between different....

Research Proposal

Education - Mathematics

The student then places it on the playing field. The system allows a chosen playing card to be dragged by means of a mouse to the playing field…

Book Review

The Golden section has a special relationship to the Fibonacci sequence. This is a mathematical sequence in which the first two numbers being 0 and 1, each subsequent number…

This will not only introduce elementary students to geometry, but also begin the complicated thinking associated with algebraic concepts. Using the formula to plug in the known degrees…

Fractal Geometry is a somewhat new branch of mathematics that was developed in 1980 by enoit . Mandelbrot, a research mathematician in I..M.'s Thomas Day Watson laboratory in New…

Natural Sciences and Geometry in Metaphysical Poetry Love in metaphysical poetry: Donne and Marvell "Metaphysical texts, primarily characterized through the conflation of traditional form with seditious linguistic techniques such as…

MATH - Measurement, Geometry, Representation Part A A standard unit of measurement offers a point of reference by which items of weight, length, or capacity can be delineated. It is a…

Mathematics -- to the Moon & Back Once upon a time, Alexander, a young man from Athens fell in love with a local girl, Adrianna, whose beauty was far greater…

Introduction Intersection theory, a fundamental aspect of algebraic geometry, serves as a bridge between various mathematical disciplines, offering insights into the geometric properties of different mathematical spaces. This essay provides…

Teach Geometry Dear Parent, This letter is in response to your question: Why are students in elementary school learning geometry when they do not yet know the basic facts and…

Research Paper

Note the distinct similarities. An examination of Escher's Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher's woodcut and the Poincare disk,…

relearn several mathematical concepts and learn how to instruct other about them. It also became necessary to learn the different components of educating students on math based upon…

National and State Subject Matter Content Standards for Math According to the California standards for high school students, the geometry curriculum contains six critical components: "to establish criteria for…

Engineering

The system also has to undergo thousands of cycles and vibrations and needs to be able to stand up to the same reliability standards as the rest of…

A2 Coursework

Transportation

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Introduction to Geometry: Points, Lines, Planes and Dimensions

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When you start studying geometry, it is important to know and understand some basic concepts.

This page will help you understand the concept of dimensions in geometry, and work out whether you are working in one, two or three dimensions.

It also explains some of the basic terminology, and points you to other pages for more information.

This page covers points, lines, and planes.

Other pages in this series explain about angles and shapes, including polygons , circles and other curved shapes , and three-dimensional shapes .

What is Geometry?

Geometry , n. that part of mathematics which treats the properties of points, lines, surfaces and solids…

Chambers English Dictionary, 1989 edition

Geometry comes from the Greek meaning ‘earth measurement’ and is the visual study of shapes, sizes and patterns, and how they fit together in space. You will find that our geometry pages contain lots of diagrams to help you understand the subject.

When you’re faced with a problem involving geometry, it can be very helpful to draw yourself a diagram.

Working in Different Dimensions

No, not the space-time continuum! We’re talking about shapes that are in one, two and three dimensions.

That is, objects that have length (one dimension), length and width (two dimensions) and length, width and depth or height (three dimensions).

Points: A Special Case: No Dimensions

A point is a single location in space. It is often represented by a dot on the page, but actually has no real size or shape.

You cannot describe a point in terms of length, width or height, so it is therefore non-dimensional . However, a point may be described by co-ordinates. Co-ordinates do not define anything about the point other than its position in space, in relation to a reference point of known co-ordinates. You will come across point co-ordinates in many applications, such as when you are drawing graphs , or reading maps.

Almost everything in geometry starts with a point, whether it’s a line, or a complicated three-dimensional shape.

Lines: One Dimension

A line is the shortest distance between two points. It has length, but no width, which makes it one-dimensional.

Wherever two or more lines meet, or intersect, there is a point, and the two lines are said to share a point:

Line segments and rays

There are two kinds of lines: those that have a defined start- and endpoint and those that go on for ever.

Lines that move between two points are called line segments . They start at a specific point, and go to another, the endpoint. They are drawn as a line between two points, as you would probably expect.

The second type of line is called a ray , and these go on forever. They are often drawn as a line starting from a point with an arrow on the other end:

Parallel and perpendicular lines

There are two types of lines that are particularly interesting and/or useful in mathematics. Parallel lines never meet or intersect. They simply go on forever side by side, a bit like railway lines. The convention for showing that lines are parallel in a diagram is to add ‘feathers’, which look like arrow heads.

Perpendicular lines intersect at a right angle, 90°:

Planes and Two-dimensional Shapes

Now that we have dealt with one dimension, it’s time to move into two.

A plane is a flat surface, also known as two-dimensional. It is technically unbounded, which means that it goes on for ever in any given direction and as such is impossible to draw on a page.

One of the key elements in geometry is how many dimensions you’re working in at any given time. If you are working in a single plane, then it’s either one (length) or two (length and width). With more than one plane, it must be three-dimensional, because height/depth is also involved.

Two-dimensional shapes include polygons such as squares, rectangles and triangles, which have straight lines and a point at each corner.

There is more about polygons in our page on Polygons . Other two-dimensional shapes include circles, and any other shape that includes a curve. You can find out more about these on our page, Curved Shapes .

Three Dimensions: Polyhedrons and Curved Shapes

Finally, there are also three-dimensional shapes , such as cubes, spheres, pyramids and cylinders.

To learn more about these see our page on Three-Dimensional Shapes .

Signs, Symbols and Terminology

The shape illustrated here is an irregular pentagon, a five-sided polygon with different internal angles and line lengths (see our page on Polygons for more about these shapes).

Degrees ° are a measure of rotation, and define the size of the angle between two sides.

Angles are commonly marked in geometry using a segment of a circle (an arc), unless they are a right angle when they are ‘squared off’. Angle marks are indicated in green in the example here. See our page on Angles for more information.

Tick marks (shown in orange) indicate sides of a shape that have equal length (sides of a shape that are  congruent  or that match). The single lines show that the two vertical lines are the same length while the double lines show that the two diagonal lines are the same length. The bottom, horizontal, line in this example is a different length to the other 4 lines and therefore not marked. Tick marks can also be called ‘ hatch marks ’.

A vertex  is the point where lines meet (lines are also referred to as rays or edges). The plural of vertex is vertices. In the example there are five vertices labelled A, B, C, D and E. Naming vertices with letters is common in geometry.

In a closed shape, such as in our example, mathematical convention states that the letters must always be in order in a clockwise or counter-clockwise direction. Our shape can be described ‘ABCDE’, but it would be incorrect to label the vertices so that the shape was ‘ADBEC’ for example. This may seem unimportant, but it is crucial in some complex situations to avoid confusion.

The angle symbol ‘∠’ is used as a shorthand symbol in geometry when describing an angle. The expression  ∠ABC  is shorthand to describe the angle between points A and C at point B. The middle letter in such expressions is always the vertex of the angle you are describing - the order of the sides is not important.  ∠ABC  is the same as  ∠CBA,  and both describe the vertex  B  in this example.

If you want to write the measured angle at point B in shorthand then you would use:

m∠ABC = 128°  (m simply means 'measure')

m∠CBA = 128°

In our example we can also say:

Why Do These Concepts Matter?

Points, lines and planes underpin almost every other concept in geometry. Angles are formed between two lines starting from a shared point. Shapes, whether two-dimensional or three-dimensional, consist of lines which connect up points. Planes are important because two-dimensional shapes have only one plane; three-dimensional ones have two or more.

In other words, you really need to understand the ideas on this page before you can move on to any other area of geometry.

Further Reading from Skills You Need

The Skills You Need Guide to Numeracy

This four-part guide takes you through the basics of numeracy from arithmetic to algebra, with stops in between at fractions, decimals, geometry and statistics.

Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.

Continue to: Angles Polygons Calculating Area

See also: Three-Dimensional Shapes Circles and Curved Shapes Area, Surface Area and Volume Reference Sheet

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1.1: Finite Geometries

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• Page ID 89842

• Mark A. Fitch
• University of Alaska Anchorage

Definition: Intersect

Two lines intersect if and only if they share a point.

Definition: Parallel

Two lines are parallel if and only if they do not intersect.

Definition: Four Point Geometry

The four point geometry is defined by the following axioms and definitions.

• There exist exactly four points.
• Any two distinct points have exactly one line on both of them.
• Each line is on exactly two points.

Explore the four point geometry as follows.

• Draw and label four points.
• Use axiom 2 to draw as many lines as possible.
• How many lines exist in this geometry?
• Find a pair of parallel lines.
• Can you find three lines that are pairwise parallel?
• Can you find a point that is on three lines?

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Geometry: A Very Short Introduction

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Geometry: A Very Short Introduction discusses the fundaments of Euclidean and non-Euclidean geometries. This topic includes curved spaces, projective geometry in Renaissance art, and the geometry of spacetime inside a black hole. The study of geometry is at least 2,500 years old, and within it is the concept of mathematical proof or deductive reasoning from a set of axioms. Geometry remained a very active area of research in mathematics, with links to science and art. The subject of geometry includes examples of mathematical objects, such as Platonic solids, or theorems like the Pythagorean theorem, as well as general principles.

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Introduction to Angles

We begin our study of angles by learning what they are, how to name them, and ways in which angles can be classified. These concepts are explained below.

An angle is formed when two rays meet at a common endpoint, or vertex. The two sides of the angle are the rays, and the point that unites them is called the vertex.

The vertices are shown in red in the diagram above

Naming Angles

Angles can be named in various ways. One way is to use the ? symbol accompanied by three letters. The first and third letters indicate points on the two rays. The letter in the middle is the vertex. Note that the first and third letters are interchangeable because they both measure the same angle. Another way to label an angle is by just using the ? symbol accompanied by the vertex point alone. However, this method only works when there is only one angle at the vertex point. If more than one angle is formed at a vertex point, we need to specify which angle we are talking about by naming it in a different way. Finally, the last way to label an angle is by using the ? symbol accompanied by the letter or number shown between the angle. The different ways of labeling an angle are shown below.

The angle above can be called ?ABC, ?CBA, ?B, or ??

Classifications of Angles

Angles can be measured in degrees or radians. For the time being, we will strictly talk about angles in terms of their degree measure. The symbol for degrees is °. Angles can measure from 0° up to 360°. Angles with no measure are called zero angles, while angles of 360° are full rotations. For our study of geometry , we will primarily focus on three important classifications of angles: acute, obtuse, and right.

Right Angle

A right angle is an angle whose measure is exactly 90°. An easy way to determine whether an angle is a right angle is by considering whether a small square could fit perfectly in the corner of the intersection of the two lines that form the angle. While you would need a protractor to give a more precise measurement, this can give you an approximation of whether or not an angle is close to 90°.

Acute Angle

An acute angle is an angle whose measure is less than 90°. For these kinds of angles, a square could not fit perfectly at the intersection of the two lines that form them.

Obtuse Angle

Obtuse angles have measures greater than 90° but less than 180°.

Straight Angle

If an angle’s measure is 180°, it is called a straight angle. Straight angles are just lines with three points on them.

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• Coordinate Geometry

Co-ordinate Geometry

Coordinate Geometry is considered to be one of the most interesting concepts of mathematics. Coordinate Geometry (or the analytic geometry ) describes the link between geometry and algebra through graphs involving curves and lines. It provides geometric aspects in Algebra and enables them to solve geometric problems. It is a part of geometry where the position of points on the plane is described using an ordered pair of numbers. Here, the concepts of coordinate geometry (also known as Cartesian geometry) are explained along with its formulas and their derivations.

Introduction to Coordinate Geometry

Coordinate geometry (or analytic geometry) is defined as the study of geometry using the coordinate points. Using coordinate geometry, it is possible to find the distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc. There are certain terms in Cartesian geometry that should be properly understood. These terms include:

What is a Co-ordinate and a Co-ordinate Plane?

You must be familiar with plotting graphs on a plane, from the tables of numbers for both linear and non-linear equations. The number line which is also known as a Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis ( horizontal line ) and the y-axis( vertical line ).

The four quadrants along with their respective values are represented in the graph below-

• Quadrant 1 : (+x, +y)
• Quadrant 2 : (-x, +y)
• Quadrant 3 : (-x, -y)
• Quadrant 4 : (+x, -y)

The point at which the axes intersect is known as the origin . The location of any point on a plane is expressed by a pair of values (x, y) and these pairs are known as the coordinates .

The figure below shows the Cartesian plane with coordinates (4,2). If the coordinates are identified, the distance between the two points and the interval’s midpoint that is connecting the points can be computed.

Coordinate Geometry Fig. 1: Cartesian Plane

Equation of a Line in Cartesian Plane

Equation of a line can be represented in many ways, few of which is given below-

(i) General Form

The general form of a line is given as Ax + By + C = 0.

(ii) Slope intercept Form 

Let x, y be the coordinate of a point through which a line passes, m be the slope of a line, and c be the y-intercept, then the equation of a line is given by:

(iii) Intercept Form of a Line

Consider a and b be the x-intercept and y-intercept respectively, of a line, then the equation of a line is represented as-

Slope of a Line: 

Consider the general form of a line Ax + By + C = 0, the slope can be found by converting this form to the slope-intercept form.

Ax + By + C = 0

⇒ By = − Ax – C

Comparing the above equation with y = mx + c,

Thus, we can directly find the slope of a line from the general equation of a line.

Coordinate Geometry Formulas and Theorems

Distance formula: to calculate distance between two points.

Let the two points be A and B, having coordinates to be (x 1 , y 1 ) and (x 2 , y 2 ), respectively.

Thus, the distance between two points is given as-

Coordinate Geometry Fig. 2: Distance Formula

Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points

Consider the same points A and B, which have coordinates (x 1 , y 1 ) and (x 2 , y 2 ), respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of point M is given as-

Angle Formula: To Find The Angle Between Two Lines

Consider two lines A and B, having their slopes m 1 and m 2, respectively.

Let “θ” be the angle between these two lines, then the angle between them can be represented as-

Special Cases:

• Case 1:  When the two lines are parallel to each other,

m 1 = m 2 = m

Substituting the value in the equation above,

• Case 2:  When the two lines are perpendicular to each other,

m 1 . m 2 = -1

Substituting the value in the original equation,

\(\begin{array}{l}\large \tan \theta = \frac{m_{1} – m_{2}}{1 + (-1)} = \frac{m_{1} – m_{2}}{0}\end{array} \) which is undefined.

Section Formula: To Find a Point Which Divides a Line into m:n Ratio

Consider a line A and B having coordinates (x 1 , y 1 ) and (x 2 , y 2 ), respectively. Let P be a point that which divides the line in the ratio m:n, then the coordinates of the coordinates of the point P is given as-

• When the ratio m:n is internal:
• When the ratio m:n is external:

Students can follow the link provided to learn more about the section formula  along its proof and solved examples.

Area of a Triangle in Cartesian Plane

The area of a triangle In coordinate geometry whose vertices are (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) is

If the area of a triangle whose vertices are (x 1 , y 1 ),(x 2 , y 2 ) and (x 3 , y 3 ) is zero, then the three points are collinear.

• Important:  Click here to Download  Co-ordinate Geometry pdf

Examples Based On Coordinate Geometry Concepts

Examples 1: Find the distance between points M (4,5) and N (-3,8).

Applying the distance formula we have,

Example 2: Find the equation of a line parallel to 3x+4y = 5 and passing through points (1,1).

For a line parallel to the given line, the slope will be of the same magnitude.

Thus the equation of a line will be represented as 3x+4y=k

Substituting the given points in this new equation, we have

k = 3 × 1 + 4 × 1 = 3 + 4 = 7

Therefore the equation is 3x + 4y = 7

Coordinate Geometry Questions For Practice

• Calculate the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).
• Find the area of the triangle having vertices at A, B, and C which are at points (2, 3), (–1, 0), and (2, – 4), respectively. Also, mention the type of triangle.
• A point A is equidistant from B(3, 8) and C(-10, x). Find the value for x and the distance BC.

Video Lesson on Coordinate Geometry Toughest Problems

Continue Learning

Frequently asked questions, what is abscissa and ordinates in coordinate geometry.

The abscissa and ordinate is used to represent the position of a point on a graph. The horizontal value or the X axis value is the abscissa while the vertical value i.e. the Y axis value is the ordinate. For example, in an ordered pair (2, 3), 2 is abscissa and 3 is ordinate.

What is a Cartesian Plane?

A Cartesian plane is a plane which is formed by two perpendicular lines known as the x-axis (horizontal axis) and the y-axis (vertical axis). The exact position of a point in Cartesian plane can be determined using the ordered pair (x, y).

Why do we Need Coordinate Geometry?

Coordinate geometry has various applications in real life. Some of the areas where coordinate geometry is an integral part include.

• In digital devices like computers, mobile phones, etc. to locate the position of cursor or finger.
• In aviation to determine the position and location of airplanes accurately.
• In maps and in navigation (GPS).
• To map geographical locations using latitudes and longitudes.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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• Introduction to Fractal Geometry

Introduction to Fractal Geometry - Essay Example

• Subject: Mathematics
• Type: Essay
• Level: College
• Pages: 2 (500 words)
• Downloads: 5
• Author: pinkie51

Extract of sample "Introduction to Fractal Geometry"

It is a matter of fact that the word “fractal” has two definite meanings: the first meaning refers to colloquial use and the second is connected with geometry. In colloquial speech, a fractal is a shape which is self-similar or recursively constructed. It means that such a shape is similar at all levels of magnification and therefore it is sometimes called “infinitely complex”. In the sphere of mathematics the word “fractal” means geometric object “that satisfies a specific technical condition, namely having a Hausdoff dimension greater than its topological dimension” (26). The simplest fractal is known to be Cantor Bar Set which was called after a German famous mathematician. It is necessary to admit that fractals refer not only to the realm of geometry and mathematics. They can be found everywhere in the natural world.

Self-similarity (recursive nature) means that fractals are constructed by iteration and they are made up of smaller parts (copies) of themselves. Examples of self-similarity are Sierpinski Gasket, the Mandelbrot Set, Self-affine fractals, etc. It must be admitted that fractals are not all self-similar objects. Scientists found three main types of self-similarity: - Exact self-similarity means that fractals are completely identical at all levels of magnification. It is the strongest type of self-similarity because fractals bear exact and complete self-similarity. - Quasi-self-similarity is a loose form of self-similarity. It means that fractals are approximately identical, but not completely. Such objects “contain small copies of the entire fractal in distorted and degenerate forms” (29). - Statistical self-similarity is known to be the weakest type of self-similarity because fractals are only statistically similar. They are only random fractals.

It is necessary to mention that fractals can be used in music, medicine, art forms, image compression, cosmology, seismology, computer design and graphics, fracture mechanisms, fractal antennas, geology, nature and even the food industry. But fractals found in nature are different from other fractals because their self-similarity is statistical, limited and approximate.

Fractals in biology, medicine and nature are new developments. Fractals are found in nature. Such fractals involve snow flacks, clouds, river networks, mountains, blood vessels, lightning, and coastlines because all of them display self-similarity over scale range. Trees and ferns are also considered to be fractal because they can be virtually modelled with the help of a recursive algorithm. The recursive nature of fractals is rather clear, for example, a branch of any tree is the miniature copy of the whole tree. The recursive nature ensures that the whole is similar to each detail in a statistical way (55).

Fractals are also found in medicine. It is necessary to mention studying lung disease with x-rays, human bronchial trees, human electroencephalograms, analysis of neurons, segmentation of cells, etc. One more example is the healthy rhythm of the human heart. Fractals are very important in biology where surface area is in the first place (the lungs). The surface of the human lungs is made up of self-similar branches. The fractal properties are also displayed by blood vessels and nerve cells within the human body (58).

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