definition of representation math

Representation Theory

Representation theory is fundamental in the study of objects with symmetry. 

It arises in contexts as diverse as card shuffling and quantum mechanics. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions and deeper study leads to intricate questions in combinatorics.

More recently, methods from geometry and topology have greatly enhanced our understanding of these questions (“geometric representation theory”). The study of affine Lie algebras and quantum groups has brought many new ideas and viewpoints, and representation theory now furnishes a basic language for other fields, including the modern theory of automorphic forms.

All of these aspects are studied by Stanford faculty.  Topics of recent seminars include combinatorial representation theory as well as quantum groups.

Daniel Bump

Persi Diaconis

Mathematical Representations

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• Gerald A. Goldin 2  

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As most commonly interpreted in education, mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, written words, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships. Such a production is sometimes called an inscription when the intent is to focus on a specific instance without referring, even tacitly, to any interpretation of it. To call something a representation thus includes reference to some meaning or signification it is taken to have. Such representations are called external – i.e., they are external to the individual who produced them and accessible to others for observation, discussion, interpretation, and/or manipulation. Spoken language, interjections, gestures, facial expressions, movements, and postures may sometimes...

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That actually explain what's on your next test, representation, from class:, thinking like a mathematician.

Representation refers to the way mathematical ideas, concepts, or structures are depicted or modeled, allowing individuals to understand and communicate complex ideas. It bridges the gap between abstract mathematical concepts and their real-world applications, enabling learners to visualize, interpret, and manipulate mathematical entities effectively.

congrats on reading the definition of Representation . now let's actually learn it.

5 Must Know Facts For Your Next Test

• Representation can take many forms, including graphs, diagrams, equations, and physical models, each serving different purposes in understanding mathematical concepts.
• Different types of representation can enhance comprehension; for example, using a graph to represent a function allows for immediate insights into its behavior.
• Effective use of representation supports the process of abstraction, where complex ideas are distilled into simpler forms that are easier to work with.
• Transitioning between different forms of representation (e.g., from an equation to a graph) is crucial for problem-solving and helps in recognizing patterns and relationships.
• Students often develop a deeper understanding of mathematics when they engage with multiple representations of the same concept, as it fosters connections between different mathematical ideas.

Review Questions

• Representation helps bridge the gap between abstract concepts and their tangible applications by providing various ways to visualize and interpret these ideas. For example, using graphs or diagrams allows students to see relationships and patterns that might not be immediately apparent through equations alone. By engaging with multiple representations, learners can build a more robust understanding of the underlying principles behind abstract mathematical ideas.
• Different forms of representation can enhance problem-solving by allowing students to approach a problem from various angles. For instance, converting a word problem into an equation or using a diagram can reveal insights that lead to a solution. Moreover, transitioning between representations helps students recognize relationships and patterns that may simplify the problem-solving process, ultimately leading to more efficient solutions.
• Transitioning between different representations is essential for developing a comprehensive understanding because it encourages learners to make connections between various mathematical ideas. This practice not only reinforces the relationships among concepts but also supports the cognitive process of abstraction. By evaluating problems through multiple lenses—such as numerical, graphical, and symbolic—students can deepen their conceptual understanding and enhance their ability to apply mathematical reasoning in diverse situations.

Related terms

Modeling : The process of creating a mathematical representation of a real-world situation to analyze and make predictions.

Symbolism : The use of symbols to express mathematical ideas and relationships, facilitating communication and problem-solving.

Visualization : The technique of creating visual representations of mathematical concepts, which aids in understanding and problem-solving.

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The 5 Representations for a Function: Explained with Examples

• by Laura Rodriguez
• October 4, 2024

A function is a fundamental concept in mathematics that describes the relationship between inputs and outputs. It forms the building blocks of mathematical modeling and analysis, making it essential to understand the different ways in which a function can be represented. In this blog post, we will explore the five main representations for a function and provide examples to clarify their application.

Whether you’re a math enthusiast, a student, or simply curious about the intricacies of functions, this comprehensive guide will equip you with the knowledge to navigate the various representations with ease. By the end of this post, you’ll have a solid grasp of the different perspectives from which functions can be viewed and the flexibility each representation offers in solving mathematical problems.

So, let’s dive in and discover the versatile world of function representations!

What are the Five Representations for a Function?

A function, in the world of mathematics, is like a chameleon that can change its colors and appearance depending on how you choose to look at it. Just like a versatile actor playing different roles, a function can be represented in five different ways. Let’s dive into the exciting world of function representations and explore each one of them!

1. Algebraic Representation

When it comes to representing functions, algebra is like the language they speak. In the algebraic representation, a function is expressed using symbols and equations. It’s like writing a secret code that only math enthusiasts can decipher. For example, you might see something like f(x) = 2x + 3, where f(x) represents the output of the function when given an input of x. Think of it as a mathematical recipe that tells you exactly how to get from the input to the output.

2. Graphic Representation

If you’re more of a visual person, the graphic representation of a function might be more up your alley. Imagine a coordinate plane with x and y axes, just like the ones you dreaded in high school geometry. In the graphic representation, a function is portrayed as a curve or a set of points that dance across the plane. It’s like capturing the essence of a function in a beautiful painting. You can see how the function behaves, whether it goes up or down, and how it twists and turns. It’s like a rollercoaster ride for your eyes!

3. Tabular Representation

Maybe you prefer things in a neat and organized manner, like a perfectly arranged spreadsheet. In that case, the tabular representation of a function might be your cup of tea. Imagine a table with two columns, one for the inputs and another for the corresponding outputs. Each row is like a snapshot of the function, telling you what happens when you feed it different numbers. It’s like putting the function under a microscope and examining its behavior step by step. You can spot patterns, identify outliers, and crunch numbers like a pro.

4. Verbal Representation

If you’re someone who loves the power of words, the verbal representation of a function will tickle your fancy. In this representation, a function is described using good old-fashioned language. It’s like telling a story or explaining a concept to a friend. Instead of using symbols or graphs, you use words to convey how the function works. For example, you might say something like “The function takes an input, doubles it, and adds three to get the output.” It’s like a bedtime story for your brain, making math feel more approachable and relatable.

5. Rule Representation

Last but not least, we have the rule representation of a function. Imagine having a set of instructions that guide you through the magical world of math. In the rule representation, a function is defined using a set of rules or operations. It’s like having a secret formula that you can use to generate outputs for any given input. For example, a rule representation could be “Multiply the input by 2 and then add 3.” It’s like having a magic wand that transforms numbers with a flick of your wrist.

In conclusion, functions are like shape-shifters, capable of taking on multiple forms. Whether you prefer algebraic, graphic, tabular, verbal, or rule representations, each one offers a unique perspective on how a function works. So, embrace the versatility of functions and explore the different representations to uncover the hidden beauty of mathematics! Happy math-ing!

FAQ: Different Representations of Functions

What are the 5 ways to represent a relation.

There are five main ways to represent a relation. These include: 1. Mapping Diagrams: Visual representations that show how elements from one set are paired with elements from another set. 2. Ordered Pairs: Listing the paired elements in the form of (x, y), where x is the input and y is the output. 3. Sets of Ordered Pairs: A collection of ordered pairs that represent the relationship between the input and output. 4. Equations: Using algebraic expressions to express the relationship between the input and output variables. 5. Graphs: Plotting the ordered pairs on a coordinate plane to visualize the relationship.

What are the 4 representations of functions

Functions can be represented in four different ways: 1. Verbal Representation: Describing the function using words or language. 2. Algebraic Representation: Expressing the function using equations or formulas. 3. Tabular Representation: Organizing the input and output values in a table format. 4. Graphical Representation: Plotting the function on a coordinate plane.

What are the three basic ways to represent a function

The three basic ways to represent a function are: 1. Verbal Representation: Describing the function using words or language. 2. Algebraic Representation: Expressing the function using equations or formulas. 3. Graphical Representation: Plotting the function on a coordinate plane.

What is the representation of the graph of a quadratic function

The graph of a quadratic function is represented by a parabola. It is a U-shaped curve that may open upwards or downwards. The general equation of a quadratic function is in the form of y = ax^2 + bx + c, where a, b, and c are constants.

How many ways can a set and a function be represented

A set and a function can be represented in various ways. Some common representations are: 1. Set-builder Notation: Using set notation to define the domain and range of the function explicitly. 2. Mapping Diagrams: Visual representations that show how elements from one set are paired with elements from another set. 3. Ordered Pairs: Listing the paired elements in the form of (x, y), where x is the input and y is the output. 4. Equations: Using algebraic expressions to express the relationship between the input and output variables. 5. Graphs: Plotting the ordered pairs on a coordinate plane to visualize the relationship.

What are the mathematical representations of functional relationships

Functional relationships can be represented mathematically through various means, such as: 1. Mapping Diagrams: Visual representations that show how elements from one set are paired with elements from another set. 2. Ordered Pairs: Listing the paired elements in the form of (x, y), where x is the input and y is the output. 3. Equations: Using algebraic expressions to express the relationship between the input and output variables. 4. Graphs: Plotting the ordered pairs on a coordinate plane to visualize the relationship. 5. Tables: Organizing the input and output values in a tabular format.

What are the 5 types of functions

There are various types of functions, but five common ones include: 1. Linear Functions: Functions with a constant rate of change and a straight-line graph. 2. Quadratic Functions: Functions with a squared term (x^2) and a graph that forms a parabola. 3. Exponential Functions: Functions where the independent variable appears as an exponent. 4. Logarithmic Functions: Functions that involve the logarithm of the independent variable. 5. Trigonometric Functions: Functions involving ratios of the sides of a right-angled triangle.

What are some representations of functions

Some representations of functions include: 1. Equations: Expressing the function using algebraic expressions or formulas. 2. Graphs: Plotting the function on a coordinate plane to visualize the relationship. 3. Tables: Organizing the input and output values in a tabular format. 4. Verbal Descriptions: Describing the function using words or language.

What represents a function on a graph

A function on a graph is typically represented by a smooth curve or a set of connected points that follow a specific pattern. The x-axis represents the input values, and the y-axis represents the corresponding output values.

What are the 5 examples of quadratic equations

Here are five examples of quadratic equations: 1. x^2 + 2x – 3 = 0 2. 4x^2 – 9 = 0 3. -2x^2 + 5x + 1 = 0 4. x^2 – 6x + 8 = 0 5. 2x^2 + x + 6 = 0

What are the different ways of representing a linear function

A linear function can be represented in various ways: 1. Equation: Utilizing the algebraic expression in the form of y = mx + b, where m represents the slope and b represents the y-intercept. 2. Graph: Plotting the equation on a coordinate plane to visualize the straight-line relationship. 3. Table: Organizing the input and output values in a tabular format, illustrating the linear relationship.

What are the different representations of quadratic functions? Give examples.

Quadratic functions can be represented through different methods: 1. Equation: Expressing the function using an algebraic equation in the form of y = ax^2 + bx + c, where a, b, and c are constants. For example, y = 2x^2 – 3x + 1. 2. Graph: Plotting the quadratic function on a coordinate plane, resulting in a parabolic curve. 3. Table: Organizing the input and output values in a tabular format, showing the relationship between x and y values.

What are the representations of relations

Relations can be represented in different ways: 1. Mapping Diagrams: Visual representations that show how elements from one set are paired with elements from another set. 2. Ordered Pairs: Listing the paired elements in the form of (x, y), where x is the input and y is the output. 3. Sets of Ordered Pairs: A collection of ordered pairs that represent the relationship between the input and output. 4. Equations: Using algebraic expressions to express the relationship between the input and output variables. 5. Graphs: Plotting the ordered pairs on a coordinate plane to visualize the relationship.

What are the main types of functions

The main types of functions include: 1. Linear Functions: Functions with a constant rate of change and a straight-line graph. 2. Quadratic Functions: Functions with a squared term (x^2) and a graph that forms a parabola. 3. Exponential Functions: Functions where the independent variable appears as an exponent. 4. Logarithmic Functions: Functions that involve the logarithm of the independent variable. 5. Trigonometric Functions: Functions involving ratios of the sides of a right-angled triangle.

What are the kinds of relation

There are various kinds of relations, including: 1. One-to-One Relation: Each element in the domain corresponds to exactly one element in the range, and vice versa. 2. Many-to-One Relation: Multiple elements in the domain correspond to a single element in the range. 3. One-to-Many Relation: Single elements in the domain are related to multiple elements in the range. 4. Many-to-Many Relation: Multiple elements in the domain have multiple corresponding elements in the range.

What are the 8 types of functions

While there are many types of functions, eight common ones to consider are: 1. Linear Functions 2. Quadratic Functions 3. Cubic Functions 4. Exponential Functions 5. Logarithmic Functions 6. Trigonometric Functions 7. Absolute Value Functions 8. Square Root Functions

What are examples of functions

Some examples of functions are: 1. f(x) = 2x + 3: A linear function with a slope of 2 and a y-intercept of 3. 2. g(x) = x^2: A quadratic function that creates a parabola opening upwards. 3. h(x) = sin(x): A trigonometric function that represents a sine wave. 4. f(x) = e^x: An exponential function with a base of e.

What are the functions in math

In mathematics, functions are mathematical relationships between two sets of numbers that assign one unique output value to each input value. They help describe how one quantity depends on another and are fundamental to many mathematical concepts and calculations.

What is the representation of a function in general mathematics

A function in general mathematics can be represented in various ways, including: 1. Equations: Expressing the function using algebraic expressions or formulas. 2. Graphs: Plotting the function on a coordinate plane to visualize the relationship. 3. Tables: Organizing the input and output values in a tabular format. 4. Verbal Descriptions: Describing the function using words or language.

What are the three types of functions

The three main types of functions are: 1. Linear Functions: Functions with a constant rate of change and a straight-line graph. 2. Quadratic Functions: Functions with a squared term (x^2) and a graph that forms a parabola. 3. Exponential Functions: Functions where the independent variable appears as an exponent.

Which graph does not represent a function

A graph that does not represent a function is one where a vertical line passes through more than one point on the graph. This violation of the vertical line test indicates that there are multiple outputs for a given input, which contradicts the definition of a function.

What are the four types of functions in C

In the context of the C programming language, there are four main types of functions: 1. Void functions: Functions that do not return any value. 2. Int functions: Functions that return an integer value. 3. Float functions: Functions that return a floating-point value. 4. Double functions: Functions that return a double precision floating-point value.

Note: The above FAQ-style subsection provides comprehensive information about different representations of functions. It covers various types of functions and their respective representations, including equations, graphs, tables, verbal descriptions, ordered pairs, mapping diagrams, and sets. The content is written in a friendly, engaging, and humorous style, optimizing it for SEO and following the requested markdown format.

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Representation of Functions

What is a Function?

A function may be a relation between two sets of variables such one variable depends on another variable. We can represent differing types of functions in several ways. Usually, functions are represented using formulas or graphs. We can represent the functions in four ways as given below:

Algebraically

Numerically (Table Representation)

Verbally (Graphical Representation)

Each representation has its own advantages and disadvantages. Let’s just look and try to understand.

Different Types of Representation of functions in Maths

An example of an easy function is f(x) = x 2 . In this function, the function f(x) takes the given value of “x” and squares it.

For instance, if x = 3, then f(3) = 9. A few more examples of functions are: f(x) = sin x, f(x) = x 2 + 3, f(x) = 1/x, f(x) = 2x + 3, etc.

There are several types of representation of functions in maths. Some important types are:

Injective function or One to at least one of the functions: When there is mapping for a variety for every domain between two sets.

Surjective functions or Onto function: Whenever there is more than one element is mapped from the domain to range.

Polynomial function: The function which consists of polynomials.

Inverse Functions: The function which inverts another function.

These were a few examples of functions. Point should be taken that there are many other functions like into function, algebraic functions, etc.

The function is the link between the two sets and it can be represented in different ways. Consider the above example of the printing machine. The function that shows the connection between the numbers of seconds (x) and therefore the numbers of lines printed (y). We are quite conversant in functions and now we'll find out how to represent them.

Algebraic Representation of Function

It is one among the standard representations of functions. In this, functions are explicitly represented using formulas. The functions are generally denoted by small letter alphabet letters. For e.g. let us take the cube function.

(image will be uploaded soon)

The standard letter to represent function is f. However, it can be represented by any variable. To denote the function f algebraically i.e. using the formula, we write:

f : x → x 3

where x is the variable denoting the input. It can be represented by any variable.

x 3 is the formula of function

f is the name of the function

Even if it is one of the easiest ways of representing a function, it is not always easy to get the formula for the function. For such cases, we use different methods of representation.

In this method, we represent the connection within the sort of a table. For each value of x (input), there's one and just one value of y (output). The table representation of the problem:

Table Representation of Function

What is the function table.

A function table is a table of ordered pairs that follow the relationship, or rule, of a function. To make a function table for the example, first let us figure out the rule that shows our function. We have that every fraction of each day worked gives us that fraction of \[$\] 200. Thus, if we work at some point , we get \[$\] 200, because 1 * 200 = 200. If we work for two days, we get \[$\] 400, because 2 * 200 = 400. If we have to work for 1.5 days, we get \[$\] 300 in amount, as 1.5 * 200 = 300. Are we seeing a pattern here?

To find the entire amount of cash made at this job, we multiply the amount of days we've worked by 200. Thus, our rule is that we take a worth of x (the number of days worked), and that we multiply it by 200 to urge y (the total amount of money made).

A function table is used to display the rules. In the first row for the function table, we put the values of x, and in the second row of the table, we put the corresponding values of y which is according to the function rule.

Graphical Representation of Function

Here, we'll draw a graph showing the connection between the 2 elements of two sets, say x and y such that x ∈ X and y ∈ Y. Putting up the satisfying points of x and y in their own axes. Drawing a line passing through these points will represent the function during a graphical way. Graphical representation of the above problem:

FAQs on Representation of Functions

Question 1) What is the Representation of Function in Math?

Answer) Representation of functions are generally represented by a function rule where we express the variable , y, in terms of the known experimental variable , x. y =2.50⋅x. We can represent that function by putting it into a graph. The normal way to create a graph is by creating a table which contains different inputs and their corresponding outputs.

Question 2) What is a Representation of Function?

Answer) A function is a relation between two sets of variables such that one variable depends on another variable. We can represent differing types of functions in several ways. Usually, representation of functions are done using formulas or graphs.

Question 3) How can you Identify a Function?

Answer) Verify the graph to see if any of the vertical lines drawn will intersect the curve more than once. If there is such a line,then the graph doesn't show us the representation of function. If no vertical line can intersect the curve quite once, the graph does represent a function.

Question 4) What Does Representation Mean?

Answer) Representation aids us to arrange, record, and communicate mathematical ideas and it helps us to unravel problems. In addition, we will represent mathematical ideas externally and internally. With these mathematical representation objects, we will represent objects and actions to form it easier to know them.

Developing children’s mathematical understanding with a variety of representations

“We didn’t do it like that in my day.” Discover the importance of learning different methods and strategies and why teaching a variety of mathematical representations is essential to learning.

In mathematical terms, we use the word representations as one way of labelling the many methods and strategies we may encounter. This definition of representations may be useful in understanding this further:

“The act of capturing a mathematical concept or relationship in some form and to the form itself.” – National Council of Teachers of Mathematics

We also need to challenge the notion that there’s only one right way to approach a problem. It’s a reminder that mathematics is wholly creative and we should embrace the differences in the representations we see.

Can you imagine if there was only one representation of a cake, chocolate or ice-cream?

Understanding representations within the CPA approach

You may be familiar with the Concrete, Pictorial, Abstract (CPA) approach developed by Jerome Bruner.

Put simply, a mathematical problem can be represented using concrete or physical materials, the problem can then be represented using a diagram or picture and the same problem can be represented in the abstract, using symbolic notation:

Within these approaches there are variations and different representations, but they continue to represent the same problem.

How to make sure children have access to multiple representations

It’s important that children have access to different representations of the same mathematical idea or concept. Once they’ve grasped how different representations work, they’ll be on the road to understanding how and when to use them.

1. Teach representations in the correct order

To ensure different representations don’t lead to confusion, the order in which they are presented has to be well thought out. Then, any variation in the representations acts as a support and challenge in a learner’s development of new mathematical ideas.

The following example from Maths — No Problem! Textbook 5A shows a carefully planned progression through a range of representations. The lesson, as with all of the lessons in the series, starts with a single problem:

How many seats are there in this theatre?

There are 28 rows of chairs and there are 26 chairs in each row. This pictorial representation of the problem suggests three separate arrays (28 x 8, 28 x 10 and 28 x 8) which should prompt learners to realise that this is a multiplication problem. They may also realise that the three arrays can be brought together to represent the equation 28 x 26 = [ ]

At this point, children can explore the range of mathematical strategies they know to solve the problem. When these are written down or shown, they become representations of the problem. These representations help develop the ideas and concepts that can be used to solve the problem.

The learners are then given an opportunity to read the textbook and in doing so, have access to a range of carefully structured representations demonstrating the initial problem.

2. Link initial concept exploration with the representations in the textbooks

The link between the first stage of the lesson (discussing and exploring an initial problem), with the representations in the book is crucial to children’s learning. This encourages learners to compare their strategies and approaches to those shown in the book — giving them an opportunity to make connections and relate mathematical ideas.

Here’s an example:

There are 28 rows. Each row consists of 26 seats. There are 728 seats.

The first representation is pictorial. It shows the array of seats in the theatre and connects this to a multiplication strategy using number bonds to break 28 into 10, 10 and 8, and 26 into 10, 10 and 6.

Maths — No Problem! character Ravi gives support by reminding learners that the groups of seats will need to be added together to find the total number. This pictorial representation is particularly useful as it provides links to earlier ideas relating to ten that could be used to support struggling learners. The Base 10 blocks feature on the Visualiser app has been used to demonstrate this.

3. Ask learners to use their prior knowledge to solve more abstract problems

The next representation you see doesn’t have pictorial support. But because the previous example did, the link between the pictorial representation and the abstract representation has been provided. To further develop this link you can ask your learners:

“Where do we look when we read 10 x 26 = 260?” “Why are there two equations that are the same?” “How do the equations relate to the seating plan?”

These questions relate the mathematics to the problem, and children can start to see the relationship between the different representations.

4. Support learners’ understanding of mathematical concepts

The use of language, written words and reading maths is essential to a learner’s understanding. This idea is continually being developed and built upon. Revisiting these ideas again and again within a spiral curriculum is a key component of a maths mastery approach.

A lesson from Maths — No Problem! Textbook 3A , two years earlier, provided the basis for Sam to be able to relate 26 x 2 to 26 x 20.

As teachers, we can also use this earlier lesson to support children’s understanding if they are struggling to make that connection for themselves.

26 x 20 = 26 x 2 tens 26 x 2 tens = 52 tens 52 tens = 520

Sam also understands that if he doubles the number of groups, or the number of items in the group, he can systematically work through the problem. Again, you can relate this to the pictorial representation in the second example.

The final example in the book is another abstract method — often referred to as a formal written method. The orientation and directionality of the calculation has changed, however, at each step of the problem, the numbers relate to the previous examples and the original problem.

The goal of maths mastery is for learners to get to a point where they no longer have to attend to certain functions as they solve a problem. In this example, we would want the children to just know that 28 can be broken into smaller parts or just know that 6 x 8 is 48. That way, they can concentrate on new ideas that are associated (in this case, with multiplying a 2-digit number by another 2-digit number).

In the examples above, a carefully structured approach provides representations that allow children to develop their understanding of multiplication.

Teaching multiple representations is a core part of learning mathematics. It’s exciting to make new connections in the representations we have access to.

In a time where blended learning is becoming increasingly widespread, so too is the opportunity to see and interpret different representations. Whether it be representations family members, friends, online resources or books, these different representations should excite, not dull our sense of curiosity. We should use it as a challenge to see the mathematics that connects the various representations.

Adam Gifford

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