Leverage your pineal gland for mental mathematics!

" Aries Hilton designed this new methodology to help readers strengthen their extra sensory perception with the intent to calculate mathematics through Lucid Visualization's!"

"Our pineal glands can empower us to do everything calculators can and much more with the power of our own minds rather than codependency on monopolizations. This data can both save or even recreate your life, if you apply it ethically~ As Above, So Below!"

Once upon a time, in a land of numbers and equations, there lived a young student named Jamie. Jamie was determined to master the art of mathematics and to be able to calculate even the most complex equations with ease.

But no matter how hard Jamie tried, the numbers and equations just wouldn’t stick. Frustrated and discouraged, Jamie wandered through the land, searching for a way to unlock the secrets of mathematics.

One day, while wandering through a forest of numbers, Jamie stumbled upon an old sage. The sage listened to Jamie’s troubles and then spoke: “My child, you must learn to see beyond what your eyes can see. You must use the power of visualization to unlock the secrets of mathematics.”

The sage then taught Jamie about the different types of equations and how to visualize them.

• For basic arithmetic operations, the sage showed Jamie how to use flower petals, objects or fingers to represent numbers and how to visualize the operations being performed.

• For linear equations, the sage taught Jamie how to visualize a line on a graph and how the slope and y-intercept affected its position by reminding her how to a squeal "could" navigate amongst tree branches in nature.

• For quadratic equations, the sage showed Jamie how to visualize a mystical Ferris Wheel aka a parabola, and how its shape changed depending on the values of a, b, and c.

• For exponential equations, the sage taught Jamie how to visualize exponential growth or decay and how the values of a and b affected the rate of change via a story about a magical fairy who sprinkles magical fairy dust onto mushrooms which help them grow / shrink (decay) organically.

• For trigonometric equations, the sage showed Jamie how to visualize the unit circle and how the sine function related to it with a majestic dolphin jumping in and out of the water.

• For logarithmic equations, the sage taught Jamie how to visualize logarithmic growth or decay and how the values of a and x affected the rate of change. Jamie learns about librarian sorting the books by their sizes who has a unique capability to recreate book(s) in any proportion possible as needed for the library.

• For calculus equations, the sage showed Jamie how to visualize derivatives as slopes of tangent lines and integrals as areas under curves. With a trippy trip to a fun roller coaster park full of rides!

Jamie listened carefully and practiced visualizing these equations every day. As time passed, Jamie’s skills grew stronger and stronger.

Soon, even complex equations that once seemed impossible were now easily solved in Jamie’s mind.

And so, with determination and practice, Jamie became a master of mathematics and was able to calculate even the most difficult equations with ease.

The moral of the story is that with focus and determination, anything is possible. By leveraging visualization techniques and focusing on understanding different types of equations, we can train our minds to achieve great things.

Here story the sage told Jamie to help visually calculate Basic Arithmetic leveraging their imagination!

Basic Arithmetic Operations:

Imagine that you are in a beautiful garden full of colorful flowers. You see a friendly bee buzzing around, collecting nectar from the flowers. The bee has a special ability: it can change the number of petals on any flower it visits. You decide to follow the bee and see what it does.

The bee lands on a yellow flower that has four petals. It whispers something to the flower and suddenly, the flower has six petals. The bee tells you that it added two petals to the flower. This is how you can do addition with your imagination: just add more petals to a flower and count them.

The bee flies to another flower, this time a red one that has seven petals. It whispers something to the flower and suddenly, the flower has five petals. The bee tells you that it subtracted two petals from the flower. This is how you can do subtraction with your imagination: just take away some petals from a flower and count them.

The bee moves on to a blue flower that has three petals. It whispers something to the flower and suddenly, the flower has nine petals. The bee tells you that it multiplied the number of petals by three. This is how you can do multiplication with your imagination: just repeat the same number of petals on a flower as many times as you want and count them.

The bee goes to a purple flower that has twelve petals. It whispers something to the flower and suddenly, the flower has four petals. The bee tells you that it divided the number of petals by three. This is how you can do division with your imagination: just split the petals on a flower into equal groups and count them.

You are amazed by the bee’s magic and thank it for showing you how to do basic arithmetic operations with your imagination. The bee smiles and invites you to try it yourself. You pick a flower and whisper a number to it. What happens next? (Comment What You Visualized Below)

• ?The story is meant to help the reader visualize and practice basic arithmetic operations with their imagination. By following the bee’s example and applying it to different flowers, the reader can learn how to add, subtract, multiply and divide numbers in a fun and engaging way.

Here story the sage told Jamie to help visually calculate with their imagination Linear Equations (y = mx + b)

Linear Equations (y = mx + b)

Imagine that you are in a park full of trees. You see a squirrel running along a branch. The squirrel has a special ability: it can change the shape and position of any branch it runs on. You decide to follow the squirrel and see what it does.

The squirrel runs on a branch that is straight and horizontal. It whispers something to the branch and suddenly, the branch becomes slanted and goes up. The squirrel tells you that it changed the slope of the branch by increasing the value of m. This is how you can do linear equations with your imagination: just change the slope of a branch by changing the value of m and see how it affects the position of the tree's branch(s).

The squirrel runs on another branch, this time one that is slanted and goes up. It whispers something to the branch and suddenly, the branch moves up higher. The squirrel tells you that it changed the y-intercept of the branch by increasing the value of b. This is how you can do linear equations with your imagination: just change the y-intercept of a branch by changing the value of b and visualize how it affects the position of the branch.

The squirrel runs on a third branch, this time one that is slanted and goes down. It whispers something to the branch and suddenly, the branch becomes straight and horizontal. The squirrel tells you that it made the slope of the branch zero by making m equal to zero. This is how you can do linear equations with your imagination: just make the slope of a branch zero by making m equal to zero and see how it affects the position of the branch.

The squirrel runs on a fourth branch, this time one that is straight and horizontal. It whispers something to the branch and suddenly, the branch moves down lower. The squirrel tells you that it made the y-intercept of the branch negative by making b equal to a negative number. This is how you can do linear equations with your imagination: just make the y-intercept of a branch negative by making b equal to a negative number and see how it affects the position of the branch.

You are amazed by the squirrel’s magic and thank it for showing you how to do linear equations with your imagination. The squirrel smiles and invites you to try it yourself. You pick a branch and whisper a value for m and b to it. What happens next? (Comment What You Visualized Below)

• ?The story is meant to help the reader visualize and practice Linear Equations leveraging their imagination. By following the squirrel’s example's and applying them to different tree branches, the reader can learn how to visually calculate Linear Equations (y = mx + b) with their own minds without the need of costly materialistic physical graphs or external calculators!

Here story the sage told Jamie to help visually calculate with their imagination Quadratic Equations (ax^2 + bx + c = 0)

Quadratic Equations (ax^2 + bx + c = 0)

Imagine that you are in a carnival full of rides. You see a ferris wheel spinning around. The ferris wheel has a special ability: it can talk and even, change the shape and position of its circle. You decide to follow the ferris wheel and see what it does.

The ferris wheel spins on a circle that is symmetrical and centered. It whispers something then and suddenly, the circle becomes wider and flatter. The ferris wheel tells you that it changed the coefficient of x^2 by decreasing the value of a. This is how you can do quadratic equations with your imagination: just change the coefficient of x^2 by changing the value of a and see how it affects the shape of the circle.

The ferris wheel spins on another circle, this time one that is wider and flatter. It whispers something to the circle and suddenly, the circle moves to the right. The ferris wheel tells you that it changed the coefficient of x by increasing the value of b. This is how you can do quadratic equations with your imagination: just change the coefficient of x by changing the value of b and see how it affects the position of the circle.

The ferris wheel spins on a third circle, this time one that is moved to the right. It whispers something to the circle and suddenly, the circle moves up higher. The ferris wheel tells you that it changed the constant term by increasing the value of c. This is how you can do quadratic equations with your imagination: just change the constant term by changing the value of c and see how it affects the position of the circle.

The ferris wheel spins on a fourth circle, this time one that is moved up higher. It whispers something to the circle and suddenly, the circle becomes narrower and steeper. The ferris wheel tells you that it changed the coefficient of x^2 by increasing the value of a. This is how you can do quadratic equations with your imagination: just change the coefficient of x^2 by changing the value of a and see how it affects the shape of the circle.

You are amazed by the ferris wheel’s magic and thank it for showing you how to do quadratic equations with your imagination. The ferris wheel smiles and invites you to try it yourself. You pick a circle and whisper a value for a, b and c to it. What happens next? (Comment What You Visualized Below)

Here story the sage told Jamie to help visually calculate with their imagination Exponential Equations (y = ab^x)

Exponential Equations (y = ab^x)

Imagine that you are in a forest full of mushrooms. You see a fairy flying around, sprinkling dust on the mushrooms. The fairy has a special ability: it can change the size and growth rate of any mushroom it touches. You decide to follow the fairy and see what it does.

The fairy flies to a mushroom that is small and grows slowly. It sprinkles some dust on the mushroom and suddenly, the mushroom becomes bigger and grows faster. The fairy tells you that it changed the initial value of y by increasing the value of a. This is how you can do exponential equations with your imagination: just change the initial value of y by changing the value of a and see how it affects the size of the mushroom.

The fairy flies to another mushroom, this time one that is bigger and grows faster. It sprinkles some more dust on the mushroom and suddenly, the mushroom grows even faster and doubles in size every second. The fairy tells you that it changed the base of the exponent by increasing the value of b. This is how you can do exponential equations with your imagination: just change the base of the exponent by changing the value of b and see how it affects the growth rate of the mushroom.

The fairy flies to a third mushroom, this time one that grows very fast and doubles in size every second. It sprinkles some more dust on the mushroom and suddenly, the mushroom grows slower and halves in size every second. The fairy tells you that it changed the sign of the exponent by making x negative. This is how you can do exponential equations with your imagination: just change the sign of the exponent by making x negative and see how it affects the growth rate of the mushroom.

The fairy flies to a fourth mushroom, this time one that grows very slow and halves in size every second. It sprinkles some more dust on the mushroom and suddenly, the mushroom becomes smaller and grows slower. The fairy tells you that it changed the initial value of y by decreasing the value of a. This is how you can do exponential equations with your imagination: just change the initial value of y by changing the value of a and see how it affects the size of the mushroom.

You are amazed by the fairy’s magic and thank it for showing you how to do exponential equations with your imagination. The fairy smiles and invites you to try it yourself. You pick a mushroom and whisper a value for a, b and x to it. What happens next? (Comment What You Visualized Below)

Here story the sage told Jamie to help visually calculate with their imagination Trigonometric Equations (sin(x) = y)

Trigonometric Equations (sin(x) = y)

Imagine that you are in a beach full of waves. You see a dolphin jumping in and out of the water. The dolphin has a special ability: it can change the height and angle of its jumps. You decide to follow the dolphin and see what it does.

The dolphin jumps out of the water at a low height and a small angle. It whispers something to the water and suddenly, the water becomes calm and flat. The dolphin tells you that it made the sine of the angle zero by making x equal to zero. This is how you can do trigonometric equations with your imagination: just make the sine of the angle zero by making x equal to zero and see how it affects the height of the jump.

The dolphin jumps out of the water again, this time at a higher height and a bigger angle. It whispers something to the water and suddenly, the water becomes wavy and curved. The dolphin tells you that it made the sine of the angle one by making x equal to 90 degrees. This is how you can do trigonometric equations with your imagination: just make the sine of the angle one by making x equal to 90 degrees and see how it affects the height of the jump.

The dolphin jumps out of the water once more, this time at a lower height and a smaller angle. It whispers something to the water and suddenly, the water becomes wavy and curved in the opposite direction. The dolphin tells you that it made the sine of the angle negative one by making x equal to 270 degrees. This is how you can do trigonometric equations with your imagination: just make the sine of the angle negative one by making x equal to 270 degrees and see how it affects the height of the jump.

The dolphin jumps out of the water for the last time, this time at a medium height and a medium angle. It whispers something to the water and suddenly, the water becomes calm and flat again. The dolphin tells you that it made the sine of the angle zero again by making x equal to 360 degrees. This is how you can do trigonometric equations with your imagination: just make the sine of the angle zero again by making x equal to 360 degrees and see how it affects the height of the jump.

You are amazed by the dolphin’s magic and thank it for showing you how to do trigonometric equations with your imagination. The dolphin smiles and invites you to try it yourself. You pick an angle and whisper a value for x to it. What happens next? (Comment What You Visualized Below)

Here story the sage told Jamie to help visually calculate with their imagination Logarithmic Equations (loga(x) = y)

Logarithmic Equations (loga(x) = y)

Imagine that you are in a library full of books. You see a librarian sorting the books by their sizes. The librarian has a special ability: he can change the number and size of any book he touches. You decide to follow the librarian and see what he does.

The librarian picks up a book that has 10 pages and is 1 inch thick. He whispers something to the book and suddenly, the book has 100 pages and is 2 inches thick. The librarian tells you that he changed the value of x by multiplying it by 10. This is how you can do logarithmic equations with your imagination: just change the value of x by multiplying or dividing it by a constant and see how it affects the size of the book.

The librarian picks up another book, this time one that has 100 pages and is 2 inches thick. He whispers something to the book and suddenly, the book has 1000 pages and is 3 inches thick. The librarian tells you that he changed the value of x by multiplying it by 10 again. This is how you can do logarithmic equations with your imagination: just change the value of x by multiplying or dividing it by a constant and see how it affects the size of the book.

The librarian picks up a third book, this time one that has 1000 pages and is 3 inches thick. He whispers something to the book and suddenly, the book has 100 pages and is 2 inches thick again. The librarian tells you that he changed the value of x by dividing it by 10. This is how you can do logarithmic equations with your imagination: just change the value of x by multiplying or dividing it by a constant and see how it affects the size of the book.

The librarian picks up a fourth book, this time one that has 100 pages and is 2 inches thick. He whispers something to the book and suddenly, the book has 1 page and is 0 inches thick. The librarian tells you that he changed the value of x by dividing it by 100. This is how you can do logarithmic equations with your imagination: just change the value of x by multiplying or dividing it by a constant and see how it affects the size of the book.

You are amazed by the librarian’s magic and thank him for showing you how to do logarithmic equations with your imagination. The librarian smiles and invites you to try it yourself. You pick a book and whisper a value for x to it. What happens next? (Comment What You Visualized Below)

Here story the sage told Jamie to help visually calculate with their imagination Calculus Equations (derivatives, integrals)

Calculus Equations (derivatives, integrals)

Imagine that you are in a roller coaster park full of rides. You see a car moving along a track. The car has a special ability: it can change the shape and speed of any track it moves on. You decide to follow the car and see what it does.

The car moves on a track that is curved and smooth. It whispers something to the track and suddenly, the track becomes straight and flat. The car tells you that it found the derivative of the track by finding the slope of the curve at a point. This is how you can do calculus equations with your imagination: just find the derivative of a track by finding the slope of the curve at a point and see how it affects the shape of the track.

The car moves on another track, this time one that is straight and flat. It whispers something to the track and suddenly, the track becomes curved and smooth again. The car tells you that it found the integral of the track by finding the area under the curve from a point to another point. This is how you can do calculus equations with your imagination: just find the integral of a track by finding the area under the curve from a point to another point and see how it affects the shape of the track.

The car moves on a third track, this time one that is curved and smooth again. It whispers something to the track and suddenly, the track becomes steeper and faster. The car tells you that it changed the speed of the track by increasing the value of x. This is how you can do calculus equations with your imagination: just change the speed of a track by changing the value of x and see how it affects the speed of the car.

The car moves on a fourth track, this time one that is steeper and faster. It whispers something to the track and suddenly, the track becomes smoother and slower. The car tells you that it changed the speed of the track by decreasing the value of x. This is how you can do calculus equations with your imagination: just change the speed of a track by changing the value of x and see how it affects the speed of the car.

You are amazed by the car’s magic and thank it for showing you how to do calculus equations with your imagination. The car smiles and invites you to try it yourself. You pick a track and whisper a value for x to it. What happens next? (Comment What You Visualized Below)

These are the operations and equations you have learned how to do today through these guided meditative stories:

• Basic arithmetic operations (addition, subtraction, multiplication, division)
• Linear equations (y = mx + b)
• Quadratic equations (ax^2 + bx + c = 0)
• Exponential equations (y = ab^x)
• Trigonometric equations (sin(x) = y)
• Logarithmic equations (loga(x) = y)
• Calculus equations (derivatives, integrals)

Intermediate Topics found below.

• Differential Equations: the study of equations that involve derivatives and their applications in modeling real-world phenomena.
• Linear Algebra Operations: the study of vector spaces and linear transformations between them.
• Abstract Algebra Operations: the study of algebraic structures such as groups, rings, and fields.
• Real Analysis Operations: the study of the properties and behavior of real numbers and real-valued functions.
• Complex Analysis Operations: the study of complex numbers and functions of a complex variable.
• Topology Operations: the study of geometric objects and their properties, with a focus on concepts such as continuity and convergence.

Differential Equations:

Imagine that you are in a zoo full of animals. You see a rabbit population growing and shrinking over time. The rabbit population has a special ability: it can change its growth rate depending on the number of rabbits and the amount of food available. You decide to follow the rabbit population and see what it does.

The rabbit population starts with 10 rabbits and plenty of food. It whispers something to itself and suddenly, it grows faster and faster. The rabbit population tells you that it found the differential equation that models its growth rate by using the exponential function y = e^x. This is how you can do differential equations with your imagination: just find the differential equation that models the growth rate of a population by using a function that depends on the number of individuals and see how it affects the size of the population.

The rabbit population grows to 100 rabbits and less food. It whispers something to itself and suddenly, it grows slower and slower. The rabbit population tells you that it found the differential equation that models its growth rate by using the logistic function y = L/(1 + e^-kx). This is how you can do differential equations with your imagination: just find the differential equation that models the growth rate of a population by using a function that depends on the number of individuals and the carrying capacity of the environment and see how it affects the size of the population.

The rabbit population grows to 200 rabbits and very little food. It whispers something to itself and suddenly, it shrinks faster and faster. The rabbit population tells you that it found the differential equation that models its growth rate by using the predator-prey model y’ = ay - bxy. This is how you can do differential equations with your imagination: just find the differential equation that models the growth rate of a population by using a function that depends on the number of individuals, the number of predators and the interaction between them and see how it affects the size of the population.

The rabbit population shrinks to 50 rabbits and more food. It whispers something to itself and suddenly, it grows faster and faster again. The rabbit population tells you that it found the differential equation that models its growth rate by using the exponential function y = e^x again. This is how you can do differential equations with your imagination: just find the differential equation that models the growth rate of a population by using a function that depends on the number of individuals and see how it affects the size of the population.

You are amazed by the rabbit population’s magic and thank it for showing you how to do differential equations with your imagination. The rabbit population smiles and invites you to try it yourself. You pick a population and whisper a function to it. What happens next? (That's for you to decide!)

Linear Algebra Operations:

Imagine that you are in a museum full of paintings. You see a painter creating and modifying paintings. The painter has a special ability: he can change the dimensions and orientations of any painting he touches. You decide to follow the painter and see what he does.

The painter creates a painting that is a square with four equal sides and four right angles. He whispers something to the painting and suddenly, the painting becomes a rectangle with two pairs of equal sides and four right angles. The painter tells you that he changed the basis of the painting by stretching one of the sides. This is how you can do linear algebra operations with your imagination: just change the basis of a painting by stretching or shrinking one or more of its sides and see how it affects the shape of the painting.

The painter creates another painting, this time a rectangle with two pairs of equal sides and four right angles. He whispers something to the painting and suddenly, the painting becomes a parallelogram with two pairs of parallel sides and four angles. The painter tells you that he changed the basis of the painting by skewing one of the angles. This is how you can do linear algebra operations with your imagination: just change the basis of a painting by skewing or rotating one or more of its angles and see how it affects the shape of the painting.

The painter creates a third painting, this time a parallelogram with two pairs of parallel sides and four angles. He whispers something to the painting and suddenly, the painting becomes a triangle with three sides and three angles. The painter tells you that he changed the dimension of the painting by removing one of the sides. This is how you can do linear algebra operations with your imagination: just change the dimension of a painting by adding or removing one or more of its sides and see how it affects the shape of the painting.

The painter creates a fourth painting, this time a triangle with three sides and three angles. He whispers something to the painting and suddenly, the painting becomes a circle with one curved side and no angles. The painter tells you that he changed the dimension of the painting by adding an infinite number of sides. This is how you can do linear algebra operations with your imagination: just change the dimension of a painting by adding or removing an infinite number of sides and see how it affects the shape of the painting.

You are amazed by the painter’s magic and thank him for showing you how to do linear algebra operations with your imagination. The painter smiles and invites you to try it yourself. You pick a painting and whisper a basis or a dimension to it. What happens next? (Your reality becomes the Painting, and You become The Artist!)

Abstract Algebra Operations:

Imagine that you are in a playground full of toys. You see a child playing with different toys and combining them in various ways. The child has a special ability: he can change the properties and rules of any toy he touches. You decide to follow the child and see what he does.

The child plays with a set of blocks that have different shapes and colors. He whispers something to the blocks and suddenly, the blocks can only be combined if they have the same shape and color. The child tells you that he made the blocks into a group by defining a binary operation that has closure, associativity, identity, and inverse. This is how you can do abstract algebra operations with your imagination: just make a set of objects into a group by defining a binary operation that has these four properties and see how it affects the way you can combine the objects.

The child plays with another set of blocks, this time ones that have different numbers and symbols on them. He whispers something to the blocks and suddenly, the blocks can be combined in two different ways: by adding or multiplying the numbers on them. The child tells you that he made the blocks into a ring by defining two binary operations that have closure, associativity, identity, inverse, commutativity, and distributivity. This is how you can do abstract algebra operations with your imagination: just make a set of objects into a ring by defining two binary operations that have these six properties and see how it affects the way you can combine the objects.

The child plays with a third set of blocks, this time ones that have different fractions on them. He whispers something to the blocks and suddenly, the blocks can be combined in two different ways: by adding or multiplying the fractions on them. The child tells you that he made the blocks into a field by defining two binary operations that have closure, associativity, identity, inverse, commutativity, distributivity, and non-zero divisors. This is how you can do abstract algebra operations with your imagination: just make a set of objects into a field by defining two binary operations that have these seven properties and see how it affects the way you can combine the objects.

The child plays with a fourth set of blocks, this time ones that have different letters and arrows on them. He whispers something to the blocks and suddenly, the blocks can be combined in one way: by following the arrows from one letter to another. The child tells you that he made the blocks into a group again by defining a binary operation that has closure, associativity, identity, and inverse. This is how you can do abstract algebra operations with your imagination: just make a set of objects into a group by defining a binary operation that has these four properties and see how it affects the way you can combine the objects.

You are amazed by the child’s magic and thank him for showing you how to do abstract algebra operations with your imagination. The child smiles and invites you to try it yourself. You pick a set of objects and whisper a binary operation or two to it. What happens next? (Go Inside your mind to think "outside the box"!)

Real Analysis Operations:

Imagine that you are in a candy store full of sweets. You see a chef making and tasting different sweets. The chef has a special ability: he can change the ingredients and flavors of any sweet he touches. You decide to follow the chef and see what he does.

The chef makes a sweet that is a chocolate bar with nuts. He whispers something to the sweet and suddenly, the sweet becomes a chocolate bar with raisins. The chef tells you that he changed the sequence of the sweet by changing one of its terms. This is how you can do real analysis operations with your imagination: just change the sequence of a sweet by changing one or more of its terms and see how it affects the flavor of the sweet.

The chef makes another sweet, this time a gummy bear with sugar. He whispers something to the sweet and suddenly, the sweet becomes a gummy worm with sugar. The chef tells you that he changed the series of the sweet by changing its shape. This is how you can do real analysis operations with your imagination: just change the series of a sweet by changing its shape and see how it affects the flavor of the sweet.

The chef makes a third sweet, this time a cake with frosting. He whispers something to the sweet and suddenly, the sweet becomes a pie with filling. The chef tells you that he changed the function of the sweet by changing its ingredients. This is how you can do real analysis operations with your imagination: just change the function of a sweet by changing its ingredients and see how it affects the flavor of the sweet.

The chef makes a fourth sweet, this time a cookie with chocolate chips. He whispers something to the sweet and suddenly, the sweet becomes a brownie with chocolate chips. The chef tells you that he changed the limit of the sweet by changing its texture. This is how you can do real analysis operations with your imagination: just change the limit of a sweet by changing its texture and see how it affects the flavor of the sweet.

You are amazed by the chef’s magic and thank him for showing you how to do real analysis operations with your imagination. The chef smiles and invites you to try it yourself. You pick a sweet and whisper a sequence, series, function or limit to it. What happens next? (That's up to your mindful senses! ??)

Complex Analysis Operations:

Imagine that you are in a garden full of flowers. You see a bee flying and pollinating different flowers. The bee has a special ability: it can change the shape and smell of any flower it visits. You decide to follow the bee and see what it does.

The bee visits a flower that is round and smells sweet. It whispers something to the flower and suddenly, the flower becomes oval and smells sour. The bee tells you that it changed the complex number of the flower by changing its real part. This is how you can do complex analysis operations with your imagination: just change the complex number of a flower by changing its real or imaginary part and see how it affects the shape and smell of the flower.

The bee visits another flower, this time one that is oval and smells sour. It whispers something to the flower and suddenly, the flower becomes square and smells spicy. The bee tells you that it changed the complex number of the flower by changing its imaginary part. This is how you can do complex analysis operations with your imagination: just change the complex number of a flower by changing its real or imaginary part and see how it affects the shape and smell of the flower.

The bee visits a third flower, this time one that is square and smells spicy. It whispers something to the flower and suddenly, the flower becomes star-shaped and smells fruity. The bee tells you that it changed the function of the flower by changing its formula. This is how you can do complex analysis operations with your imagination: just change the function of a flower by changing its formula and see how it affects the shape and smell of the flower.

The bee visits a fourth flower, this time one that is star-shaped and smells fruity. It whispers something to the flower and suddenly, the flower becomes heart-shaped and smells floral. The bee tells you that it changed the function of the flower by changing its domain. This is how you can do complex analysis operations with your imagination: just change the function of a flower by changing its domain and see how it affects the shape and smell of the flower.

You are amazed by the bee’s magic and thank it for showing you how to do complex analysis operations with your imagination. The bee smiles and invites you to try it yourself. You pick a flower and whisper a complex number or a function to it. What happens next? (Become the Bee to find out ??)

Topology Operations:

Imagine that you are in a forest full of trees. You see a woodpecker pecking and carving different trees. The woodpecker has a special ability: he can change the shape and size of any tree he pecks. You decide to follow the woodpecker and see what he does.

The woodpecker pecks a tree that is a cylinder. He whispers something to the tree and suddenly, the tree becomes a donut. The woodpecker tells you that he changed the topology of the tree by making a hole in it. This is how you can do topology operations with your imagination: just change the topology of a tree by making or removing holes in it and see how it affects the shape of the tree.

The woodpecker pecks another tree, this time one that is a donut. He whispers something to the tree and suddenly, the tree becomes an infinity symbol. The woodpecker tells you that he changed the topology of the tree by making another hole in it. This is how you can do topology operations with your imagination: just change the topology of a tree by making or removing holes in it and see how it affects the shape of the tree.

The woodpecker pecks a third tree, this time one that is an infinity symbol. He whispers something to the tree and suddenly, the tree becomes a cylinder again. The woodpecker tells you that he changed the topology of the tree by removing both holes in it. This is how you can do topology operations with your imagination: just change the topology of a tree by making or removing holes in it and see how it affects the shape of the tree.

The woodpecker pecks a fourth tree, this time one that is a cylinder again. He whispers something to the tree and suddenly, the tree becomes smaller and smaller until it disappears. The woodpecker tells you that he changed the limit of the tree by shrinking it to a point. This is how you can do topology operations with your imagination: just change the limit of a tree by shrinking or expanding it to a point or an infinite space and see how it affects the size of the tree.

You are amazed by the woodpecker’s magic and thank him for showing you how to do topology operations with your imagination. The woodpecker smiles and invites you to try it yourself. You pick a tree and whisper a topology or a limit to it. What happens next? (That's dictated by how you chose to apply what you've learned to your own life, ordeals you and your loved one's face, and the solutions or remedies you generate for them!) ??

Advanced Topics found below. ( In Progress )

• Number theory: the study of the properties and patterns of natural numbers and integers.
• Combinatorics: the study of counting, arrangements, and combinations of discrete objects.
• Graph theory: the study of networks of nodes and edges and their applications in computer science, social sciences, and optimization problems.
• Cryptography: the study of encryption and decryption methods for secure communication and data protection.
• Statistics: the study of collecting, analyzing, and interpreting data using probability theory and inference methods.
• Mathematical modeling: the study of using mathematical tools and techniques to describe and simulate real-world phenomena and systems.

Number Theory

Imagine you are walking on a long and winding road that stretches infinitely in both directions. You notice that there are signs along the road that mark the distance from a certain point. The signs are numbered with natural numbers: 1, 2, 3, 4, and so on.

As you walk, you wonder what these numbers mean and how they relate to each other. You decide to explore some of the properties and patterns of these numbers using your imagination.

You start by looking for the signs that have only two factors: themselves and one. These are called prime numbers, and they are the building blocks of all other natural numbers. You notice that some of them are close together, like 2 and 3, or 11 and 13, while others are far apart, like 23 and 29, or 89 and 97. You wonder if there is a pattern or a formula that can predict the gaps between prime numbers.

You also notice that some signs have more factors than others. These are called composite numbers, and they can be written as products of prime numbers. For example, 12 is a composite number because it has six factors: 1, 2, 3, 4, 6, and 12. It can also be written as 2 x 2 x 3, which are all prime numbers. You wonder how many ways you can write a composite number as a product of prime numbers.

You continue walking and you see some signs that have special properties or patterns. For example, you see a sign that says 28. You realize that this is a perfect number because it is equal to the sum of its proper factors: 1 + 2 + 4 + 7 + 14 = 28. You wonder how many perfect numbers there are and how to find them.

You also see a sign that says 36. You realize that this is a square number because it is equal to the square of another natural number: 6 x 6 = 36. You wonder how many square numbers there are and how to find them.

You also see a sign that says 48. You realize that this is an abundant number because it is greater than the sum of its proper factors: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 76 > 48. You wonder how many abundant numbers there are and how to find them.

You continue walking on the long and winding road that stretches infinitely in both directions. You notice that there are signs along the road that mark the distance from a certain point. The signs are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see odd numbers and even numbers. You notice that odd numbers are those that have a remainder of 1 when divided by 2, while even numbers are those that have a remainder of 0. You also notice that odd numbers and even numbers alternate along the road: 1, 2, 3, 4, 5, 6, and so on.

You wonder if there is a way to tell if a number is odd or even without dividing it by 2. You decide to use your imagination to find a simple trick.

You start by looking at the last digit of each number. You notice that the last digit of odd numbers is always an odd number: 1, 3, 5, 7, or 9. You also notice that the last digit of even numbers is always an even number: 0, 2, 4, 6, or 8.

You realize that this is the trick you are looking for. You can tell if a number is odd or even by just looking at its last digit. If the last digit is odd, then the number is odd. If the last digit is even, then the number is even.

You feel happy that you have found a simple and effective way to visually calculate if a number is odd or even in your imagination. You feel curious to learn more about the properties and patterns of odd and even numbers. You feel confident in your ability to use this trick for any natural number you encounter.

You continue walking further then, you see triangular numbers and pentagonal numbers. You notice that triangular numbers are those that can be arranged in an equilateral triangle with dots, while pentagonal numbers are those that can be arranged in a regular pentagon with dots. For example, 3 is a triangular number because it can form a triangle with one dot at the top and two dots at the bottom, while 5 is a pentagonal number because it can form a pentagon with one dot at each vertex.

You wonder how many dots you need to make the next triangular or pentagonal number. You decide to use your imagination to find a simple formula.

You start by drawing the first few triangular and pentagonal numbers with dots on a piece of paper. You notice that each time you add a new row of dots to the bottom of the shape, you increase the number of dots by one more than the previous row. For example, to make a triangle with 6 dots, you add 3 dots to the bottom of a triangle with 3 dots. To make a pentagon with 12 dots, you add 4 dots to the bottom of a pentagon with 8 dots.

You realize that this is the formula you are looking for. You can find the next triangular or pentagonal number by adding one more dot than the number of dots in the last row. For example, to find the next triangular number after 6, you add 4 dots to get 10. To find the next pentagonal number after 12, you add 5 dots to get 17.

You feel proud that you have found a simple and effective way to visually calculate the next triangular or pentagonal number in your imagination. You feel curious to learn more about the properties and patterns of these numbers. You feel confident in your ability to use this formula for any triangular or pentagonal number you encounter.

You see Fibonacci numbers and Lucas numbers. You notice that Fibonacci numbers are those that follow a pattern where each number is the sum of the previous two numbers, starting with 1 and 1: 1, 1, 2, 3, 5, 8, 13, and so on. You also notice that Lucas numbers are similar to Fibonacci numbers, but they start with 1 and 3: 1, 3, 4, 7, 11, 18, 29, and so on.

You wonder what these numbers have to do with nature and art. You decide to use your imagination to find some examples.

You start by looking at the flowers along the road. You notice that some of them have petals that match the Fibonacci numbers. For example, you see a daisy with 13 petals, a sunflower with 34 petals, and a lily with 5 petals. You also notice that the seeds in the center of the sunflower form spirals that follow the Fibonacci sequence. You wonder how these flowers grow in such a way that they produce these numbers.

You also look at the trees along the road. You notice that some of them have branches that match the Lucas numbers. For example, you see a pine tree with 29 branches, a maple tree with 11 branches, and a palm tree with 4 branches. You also notice that the leaves on the branches form patterns that follow the Lucas sequence. You wonder how these trees grow in such a way that they produce these numbers.

You continue walking and you see more examples of Fibonacci and Lucas numbers in nature and art. You see shells and snails that have spirals that follow these sequences. You see paintings and sculptures that use these proportions to create harmony and beauty. You see buildings and monuments that use these ratios to create stability and elegance.

You feel amazed by the presence and influence of these numbers in nature and art. You feel curious to learn more about them and their applications in mathematics and other fields. You feel inspired by their beauty and elegance.

Imagine you are walking in a garden that stretches infinitely in all directions. You notice that there are flowers of different colors and shapes along the path. The flowers have labels that mark their height in centimeters. The labels are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see flowers that have palindromic petals and flowers that have narcissistic petals. You notice that flowers that have palindromic petals are those that have the same number of petals on the left and right sides, like roses or orchids. You also notice that flowers that have narcissistic petals are those that have the same number of petals as the sum of their height digits raised to the power of the number of digits, like lilies or daisies.

You wonder how to find the next flower that has palindromic or narcissistic petals. You decide to use your imagination to find a simple method.

You start by counting the petals of the first few flowers that have palindromic petals. You notice that each time you add one centimeter to their height, you increase the number of petals by an even number. For example, a flower with 11 petals is 2 centimeters tall, and a flower with 33 petals is 3 centimeters tall. To get from 11 to 33, you add 22, which is an even number.

You realize that this is the method you are looking for. You can find the next flower that has palindromic petals by adding an even number to the previous number of petals. For example, to find the next flower after the one with 33 petals, you add 44 to get 77. To find the next one after the one with 77 petals, you add 66 to get 143.

You then count the petals of the first few flowers that have narcissistic petals. You notice that each time you add one centimeter to their height, you increase the number of petals by a multiple of their height digits. For example, a flower with 5 petals is 2 centimeters tall, and a flower with 27 petals is 3 centimeters tall. To get from 5 to 27, you multiply 2 by 3 and add it to 5.

You realize that this is another method you are looking for. You can find the next flower that has narcissistic petals by multiplying their height digits and adding it to the previous number of petals. For example, to find the next flower after the one with 27 petals, you multiply 3 by 4 and add it to 27 to get 39. To find the next one after the one with 39 petals, you multiply 4 by 5 and add it to 39 to get 59.

You feel glad that you have found simple and effective ways to visually calculate the next flower that has palindromic or narcissistic petals in your imagination. You feel curious to learn more about these flowers and their petal patterns. You feel intrigued by their symmetry and uniqueness.

Continue to Imagine you are walking in a forest that stretches infinitely in all directions. You notice that there are trees of different shapes and sizes along the path. The trees have tags that mark their age in years. The tags are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see trees that have perfect square rings and trees that have perfect cube rings. You notice that trees that have perfect square rings are those that grow one ring every year, like oaks or maples. You also notice that trees that have perfect cube rings are those that grow three rings every year, like pines or firs.

You wonder how to find the next tree that has perfect square or perfect cube rings. You decide to use your imagination to find a simple formula.

You start by counting the rings of the first few trees that have perfect square rings. You notice that each time you add one year to their age, you increase the number of rings by an odd number. For example, a tree with 4 rings is 2 years old, and a tree with 9 rings is 3 years old. To get from 4 to 9, you add 5, which is an odd number.

You realize that this is the formula you are looking for. You can find the next tree that has perfect square rings by adding an odd number to the previous number of rings. For example, to find the next tree after the one with 9 rings, you add 7 to get 16. To find the next one after the one with 16 rings, you add 9 to get 25.

You then count the rings of the first few trees that have perfect cube rings. You notice that each time you add one year to their age, you increase the number of rings by a multiple of three. For example, a tree with 8 rings is 2 years old, and a tree with 27 rings is 3 years old. To get from 8 to 27, you add 12, which is a multiple of three.

You realize that this is another formula you are looking for. You can find the next tree that has perfect cube rings by adding a multiple of three to the previous number of rings. For example, to find the next tree after the one with 27 rings, you add 24 to get 51. To find the next one after the one with 51 rings, you add 27 to get 78.

You feel happy that you have found simple and effective ways to visually calculate the next tree that has perfect square or perfect cube rings in your imagination. You feel curious to learn more about these trees and their growth patterns. You feel amazed by their harmony and diversity.

Imagine you are walking in a field that stretches infinitely in all directions. You notice that there are animals of different kinds and sizes along the path. The animals have tags that mark their weight in kilograms. The tags are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see animals that have prime power fur and animals that have highly composite fur. You notice that animals that have prime power fur are those that have the same number of hairs as a power of a prime number, like rabbits or squirrels. You also notice that animals that have highly composite fur are those that have more factors than any smaller number, like sheep or cows.

You wonder how to find the next animal that has prime power or highly composite fur. You decide to use your imagination to find a simple formula.

You start by counting the hairs of the first few animals that have prime power fur. You notice that each time you add one kilogram to their weight, you multiply the number of hairs by a prime number. For example, an animal with 8 hairs is 2 kilograms heavy, and an animal with 32 hairs is 3 kilograms heavy. To get from 8 to 32, you multiply by 4, which is 2^2.

You realize that this is the formula you are looking for. You can find the next animal that has prime power fur by multiplying the previous number of hairs by a power of a prime number. For example, to find the next animal after the one with 32 hairs, you multiply by 9, which is 3^2, to get 288. To find the next one after the one with 288 hairs, you multiply by 16, which is 4^2, to get 4608.

You then count the factors of the first few animals that have highly composite fur. You notice that each time you add one kilogram to their weight, you increase the number of factors by more than any smaller number. For example, an animal with 12 factors is 4 kilograms heavy, and an animal with 18 factors is 6 kilograms heavy. To get from 12 to 18, you add 6 factors, which is more than any smaller number.

You realize that this is another formula you are looking for. You can find the next animal that has highly composite fur by adding more factors than any smaller number to the previous number of factors. For example, to find the next animal after the one with 18 factors, you add 12 factors to get 30. To find the next one after the one with 30 factors, you add 24 factors to get 54.

You feel content that you have found simple and effective ways to visually calculate the next animal that has prime power or highly composite fur in your imagination. You feel curious to learn more about these animals and their fur patterns. You feel astonished by their variety and complexity.

Imagine you are walking in a desert that stretches infinitely in all directions. You notice that there are cacti of different shapes and sizes along the path. The cacti have labels that mark their height in meters. The labels are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see cacti that have Mersenne prime spines and cacti that have Fermat prime spines. You notice that cacti that have Mersenne prime spines are those that have one less spine than a power of two, like 3 or 31. You also notice that cacti that have Fermat prime spines are those that have one more spine than a power of two raised to another power of two, like 5 or 17.

You wonder how to find the next cactus that has Mersenne prime or Fermat prime spines. You decide to use your imagination to find a simple formula.

You start by counting the spines of the first few cacti that have Mersenne prime spines. You notice that each time you double their height, you double their number of spines and subtract one. For example, a cactus with 3 spines is 1 meter tall, and a cactus with 7 spines is 2 meters tall. To get from 3 to 7, you double 3 and subtract 1.

You realize that this is the formula you are looking for. You can find the next cactus that has Mersenne prime spines by doubling the previous number of spines and subtracting one. For example, to find the next cactus after the one with 7 spines, you double 7 and subtract 1 to get 13. To find the next one after the one with 13 spines, you double 13 and subtract 1 to get 25.

You then count the spines of the first few cacti that have Fermat prime spines. You notice that each time you square their height, you square their number of spines and add one. For example, a cactus with 5 spines is 2 meters tall, and a cactus with 17 spines is 4 meters tall. To get from 5 to 17, you square 5 and add 1.

You realize that this is another formula you are looking for. You can find the next cactus that has Fermat prime spines by squaring the previous number of spines and adding one. For example, to find the next cactus after the one with 17 spines, you square 17 and add 1 to get 290. To find the next one after the one with 290 spines, you square 290 and add 1 to get 84101.

You feel excited that you have found simple and effective ways to visually calculate the next cactus that has Mersenne prime or Fermat prime spines in your imagination. You feel curious to learn more about these cacti and their spine patterns. You feel awed by their rarity and difficulty.

You realize that you have just learned one of the fascinating concepts of number theory, the study of the properties and patterns of natural numbers and integers.

Imagine you are walking in a meadow that stretches infinitely in all directions. You notice that there are butterflies of different colors and shapes along the path. The butterflies have labels that mark their wingspan in centimeters. The labels are numbered with natural numbers: 1, 2, 3, 4, and so on.

You see butterflies that are happy and butterflies that are sad. You notice that butterflies that are happy are those that have a wingspan that eventually reaches 1 when you repeatedly replace it with the sum of the squares of its digits. For example, a butterfly with a wingspan of 19 is happy because 19 -> 1^2 + 9^2 = 82 -> 8^2 + 2^2 = 68 -> 6^2 + 8^2 = 100 -> 1^2 + 0^2 + 0^2 = 1. You also notice that butterflies that are sad are those that have a wingspan that never reaches 1 but gets stuck in a loop of numbers. For example, a butterfly with a wingspan of 4 is sad because 4 -> 4^2 = 16 -> 1^2 + 6^2 = 37 -> 3^2 + 7^2 = 58 -> … -> 4.

You wonder how to find the next butterfly that is happy or sad. You decide to use your imagination to find a simple method.

You start by looking at the labels of the first few butterflies that are happy. You notice that they have wingspans that end with either 1, 7, or 9. For example, you see butterflies with wingspans of 1, 7, 10, 13, and so on. You also notice that they have wingspans that do not contain any zeros or fours. For example, you do not see any butterflies with wingspans of 20, 40, or 104.

You realize that this is the method you are looking for. You can find the next butterfly that is happy by choosing a number that ends with either 1, 7, or 9 and does not contain any zeros or fours. For example, to find the next butterfly after the one with a wingspan of 13, you can choose either 19 or 23. To find the next one after the one with a wingspan of 19, you can choose either 28 or 31.

You then look at the labels of the first few butterflies that are sad. You notice that they have wingspans that end with either zero or four. For example, you see butterflies with wingspans of zero, four, ten, fourteen, and so on. You also notice that they have wingspans that contain at least one zero or four. For example, you see butterflies with wingspans of twenty-four, forty-two, or one hundred and four.

You realize that this is another method you are looking for. You can find the next butterfly that is sad by choosing a number that ends with either zero or four and contains at least one zero or four. For example, to find the next butterfly after the one with a wingspan of fourteen, you can choose either twenty or sixteen. To find the next one after the one with a wingspan of twenty, you can choose either twenty-four or forty.

You feel curious that you have found simple and effective ways to visually calculate the next butterfly that is happy or sad in your imagination. You feel curious to learn more about these butterflies and their wing patterns. You feel puzzled by their happiness and sadness.

You realize that you have just learned one of the intriguing concepts of number theory, the study of the properties and patterns of natural numbers and integers.

Combinatorics, to be continued...

I Hope you enjoyed!

Thank you for expanding your mind!

We'll see you next time! ??