The magic of Fibonacci numbers | Arthur Benjamin | TED

Summary

The speaker explores the importance of learning mathematics, emphasizing its roles in calculation, application, and inspiration. While math is often taught for practical reasons like future tests, the speaker advocates for appreciating its beauty and the patterns it reveals. Using Fibonacci numbers as an example, they demonstrate how math can be fascinating and how these patterns can inspire logical, critical, and creative thinking. The talk concludes by urging that mathematics should be seen not just as solving problems but as a way to understand the world.

Takeaways

• 📚 Mathematics is studied for three key reasons: calculation, application, and inspiration.
• 🧠 Mathematics helps us learn to think logically, critically, and creatively.
• 🎓 Much of the math taught in schools isn't effectively motivated, leading students to question its relevance.
• 🌟 Mathematics can be appreciated for its beauty, fun, and ability to excite the mind.
• 🌸 Fibonacci numbers are a powerful example of math in nature, appearing in flowers, sunflowers, and pineapples.
• 🔢 The Fibonacci sequence is easy to understand and can be used to demonstrate interesting patterns.
• 📐 The speaker demonstrates a pattern where the sum of squares of Fibonacci numbers equals the product of certain Fibonacci numbers.
• 🔍 The Golden Ratio, approximately 1.618, emerges from the Fibonacci sequence and has fascinated many throughout history.
• 🎨 Mathematics has a beautiful side that often gets overlooked in traditional education.
• 📝 Mathematics is not just about solving for x, but also about understanding why.

Q & A

Why do we learn mathematics according to the speaker?

We learn mathematics for three main reasons: calculation, application, and inspiration. The speaker emphasizes that although inspiration is often the least prioritized, it is an important aspect of mathematics.

How does the speaker describe mathematics?

The speaker describes mathematics as the science of patterns, which helps us learn to think logically, critically, and creatively.

What issue does the speaker highlight regarding the way mathematics is taught in schools?

The speaker points out that mathematics in schools is often not effectively motivated. When students ask why they are learning certain mathematical concepts, they are often told it is for future tests or upcoming classes, rather than for its beauty or excitement.

How does the speaker suggest mathematics could be taught differently?

The speaker suggests that mathematics should occasionally be taught simply because it is fun, beautiful, or intellectually stimulating. This approach would help students see the inspirational side of mathematics.

What example does the speaker use to illustrate the beauty of mathematics?

The speaker uses the Fibonacci numbers as an example. Fibonacci numbers are simple to understand and appear frequently in nature, but they also display beautiful mathematical patterns that can be explored and appreciated.

What is the significance of Fibonacci numbers in nature?

Fibonacci numbers appear in various natural phenomena, such as the number of petals on flowers and the arrangement of spirals on sunflowers and pineapples.

What pattern does the speaker highlight when adding the squares of Fibonacci numbers?

The speaker shows that when you add the squares of consecutive Fibonacci numbers, you get another Fibonacci-related pattern. For example, 1 + 1 = 2, 1 + 4 = 5, 4 + 9 = 13, and so on.

How does the speaker explain the relationship between the squares of Fibonacci numbers and geometry?

The speaker explains the relationship by drawing a geometric picture of squares whose side lengths are Fibonacci numbers. The combined area of these squares forms a larger rectangle, whose area can also be calculated by multiplying two Fibonacci numbers.

What is the connection between the Fibonacci sequence and the Golden Ratio?

As the Fibonacci sequence progresses, the ratio of consecutive Fibonacci numbers approaches the Golden Ratio, approximately 1.618, which has fascinated mathematicians, scientists, and artists for centuries.

How does the speaker summarize the broader importance of learning mathematics?

The speaker summarizes that mathematics is not just about solving for variables, like 'x,' but also about understanding the underlying reasons and logic behind the concepts. Learning mathematics teaches us how to think.

Outlines

00:00

🎓 Why We Study Mathematics: Beyond Just Calculation

This paragraph discusses the reasons for learning mathematics, emphasizing calculation, application, and inspiration. Mathematics is described as the science of patterns that teaches logical, critical, and creative thinking. However, it is often not effectively motivated in schools, where students may wonder about its relevance. The speaker advocates for appreciating mathematics for its fun, beauty, and mental stimulation. The Fibonacci sequence is introduced as an example of this, illustrating its simplicity in calculation, frequent appearance in nature, and the beautiful patterns it creates.

05:01

🔍 Discovering the Beauty of Fibonacci and the Golden Ratio

This paragraph delves deeper into the Fibonacci sequence, explaining how the sum of squares of the first few Fibonacci numbers equals the product of two Fibonacci numbers. The concept is demonstrated through the creation of a rectangle using squares of Fibonacci dimensions. The paragraph concludes by connecting the Fibonacci sequence to the Golden Ratio, a number that has captivated mathematicians, scientists, and artists for centuries. The speaker emphasizes that mathematics is not just about solving equations but understanding the underlying beauty and logic, advocating for a broader appreciation of mathematics beyond mere calculations.

Keywords

💡Mathematics

Mathematics is the science of patterns and logical reasoning. In the video, it is portrayed as a subject that goes beyond mere calculation and application; it is also a source of inspiration and beauty. The speaker emphasizes that mathematics should not just be seen as a tool for solving problems or passing tests, but also as a way to develop critical and creative thinking skills.

💡Fibonacci Numbers

Fibonacci numbers are a sequence where each number is the sum of the two preceding ones. The video highlights their simplicity in calculation and their surprising appearances in nature and art. The speaker uses Fibonacci numbers to demonstrate the beauty of mathematical patterns, showing how they appear in flower petals, sunflower spirals, and even in abstract mathematical equations.

💡Golden Ratio

The Golden Ratio is a special number, approximately 1.618, which is often found in nature, art, and architecture. In the video, the Golden Ratio is linked to the Fibonacci sequence, where the ratio of consecutive Fibonacci numbers approximates this value. The speaker uses it as an example of the deep, inherent beauty in mathematics that transcends simple calculation.

💡Patterns

Patterns in mathematics refer to regularities or structures that can be observed and analyzed. The speaker argues that mathematics is fundamentally about discovering and understanding patterns. The Fibonacci sequence is used as a prime example, showing how these numbers form patterns in nature and within mathematical operations, such as the summation of squares.

💡Inspiration

Inspiration in the context of the video refers to the motivational and aesthetic aspects of mathematics that are often overlooked in traditional education. The speaker suggests that mathematics can inspire creativity and wonder, as demonstrated by the beauty of Fibonacci patterns and the elegance of mathematical proofs. This inspirational side of mathematics is contrasted with the more common focus on calculation and application.

💡Calculation

Calculation refers to the process of computing or determining something mathematically. The speaker mentions that much of school mathematics focuses on this aspect, such as adding numbers or finding squares. However, they argue that while calculation is important, it should not overshadow the inspirational and creative aspects of mathematics.

💡Application

Application in mathematics refers to the practical use of mathematical concepts to solve real-world problems. The video illustrates this with examples of Fibonacci numbers appearing in nature, like the number of petals on a flower or the spirals on a sunflower. The speaker emphasizes that understanding how mathematics applies to the world around us is crucial, but it should be balanced with an appreciation for mathematics as an art form.

💡Logical Thinking

Logical thinking involves reasoning in a structured and coherent manner. The video suggests that one of the primary reasons for learning mathematics is to develop this type of thinking. By studying patterns and relationships in mathematics, students can improve their ability to think logically and solve complex problems both in and out of mathematical contexts.

💡Critical Thinking

Critical thinking in mathematics involves analyzing and evaluating information or arguments in a rigorous way. The speaker highlights that mathematics teaches us to think critically by challenging us to understand why certain patterns and relationships hold true. For example, the video explains why the squares of Fibonacci numbers follow a particular pattern, encouraging viewers to engage in deeper analysis.

💡Creative Thinking

Creative thinking in mathematics is about approaching problems and patterns in innovative and non-traditional ways. The video advocates for the inclusion of creative thinking in mathematical education, suggesting that it allows students to explore the beauty of mathematics. The exploration of Fibonacci patterns and the discovery of unexpected relationships, such as in the summation of squares, are examples of how mathematics can inspire creativity.

Highlights

Mathematics is essential for calculation, application, and inspiration, though inspiration is often given the least time.

Mathematics teaches us to think logically, critically, and creatively, yet many students struggle to see its relevance.

Fibonacci numbers are a favorite example that demonstrates the beauty and patterns in mathematics.

The Fibonacci sequence is simple to understand: each number is the sum of the two preceding ones.

Fibonacci numbers frequently appear in nature, such as the number of petals on a flower or the spirals on a sunflower.

The speaker highlights a pattern where the sum of the squares of Fibonacci numbers relates to other Fibonacci numbers.

Adding the squares of Fibonacci numbers creates a pattern that reveals hidden Fibonacci numbers within the sums.

The area of a rectangle formed by squares with Fibonacci side lengths can be calculated in two ways, showing the connection between the sums of squares and Fibonacci numbers.

Dividing consecutive Fibonacci numbers gives a ratio that approaches the Golden Ratio, a number that has fascinated people for centuries.

The Golden Ratio, approximately 1.618, is found in many aspects of mathematics, science, and art.

The speaker emphasizes that mathematics should not just be about calculation but also about finding beauty and understanding patterns.

Mathematics is about more than solving for x; it's also about figuring out why things work the way they do.

The speaker encourages a broader appreciation of mathematics beyond its practical applications, highlighting its role in developing thinking skills.

The Fibonacci sequence and its applications are used as an example of how mathematics can be both practical and inspirational.

The talk concludes by reiterating that mathematics has a beautiful side that deserves more attention in education.

Transcripts

So why do we learn mathematics?

Essentially, for three reasons:

calculation,

application,

and last, and unfortunately least

in terms of the time we give it,

inspiration.

Mathematics is the science of patterns,

and we study it to learn how to think logically,

critically and creatively,

but too much of the mathematics that we learn in school

is not effectively motivated,

and when our students ask,

"Why are we learning this?"

then they often hear that they'll need it

in an upcoming math class or on a future test.

But wouldn't it be great

if every once in a while we did mathematics

simply because it was fun or beautiful

or because it excited the mind?

Now, I know many people have not

had the opportunity to see how this can happen,

so let me give you a quick example

with my favorite collection of numbers,

the Fibonacci numbers. (Applause)

Yeah! I already have Fibonacci fans here.

That's great.

Now these numbers can be appreciated

in many different ways.

From the standpoint of calculation,

they're as easy to understand

as one plus one, which is two.

Then one plus two is three,

two plus three is five, three plus five is eight,

and so on.

Indeed, the person we call Fibonacci

was actually named Leonardo of Pisa,

and these numbers appear in his book "Liber Abaci,"

which taught the Western world

the methods of arithmetic that we use today.

In terms of applications,

Fibonacci numbers appear in nature

surprisingly often.

The number of petals on a flower

is typically a Fibonacci number,

or the number of spirals on a sunflower

or a pineapple

tends to be a Fibonacci number as well.

In fact, there are many more applications of Fibonacci numbers,

but what I find most inspirational about them

are the beautiful number patterns they display.

Let me show you one of my favorites.

Suppose you like to square numbers,

and frankly, who doesn't? (Laughter)

Let's look at the squares

of the first few Fibonacci numbers.

So one squared is one,

two squared is four, three squared is nine,

five squared is 25, and so on.

Now, it's no surprise

that when you add consecutive Fibonacci numbers,

you get the next Fibonacci number. Right?

That's how they're created.

But you wouldn't expect anything special

to happen when you add the squares together.

But check this out.

One plus one gives us two,

and one plus four gives us five.

And four plus nine is 13,

nine plus 25 is 34,

and yes, the pattern continues.

In fact, here's another one.

Suppose you wanted to look at

adding the squares of the first few Fibonacci numbers.

Let's see what we get there.

So one plus one plus four is six.

Add nine to that, we get 15.

Add 25, we get 40.

Add 64, we get 104.

Now look at those numbers.

Those are not Fibonacci numbers,

but if you look at them closely,

you'll see the Fibonacci numbers

buried inside of them.

Do you see it? I'll show it to you.

Six is two times three, 15 is three times five,

40 is five times eight,

two, three, five, eight, who do we appreciate?

(Laughter)

Fibonacci! Of course.

Now, as much fun as it is to discover these patterns,

it's even more satisfying to understand

why they are true.

Let's look at that last equation.

Why should the squares of one, one, two, three, five and eight

add up to eight times 13?

I'll show you by drawing a simple picture.

We'll start with a one-by-one square

and next to that put another one-by-one square.

Together, they form a one-by-two rectangle.

Beneath that, I'll put a two-by-two square,

and next to that, a three-by-three square,

beneath that, a five-by-five square,

and then an eight-by-eight square,

creating one giant rectangle, right?

Now let me ask you a simple question:

what is the area of the rectangle?

Well, on the one hand,

it's the sum of the areas

of the squares inside it, right?

Just as we created it.

It's one squared plus one squared

plus two squared plus three squared

plus five squared plus eight squared. Right?

That's the area.

On the other hand, because it's a rectangle,

the area is equal to its height times its base,

and the height is clearly eight,

and the base is five plus eight,

which is the next Fibonacci number, 13. Right?

So the area is also eight times 13.

Since we've correctly calculated the area

two different ways,

they have to be the same number,

and that's why the squares of one, one, two, three, five and eight

add up to eight times 13.

Now, if we continue this process,

we'll generate rectangles of the form 13 by 21,

21 by 34, and so on.

Now check this out.

If you divide 13 by eight,

you get 1.625.

And if you divide the larger number by the smaller number,

then these ratios get closer and closer

to about 1.618,

known to many people as the Golden Ratio,

a number which has fascinated mathematicians,

scientists and artists for centuries.

Now, I show all this to you because,

like so much of mathematics,

there's a beautiful side to it

that I fear does not get enough attention

in our schools.

We spend lots of time learning about calculation,

but let's not forget about application,

including, perhaps, the most important application of all,

learning how to think.

If I could summarize this in one sentence,

it would be this:

Mathematics is not just solving for x,

it's also figuring out why.

Thank you very much.

(Applause)