Have you ever worked through a proof, understood and confirmed each step, yet still not believed the theorem? You realize *that *the theorem is true, but not *why *it is true.

To see the same contrast in a familiar example, imagine learning that your child has a fever and hearing the temperature in Fahrenheit or Celsius degrees, whichever is less familiar. In my everyday experience, temperatures are mostly in Fahrenheit. When I hear about a temperature of (40^{◦}C), I therefore react in two stages:

- I convert (40^{◦}C) to Fahrenheit: (40 imes 1.8 + 32 = 104.)
- I react: “Wow, (104^{◦}F). That’s dangerous! Get thee to a doctor!”

The Celsius temperature, although symbolically equivalent to the Fahrenheit temperature, elicits no reaction. My danger sense activates only after the temperature conversion connects the temperature to my experience.

A symbolic description, whether a proof or an unfamiliar temperature, is unconvincing compared to an argument that speaks to our perceptual system. The reason lies in how our brains acquired the capacity for symbolic reasoning. (See *Evolving Brains *[2] for an illustrated, scholarly history of the brain.) Symbolic, sequential reasoning requires language, which has evolved for only (10^{5}) yr. Although (10^{5}) yr spans many human lifetimes, it is an evolutionary eyeblink. In particular, it is short compared to the time span over which our perceptual hardware has evolved: For several hundred million years, organisms have refined their capacities for hearing, smelling, tasting, touching, and seeing.

Evolution has worked 1000 times longer on our perceptual abilities than on our symbolic-reasoning abilities. Compared to our perceptual hardware, our symbolic, sequential hardware is an ill-developed latecomer. Not surprisingly, our perceptual abilities far surpass our symbolic abilities. Even an apparently high-level symbolic activity such as playing grand master chess uses mostly perceptual hardware [16]. *Seeing *an idea conveys to us a depth of understanding that a symbolic description of it cannot easily match.

Multiple problems

**Problem 4.1 Computers versus people**

At tasks like expanding ((x + 2y)^{50}), computers are much faster than people. At tasks like recognizing faces or smells, even young children are much faster than current computers. How do you explain these contrasts?

**Problem 4.2 Linguistic evidence for the importance of perception**

In your favorite language(s), think of the many sensory synonyms for under- standing (for example, grasping).

## Adding odd numbers

To illustrate the value of pictures, let’s find the sum of the first (n) odd numbers (also the subject of Problem 2.25):

[S_{n} = underbrace{1 + 3 + 5 + ... + (2n - 1).}_{n ext{terms}} label{4.1}]

Easy cases such as (n) = 1, 2, or 3 lead to the conjecture that (S_{n} = n^{2}). But how can the conjecture be proved? The standard symbolic method is proof by induction:

1. Verify that (S_{n}) = (n^{2}) for the *base case* (n = 1). In that case, (S_{1}) is 1, as is (n_{2}), so the base case is verified.

2. Make the *induction hypothesis: *Assume that (S_{m}) = (m^{2}) for m less than or equal to a maximum value n. For this proof, the following, weaker induction hypothesis is sufficient:

[sum_{1}^{n} (2k - 1) = n^{2} label{4.2}]

In other words, we assume the theorem only in the case that (m = n).

3. Perform the *induction step: *Use the induction hypothesis to show that (S_{n+1}) = (n + 1)(^{2}). The sum (S_{n+1}) splits into two pieces:

[S_{n+1}=sum_{1}^{n+1}(2 k-1)=(2 n+1)+sum_{1}^{n}(2 k-1)label{4.3}]

Thanks to the induction hypothesis, the sum on the right is (n^{2}). Thus

[S_{n+1} = (2n + 1) + n^{2}, label{4.4}]

which is ((n + 1)^{2}); and the theorem is proved.

Although these steps prove the theorem, *why *the sum (S_{n}) ends up as (n^{2}) still feels elusive.

That missing understanding the kind of gestalt insight described by Wertheimer [48] requires a pictorial proof. Start by drawing each odd number as an L-shaped puzzle piece:

[label{4.5}]Question

How do these pieces fit together?

Then compute (S_{n}) by fitting together the puzzle pieces as follows:

[label{4.6}]Each successive odd number each piece extends the square by 1 unit in height and width, so the (n) terms build an (n imes n) square. [Or is it an ((n − 1) imes (n − 1)) square?] Therefore, their sum is (n^{2}). After grasping this pictorial proof, you cannot forget why adding up the first n odd numbers produces (n^{2}).

Multiple problems

**Problem 4.3 Triangular numbers**

Draw a picture or pictures to show that

[1 + 2 + 3 + ··· + n + ··· + 3 + 2 + 1 = n^{2}. label{4.7}]

Then show that

[1 + 2 + 3 + ··· + n = frac{n(n+1)}{2}.label{4.8}]

**Problem 4.4 Three dimensions**

Draw a picture to show that

[sum_{0}^{n} (3k^{2} + 3k + 1) = (n + 1)^{3}. label{4.9}]

Give pictorial explanations for the 1 in the summand (3k^{2} + 3k + 1); for the 3 and the (k^{2}) in (3k^{2}); and for the 3 and the k in 3k.

## ODD AND EVENNUMBERS

I T IS POSSIBLE TO CLASSIFY natural numbers in many different ways. Square numbers, prime numbers, powers of 10. But the most fundamental classification is odd or even.

The demonstrations in this lesson could easily be done with algebra&mdashthat symbolic calculator. But this is arithmetic, and we can deal with numbers themselves.

A natural number is not simply a symbol that obeys formal rules.

We say that a number is even if it can be "evenly" divided into two equal parts:

A number is odd if it cannot.

In terms of division, an even number is divisible by 2, but an odd number is not. There is a remainder of 1.

It is easy to see that the sum of two even number is even:

The sum of two odd numbers is also even.

Only the sum of an odd number and an even number is odd.

We call a number a square number when it can take the form of a square.

How is a square number produced?

By adding consecutive odd numbers.

To the first odd number 1 we add the odd number 3 to produce 4.

To 4 we add 5 to produce 9.

To 9 we add 7 to produce 16.

To 16 we add 9 to produce 25.

Every square number is a sum of consecutive odd numbers.

1. | ||

1 + 3 | = | 4. |

1 + 3 + 5 | = | 9. |

1 + 3 + 5 + 7 | = | 16. |

1 + 3 + 5 + 7 + 9 | = | 25. |

An even square is the sum of an even number of odd numbers.

An odd square is the sum of an odd number of odd numbers.

For, the sum of any two consecutive odd numbers is even.

Therefore the sum of an odd number of odd numbers will be odd.

As for a square number being a number multiplied by itself, that follows from it being in a square array.

5 × 5 is 5 repeatedly added five times. We see that in the five horizontal rows.

Now, when the odd number that is added to the previous square is itself a square, then

the sum of two square numbers will equal a square number.

3-4-5 is called a Pythagorean triple. They are three whole numbers such that the sum of the squares of two of them is equal to the square of the third.

For every odd square, then, there will be a Pythagorean triple.

1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 | 256 | |

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 |

The next odd square is 25 . And we see that

The Pythagorean triple is 5-12-13.

Because 5-12-13 have no common divisors, they are said to be a primitive triple. So are 3-4-5.

In every primitive triple, two numbers are odd and one is even. In our examples, an odd square was added to an even square to produce an odd square. The sum two even squares producing an even square would not be a primitive triple.

Now the next odd square is 49 . What square number will it be added to? What Pythagorean triple will be produced?

To answer, consider that each odd number is 1 more than an even number. Let us call that even number the even part . Thus the even part of 3 is 2. The even part of 5 is 4. And so on.

Each odd number will be added to the square of half of its even part.

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 |

The even part | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 |

Half | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

The even parts are the sequence of even numbers.

The halves are the sequence of the numbers that are squared.

1 2 , 2 2 , 3 2 , 4 2 . And so on.

Beginning with 3, each odd number is added to the previous square.

1 + 3 | = | 4. |

3 is added to the square of 1. | ||

1 + 3 + 5 | = | 9. |

5 is added to the square of 2. | ||

1 + 3 + 5 + 7 | = | 25. |

7 is added to the square of 3. |

Each odd number will be added to the square of half of its even part.

As for the odd square 49, which is the square of 7:

49 will be added to the square of 24, producing the square of 25. The Pythagorean triple is 7-24-25.

Problem. The next odd square is 81. What Pythagorean triple will it produce?

To see the answer, pass your mouse over the colored area.

To cover the answer again, click "Refresh" ("Reload").

81 will be added to the square of 40.

The Pythagorean triple is 9-40-41..

Odd and even are how we classify numbers upon division by 2. Classifying numbers upon division by 4 leads to interesting results.

Thus a number will either be divisible by 4, or it will have a remainder of 1, or 2, or 3.

In other words, a number will either be a multiple of 4, or 1 more than a multiple, or 2 more, or 3 more&mdashwhich is to say, 1 less.

2 4 6 8 10 12 14 16 18 20

All multiples of 4 are even. A number two more than a multiple of 4 will also be even, because that is the next even number.

1 3 5 7 9 11 13 15 17 19 21

Odd numbers are either 1 more than a multiple of 4 or 1 less, and they alternate. If we count 0 as a multiple of 4 (0 = 0 × 4), then 1 is 1 more than a multiple of 4. 3 is 1 less. 5 is 1 more. 7 is 1 less. And so on.

Which odd numbers will be 1 more, and which will be 1 less?

1 is the 1st odd number. 3 is the 2nd. 5 is the 3rd. And so on.

Each number in an odd position&mdash1st, 3rd, 5th, etc.&mdashwill be 1 more than a multiple of 4, which are every 4 numbers beginning with 1:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, . . .

We will now see that every odd square is in that sequence.

Every odd square is 1 more than a multiple of 4.

For, the sum of any two consecutive odd numbers is a multiple of 4, a sum of 4's.

Therefore, an even number of consecutive odd numbers is a multiple of 4.

1 . | ||

1 + 3 | = | 4. |

1 + 3 + 5 | = | 9. |

1 + 3 + 5 + 7 | = | 16. |

1 + 3 + 5 + 7 + 9 | = | 25. |

1 + 3 + 5 + 7 + 9 + 11 | = | 36. |

1 + 3 + 5 + 7 + 9 + 11 + 13 | = | 49. |

Every even square is a multiple of 4.

An odd square is produced by adding the next odd number. And that odd number is in an odd position&mdashbecause an odd square is the sum of an odd number of odd numbers.

That odd number is 1 more than a multiple of 4. And a multiple of 4&mdashan even square&mdashplus a number 1 more than a multiple 4, is itself 1 more than a multiple of 4.

Therefore every odd square&mdash

&mdashis 1 more than a multiple of 4. When that odd number is added, we will have a Pythagorean triple. And the square that is produced is odd. It is 1 more than a multiple of 4.

In other words, in every Pythagorean triple produced in this way:

The sum of two squares is 1 more than a multiple of 4.

16 + 9 | = | 25. |

144 + 25 | = | 169. |

576 + 49 | = | 625. |

1600 + 81 | = | 1681. |

Why have we gone to the trouble to show that the sum of two squares is 1 more than a multiple of 4? Because there is another class of numbers for which that is true. Namely, prime numbers that are 1 more than a multiple of 4.

5 13 17 29 37 41 53 57 61 73 89

Every one of those primes is the sum of two squares Every one could be the square drawn on the hypotenuse of a right-triangle. Every one is the sum of two consecutive sums of odd numbers.

5 | = | 4 + 1. |

13 | = | 4 + 9. |

17 | = | 16 + 1. |

29 | = | 4 + 25. |

37 | = | 36 + 1. |

41 | = | 16 + 25. |

53 | = | 4 + 49. |

That is not true for a prime that is 1 less than a multiple of 4 which is the other possibility for an odd prime. It is true only for a prime that is 1 more.

That fact is not difficult to understand. But it is quite difficult to prove. It has been called Fermat's Theorem on the Sum of Two Squares although the fact has been known since ancient times.

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## Even and Odd Functions

In other words there is symmetry about the y-axis (like a reflection):

This is the curve f(x) = x 2 +1

They got called "even" functions because the functions x 2 , x 4 , x 6 , x 8 , etc behave like that, but there are other functions that behave like that too, such as cos(x):

Cosine function: f(x) = cos(x)

It is an even function

But an even exponent does not always make an even function, for example (x+1) 2 is **not** an even function.

## 9 Answers 9

So all you're missing is the addition of your two numbers when you finish your for loop. Try this:

Edit: as per your comment, you won't want to add your limits twice if they're odd. The y+1 ensures you're capturing the whole range, and the check for i == x or i == y skips those values in the range, since we've already added them at the start.

Just check the bounds separately.

The base loop should include any odd limits, so you only have to add the limits if they are odd.

First I would extend the range to y+1 then take all the odds in that range, after I would check if x and y were even if so i would add them to the list.

In a range(..) object the "upper bound" (second parameter) is *exclusive*. So in order to fix this, using range(x, y+1) is sufficient, like:

Note however that we can improve the speed, since the sum can be calculated with a formula:

So we can calculate this as:

The advantage of this approach that it works in *O(1)* for small to not so small numbers, and in *O(log m + log n)* for huge numbers (since multiplication can then take more time).

As a result we can calculate the sum of huge numbers quite fast, for example:

So calculating the sum of odd elements between 12'345'678'901'234'567'890 and 98'765'432'109'876'543'210 can be calculated in 503 nanoseconds. An iterative approach will take linear time, and will probably not obtain a result within reasonable time.

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## Comments

### Congratulated?

I appreciate this piece, and it's helped me make sense of a lot of what's going on here. However, I have to part ways in calling for the Numberphile folks to be "congratulated." As a math teacher, and one who struggles every day to counter the deeply-ingrained notion that math makes no sense whatsoever, I can't stand it when that notion is spread across a wide audience and further ingrained into our culture. Sure, those who are already somewhat mathematically inclined are intrigued and want to know more. But those who are not see a video like this and say to themselves, "Further confirmation that math makes absolutely no sense." So I'm not about to congratulate them.

### Thank you for your comments,

Thank you for your comments, its something that many math teachers have expressed to me including lecturers at university which is what prompted us to write the piece Its clear to me where I would draw the moral line and it isn't in the same place they have .

### Non mathematician's view

Well, I think the Numberphile guys made an entertaining video and I wish that my school maths teachers had been even half as good with their explanations as these guys are. For me it was the teachers, with their leaps of faith, blasting through the set books and omitting whole chapters (homework : study chapter so and so and do the exercises) who instilled the fear of maths and a sense of futility into all but the three or four kids, out of a class of 25, who could figure out what was going on. Maths teachers take a look in the mirror!

### Isn't it more important to

Isn't it more important to recognise that maths is about adventure, discovery, fun? This is simply an example of an important aspect of in mathematics - you take some concept, abstract it and extend it, and see where it leads. Shouldn't more maths teaching and learning emphasise this?

A nice example that I use in my classroom, far simpler than analytic continuation and one that survives the journey more intact, concerns the index laws. You take the well understood concept that a times a times a. n times is a^n, and uncover the fact that a^m times a^n = a^(m+n). Then generalise the concept of power so you can consider things that "don't make sense" like a^(1/2) or a^(-3). These are not meaningful under the initial view that "power means repeated multiplication", but the extension and subsequent exploration lead to important and very meaningful results. Learning the index laws can be an exercise in rote memorisation, or it can be a wonderful journey of discovery, where seeming "nonsense" becomes clarified and empowering!

Numberphile demonstrate this side of maths, and should absolutely be congratulated.

### Adventure, discovery, fun

"More important" than what? More important than truth, precision, clarity, correctness, understanding, etc.? Many Numberphile videos are fun and all that in addition to being essentially correct. However, they really blew it with the nonconvergent-series videos.

You might consider steering students to Martin Gardner's books based on his Scientific American column "Mathematical Games" even though he wasn't a mathematician, he had lots of contacts in the mathematical community, and his writing was clear, entertaining, and essentially correct.

### I have to disagree, and am

I have to disagree, and am quite bemused by the moral tone taken up by some of the detractors of this work. It's too easy to dismiss these videos as incorrect/untruthful etc. In context, they are a valid exploration of mathematical ideas, and thus important and valuable.

In the history of maths, this happened with irrational numbers, negative numbers, imaginary numbers. Formal manipulations that included "nonsense" ended up enriching mathematics. And in modern maths, look at p-adics where, for example. in the 5-adics, 5+5^2+5^3+. converges but 1/5 + 1/5^2+1/5^3+. does not. There's the Umbral Calculus where the formal basis is still only being constructed. Also, the extended complex plane where infinity is just a point like any other.

Now you can say they sneakily departed from real numbers, but the whole question naturally goes beyond the reals because it involves infinity. Infinity is not a real number, but more relevant, it is not just a "really big number". That it is qualitatively different is important to learn, and apparent from ideas like divergent and non-absolutely convergent series. As a 15 year old I was introduced to 1+2+4+8+. =-1 by a maths professor on an excursion, and it made a tremendous and positive impression on me and my fellow students. And I use similar things to both communicate my love of maths, and to encourage others to look at it differently and find their own sources of wonder.

And it is not so far removed from school maths. We teach that infinity - infinity is undefined arithmetically, but in terms of sets we often show that it can have many values (easiest example is remove all even numbers from 1,2,3,4,5. and you are left with an infinite set, compared to match up 1<->2, 2<->4, 3<->6 etc and show that none are left behind). Exposure to such conflicts is a great and fun way to learn about the limits and context of maths. And a major source of error is not learning the limits, applying things unthinkingly out of context, because too much exposure is only to "nice" examples. There is loads of education research on this. These paradoxical results force students to confront the limits, and thus can be used to enhance their mathematical thinking.

The authors of the article recognise the broader context (although were clearly not entirely happy with the presentation). The commenter above, Matt E, a maths teacher, seems to miss this broader context. Of course, approaching and crossing boundaries may well mean things need to be redefined, concepts generalised, but that's mathematics. Use it to generate interest, provide historical and real-world context, and thus enrich teaching.

Thinking about Grandi's series is like a first step on a journey. Enjoy it, and let students enjoy it too. Relate it to Thomson's Lamp. Bring mathematics to life!

## Mathematics. Grade 1. Addition worksheets. Adding odd numbers up to 100.

#### Additional resources for free download (Google Drive).

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## 4 comments

How to solve 1^2+3^2+5^2+….19^2=?

Please reply fast

Answer is 1300 how i don’t have any idia

find sigma( 2n -1)^2 limits from 1 to 19

since the base are odd numbers ,and general form of odd number is 2n-1 ,if we consider n start from 1 or 2n+1 if we consider n starts from 0,

so, lower limit is 1 if we take 2n-1,and upper limit will be 2n-1=19 so, n=10 that is the upper limit not

using sigma operation add this

like summation of (2n-1)(2n-1)=4n^2-4n+1 (A)

now put n=1,2,3 and then add for every n in the eqn(A)

## Consecutive Numbers

Kayley from Arnhem Wharf Primary School in London sent in this very good solution to the task:

When you use an even number of consecutive numbers, your answer will be even because there are two odd numbers and two even numbers in the equation. An odd number plus or minus another odd number equals an even number an even number plus or minus an even number equals an even number therefore the two answers (odd +/- odd and even +/- even) are even numbers and if you add it all together it will still be an even number.

I noticed that if you use the sequences +--, --+ or -+- with four consecutive numbers, your answer will be the same, no matter what consecutive numbers you use, if both sets numbers are ordered in the same way (e.g. from smallest to biggest). A set of three consecutive numbers will have odd answers if the starting number is even because an even number add or subtract an odd number will be odd, then when you add or subtract another odd number the answer will be odd.

I also found out that if there is a set of three consecutive numbers, and the starting number is odd, all the answers will be even because an odd plus/minus an even number is odd and if you plus/minus an odd number, the answer will be even.

These pictures show her work, the red writing being her thoughts. Larger versions can be viewed here: First page and Second page

Iris, Hannah, Hayden and Tawana from Fenstanton and Hilton Primary School said:

We worked out the calculations for 1, 2, 3, 4 as our consecutive numbers. These are the calculations we worked out:

1+2+3+4= 10 1+2+3-4= 2 1+2-3-4= -4 1-2-3-4= 4

1-2-3+4= -2 1-2-3-4= -8 1-2-3+4= 0 1-2+3+4= 6

*Are you surprised by anything you noticed?* We noticed that all the numbers are even, which would make sense because if you are adding consecutive numbers there will always be two odd numbers and two even, which means they will always add up to an even number.

*What would happen if you had the numbers going in descending order instead of ascending order?* We found out that when we put the numbers in descending order they were all still even and that some of the numbers were there same but mostly different.

We got:

4+3+2+1=10 4+3+2-1=8 4+3-2-1=4 4-3-2-1=-2

4-3-2+1=0: 4-3+2+1=4 4+3-2+1=6 4-3+2-1=2

James and Polly from the very small Meavy CE Primary School sent in this excellent solution:

Our answer to the consecutive numbers challenge:

30-31-32-33=-66 and 30+31+32+33=126 are some of the questions only using adding and subtraction using the numbers 30, 31, 32, 33. You can only get eight different calculations:

1. + + + 2. + + - 3. + - + 4. + - -

5. - + + 6. - + - 7. - - + 8. - - -

We found out that the answers you always get are 0, -2 and -4 using - - + (in any order).

If you put something in the middle of the eight answers, once you have put them in order, the gap between the answers will be like a mirror image.

We did the problem four times so that we could make sure that our answer was right. These were the groups of numbers we used:

The numbers we started with | The eight answers | The gaps between the answers |

1,2,3,4 | 10,6,4,2,0,-2,-4,-8 | 4,2,2,2,2,2,4 |

7,8,9,10 | 34,18,16,14,0,-2,-4,-20 | 16,2,2,14,2,2,16 |

20,21,22,23 | 86,44,42,40,0,-2,-4,-46 | 42,2,2,40,2,2,42 |

30,31,32,33 | 126,64,62,60,0,-2,-4,-66 | 62,2,2,60,2,2,62 |

James thinks that the middle number in the gaps is double the start number and the outside numbers are double the second number. Polly thinks the middle number in the gaps is double the start number but the outside numbers are double the first number and add two. Both of our suggestions are right.

Mrs Bromley a teacher at Skelton Primary School, York sent in Joseph's work, saying,

This morning we really enjoyed investigating consecutive numbers using your resource.

She remembered that Joseph said:

I decided to investigate odd and even numbers. I was surprised that even amounts of numbers give an even answer. I predicted that using three consecutive numbers would always give odd answers but I was proved wrong! Next, I would like to look at what happens if you start with an odd or even number.

You can see Joseph's work more clearly as .jpg files here and here.

Thank you all for these really helpful solutions which may encourage others to pursue the same task.