Make Maths Fun: Exploring Interesting Perspectives

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In summary, the conversation discusses the speaker's interest in mathematics and their struggle with understanding it from a technical perspective. They ask for book recommendations that explore math from a more interesting angle and explain theory behind concepts. One suggestion is "An Imaginary Tale: The story of \sqrt{-1}" or "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills". Another book recommended is "Algebra" by Gelfand. The conversation also includes a brief explanation of the quadratic formula and complex numbers.
  • #1
thayes93
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Well basically, I've always been thought maths from a technical (operations) point of view. For example, we were taught the quadratic equation but I have no idea how or why it works. However, I am amazed and fascinated by maths at times when I wander off and mess about with numbers and theories, or at things like the omnipresence of e or the trigonometric functions.

I suppose what I'd like to know is whether there are any books you can recommend that I should read that explore maths from an interesting perspective and explain the theory behind things and whatnot, I'm sure you know what I mean!

Thanks in advance.

edit: in case it helps, I am 16 and in school, I have a strong interest in physics and have represented Ireland at Science Olympiads, just if that makes it any easier to gauge my interests and ability.
 
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  • #2
Hmm... well, I'm 17, and I struggle to understand most of his books, but you could try taking a look at "An Imaginary Tale: The story of [tex]\sqrt{-1}[/tex]" or "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills".

But like I said, I can hardly understand them. They don't deal with physics much, I think. Mainly pure math.
 
  • #3
Mathematics and the Search for Knowledge - Morris Kline
 
  • #4
fourier jr said:
Mathematics and the Search for Knowledge - Morris Kline

I found Kline's borderline racism in his account of the history of mathematics quite distasteful.

Anyways, to the OP: maybe you should try something like Algebra by Gelfand? I think it might help answer some of your questions.
 
  • #5
Werg22 said:
I found Kline's borderline racism in his account of the history of mathematics quite distasteful.

...what borderline racism? I've never heard of that before :confused:
 
  • #6
Hi,

I'm 17 and will be going into year 12 next year in Australia. To understand the quadratic formula, it really is quite simple: take the general form for [tex]a\neq0[/tex] of [tex] ax^2 + bx + c = 0 [/tex] and divide by the factor a. We then get the following: [tex]x^2 + \frac{b}{a}x + \frac{c}{a} = 0.[/tex] This then continues with the "completing the square" method, as follows: [tex](x+\frac{b}{2a})^2 = x^2 + \frac{b}{a}x + (\frac{b}{2a})^2.[/tex]

To return to our original function, after dividing by a, we must equate both sides as such: [tex]x^2 + \frac{b}{a}x + \frac{c}{a} = (x+\frac{b}{2a})^2 + \frac{c}{a} - (\frac{b}{2a})^2.[/tex] We are well aware that both equations are equal to zero, so then we can use simple mathematics to find that this holds:
[tex]\text{As } (x+\frac{b}{2a})^2 + \frac{c}{a} - (\frac{b}{2a})^2 = 0\text{, then }(x+\frac{b}{2a})^2 = (\frac{b}{2a})^2 - \frac{c}{a}.[/tex]

We now get to the stage where we begin to recognise the steps as bringing about the quadratic formula: [tex]x + \frac{b}{2a} = \pm\sqrt{(\frac{b}{2a})^2 - \frac{c}{a}}.[/tex]

Simple steps of manipulation follow: [tex]x = -\frac{b}{2a} \pm\sqrt{(\frac{b}{2a})^2 - \frac{c}{a}}.[/tex]

And again: [tex]x = -\frac{b}{2a} \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}.[/tex]

Rearrange this as follows (slightly more complex): [tex]x = -\frac{b}{2a} \pm\sqrt{b^2-4ac}.[/tex]

One more step in the process and you're done!

Here's the derived Quadratic Formula: [tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} .[/tex]

Now, the [tex]\sqrt{b^2 - 4ac}[/tex] element of the equation is called the discriminant, and when it is less than zero, there exist no real solutions to the quadratic - note though that the operative word in this sentence is 'real' with respect to the 'real numbers'. When it is equal to zero, there is one real solution (see the formula - this root is obvious), and when it is greater than zero, there are clearly two real solutions.

As a side note, quadratics become much more interesting when you deal with complex numbers (in the form of x + iy where [tex]i = \sqrt{-1}[/tex] and [tex]x, y \in \Re[/tex]). This occurs for a quadratic when the discriminant is less than zero i.e you take the square root of a negative number, an operation clearly not defined in the reals. If you would like to discuss complex numbers, let me know, you're more than welcome to (and if you're interested, take a look at the polar and exponential forms of a complex number, and most definitely, Euler's identity)! :wink:

Just so that my post can demonstrate some sheer mathematical beauty, here is Euler's identity: [tex]{e^{i\pi} + 1 = 0.}[/tex]

Cheers :biggrin:,
Davin
 

Related to Make Maths Fun: Exploring Interesting Perspectives

1. How can we make maths fun for students?

There are many ways to make maths fun for students. One effective strategy is to use hands-on activities and games to teach mathematical concepts. This allows students to engage with the material in a more interactive and enjoyable way. Another approach is to relate maths to real-world situations and applications, making it more relevant and interesting for students. Additionally, incorporating technology and visual aids, such as videos or interactive software, can also make maths more fun and engaging for students.

2. What are some interesting perspectives to explore in maths?

There are many interesting perspectives to explore in maths, such as geometry, algebra, statistics, and probability. You could also look at the history of maths and how it has evolved over time, or explore the applications of maths in various fields, such as finance, engineering, or computer science. Another interesting perspective is to approach maths from a problem-solving mindset, focusing on puzzles and brain teasers to develop critical thinking skills.

3. How can we engage students who are not naturally interested in maths?

One way to engage students who are not naturally interested in maths is to show them the practical applications of mathematical concepts in everyday life. This can help students see the relevance of maths and make it more interesting for them. Additionally, incorporating fun and interactive activities, as well as incorporating technology and visual aids, can help make maths more engaging for these students.

4. How can we make maths fun for all ages?

Making maths fun for all ages can be achieved by adapting teaching strategies and activities to suit the age and learning style of the students. For younger students, hands-on activities and games can be effective, while older students may enjoy more challenging puzzles and real-world applications. Additionally, incorporating elements of creativity and problem-solving can make maths more enjoyable for students of all ages.

5. How can we measure the success of making maths fun?

The success of making maths fun can be measured in various ways, such as improved student engagement and participation, increased interest and enthusiasm for maths, and improved academic performance. Additionally, feedback and assessments from students can also help measure the success of making maths fun and identify areas for improvement.

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