Part Two: RELATIONSHIPS
Chapter 7. The Joy of x
Arithmetic becomes algebra when we begin working with unknowns and formulas.
Leaving arithmetic and going into algebra, this chapter explains this transition and why algebra is an art (we can express patterns about numbers for their own sake) and a science (we can express relationships between numbers in the real world). Then, he explains some misconceptions and bad uses we have with algebra and explain some rules of factorization and their true meaning.
Chapter 8. Finding Your Roots
Complex numbers, a hybrid of the imaginary and the real, are the pinnacle of number systems.
As simple as it appears at the beginning of this chapter, it’s not. First, he explains how the Greeks started to have problems with negative roots that later became the imaginary numbers, and later how this lead to the creation of complex numbers.
Complex Numbers: Two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number.
“Complex numbers are magnificent, the pinnacle of number systems… Better yet, a grand statement called the fundamental theorem of algebra says that the roots of any polynomial are always complex numbers. In that sense they’re the end of the quest, the holy grail. The universe of numbers need never expand again. Complex numbers are the culmination of the journey that began with 1.”
Then, he explains some of the uses of complex numbers in electrical, aerospace, civil, and mechanical engineer. He also explains how John Hubbard, when experimenting with these numbers, got to the discovery of fractals (an intricate shape whose inner structure repeated at finer and finer scales), and how this lead to the formation of complex dynamics, chaos theory, complex analysis, and fractal geometry.
Chapter 9. My Tub Runneth Over
Turning peril to pleasure in word problems.
In here, he explains how easy it is to get confused with word problems leading to a wrong solution based on our intuition, but when we put them on paper we find that the solution is sometimes counterintuitive. He puts the example of filling a bathtub with a cold- and hot-water faucet running together, but each has different rates of filling the tub. One of the things that help us get to a solution is of thinking in two scenarios, the extremes, and therefore the answer would be in the middle of those two.
“The undistracted reasoning that this problem (three men painting) requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.
Perhaps even more important, word problems give us practice in thinking not just about numbers, but about relationships between numbers.”
Chapter 10. Working Your Quads
The quadratic formula may never win any beauty contest, but the ideas behind it are ravishing.
This chapter goes on explaining in more depth how the quadratic formula (x = (-b +- sq(b2 – 4ac))/2a was created. By doing this, he says we would really appreciate its inner beauty. In order to do this, he explains the origins of algebra with Muhammad ibn Musa al-Khwarizmi. One example he works with is x2 + 10x = 39. In order to solve this, we must apply the method of completing the square. This can be perfectly seen when putting the equation geometrically, so that it would be evident that a square is missing to complete a larger one. This exercise can be generalized with any numbers in a quadratic equation by using the quadratic formula, so when using it you now know how we can arrive at the same answer by constructing some squares and a little extra geometry.
Chapter 11. Power Tools
In math, the function of functions is to transform.
Logarithm is a big word for some people, but in here he tries to take away that bad conception of logarithms by explaining them. So first, he explains the power or exponential function, which we are more familiar. He explains a little how to graph them and makes analogies of some parts of the function in the graph. He says that functions are the handy tools a mathematician uses to transform something. After explaining exponential functions, he explains that logarithms are just the inverse functions of the exponential functions and that we are using them in our daily life, but are just not aware of this. Finally, he gives a tiny introduction to the next part of the book by stating that these functions (the mathematicians toolbox) can only do so much, meaning that there are other tools we would explore in the next chapters.
Arithmetic becomes algebra when we begin working with unknowns and formulas.
Leaving arithmetic and going into algebra, this chapter explains this transition and why algebra is an art (we can express patterns about numbers for their own sake) and a science (we can express relationships between numbers in the real world). Then, he explains some misconceptions and bad uses we have with algebra and explain some rules of factorization and their true meaning.
Chapter 8. Finding Your Roots
Complex numbers, a hybrid of the imaginary and the real, are the pinnacle of number systems.
As simple as it appears at the beginning of this chapter, it’s not. First, he explains how the Greeks started to have problems with negative roots that later became the imaginary numbers, and later how this lead to the creation of complex numbers.
Complex Numbers: Two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number.
“Complex numbers are magnificent, the pinnacle of number systems… Better yet, a grand statement called the fundamental theorem of algebra says that the roots of any polynomial are always complex numbers. In that sense they’re the end of the quest, the holy grail. The universe of numbers need never expand again. Complex numbers are the culmination of the journey that began with 1.”
Then, he explains some of the uses of complex numbers in electrical, aerospace, civil, and mechanical engineer. He also explains how John Hubbard, when experimenting with these numbers, got to the discovery of fractals (an intricate shape whose inner structure repeated at finer and finer scales), and how this lead to the formation of complex dynamics, chaos theory, complex analysis, and fractal geometry.
Chapter 9. My Tub Runneth Over
Turning peril to pleasure in word problems.
In here, he explains how easy it is to get confused with word problems leading to a wrong solution based on our intuition, but when we put them on paper we find that the solution is sometimes counterintuitive. He puts the example of filling a bathtub with a cold- and hot-water faucet running together, but each has different rates of filling the tub. One of the things that help us get to a solution is of thinking in two scenarios, the extremes, and therefore the answer would be in the middle of those two.
“The undistracted reasoning that this problem (three men painting) requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.
Perhaps even more important, word problems give us practice in thinking not just about numbers, but about relationships between numbers.”
Chapter 10. Working Your Quads
The quadratic formula may never win any beauty contest, but the ideas behind it are ravishing.
This chapter goes on explaining in more depth how the quadratic formula (x = (-b +- sq(b2 – 4ac))/2a was created. By doing this, he says we would really appreciate its inner beauty. In order to do this, he explains the origins of algebra with Muhammad ibn Musa al-Khwarizmi. One example he works with is x2 + 10x = 39. In order to solve this, we must apply the method of completing the square. This can be perfectly seen when putting the equation geometrically, so that it would be evident that a square is missing to complete a larger one. This exercise can be generalized with any numbers in a quadratic equation by using the quadratic formula, so when using it you now know how we can arrive at the same answer by constructing some squares and a little extra geometry.
Chapter 11. Power Tools
In math, the function of functions is to transform.
Logarithm is a big word for some people, but in here he tries to take away that bad conception of logarithms by explaining them. So first, he explains the power or exponential function, which we are more familiar. He explains a little how to graph them and makes analogies of some parts of the function in the graph. He says that functions are the handy tools a mathematician uses to transform something. After explaining exponential functions, he explains that logarithms are just the inverse functions of the exponential functions and that we are using them in our daily life, but are just not aware of this. Finally, he gives a tiny introduction to the next part of the book by stating that these functions (the mathematicians toolbox) can only do so much, meaning that there are other tools we would explore in the next chapters.