Struggling with Math: Any Book Recommendations?

[Atran]

Hi, I've studied mathematics at high school and I will start university next year. I'm worried about my mathematical knowledge in general. There are many "gaps" which lead to confusion.

To give you some examples:
1) I've difficulty understanding why, for instance, 5 - (-2) = 5 + 2. The only explanation I have is: 5 - (-2) = 5 + -1*(-1*2) = 5 + (-1*-1)*2 = 5+2.
2) I can't really intuitively understand implicit differentiation.
3) I've difficulty grasping the basic counting principle, etc...

I need a book which starts from the very bottom (covering many topics) and explains why some rules are the way they are. I'm thinking to buy Finite Math and Applied Calculus by Stefan Waner and Steven Costenoble. What's your opinion about the book?

Can you recommend me other books?
Thanks for reading.

 

[sgfw]

I can't really think of any books, but I think Khan Academy could help whith a lot of your problems. It's free, and there are a lot of videos that can help make ideas more intuitive. I recommend you try it out.

 

[Mark44]

Hi, I've studied mathematics at high school and I will start university next year. I'm worried about my mathematical knowledge in general. There are many "gaps" which lead to confusion.

To give you some examples:
1) I've difficulty understanding why, for instance, 5 - (-2) = 5 + 2. The only explanation I have is: 5 - (-2) = 5 + -1*(-1*2) = 5 + (-1*-1)*2 = 5+2.

This is really the long way around. For subtraction, all you need to know is that subtracting a number is the same as (i.e., gives the same result) as adding the opposite of the number.

Some examples
6 - 4 = 6 + (-4) = 2 Here, -4 is the "opposite" (additive inverse is the proper term) of 4.
5 - (-2) = 5 + [-(-2)] = 5 + 2 = 7 The opposite of -2 is 2.

2) I can't really intuitively understand implicit differentiation.

This is a long way in complexity from adding and subtracting signed numbers, but it's not really that complicated. In implicit differentiation you are making the assumption that all variables are differentiable functions of some independent variable, typically x.

For example, starting from xy² = 3, we are assuming that y is a differentiable (its derivative exists) function of x.

Differentiating both sides with respect to x yields:
d/dx(xy²) = d/dx(3)
==> x * 2y * dy/dx + y² * dx/dx = 0
==> 2xy * dy/dx + y² = 0

Now all that's left is to solve the equation above for dy/dx.

3) I've difficulty grasping the basic counting principle, etc...

Can you be more specific?

Math books typically don't start from very basic algebra (such as addition of signed numbers) and go all the way through calculus. You might need a book for basic algebra, one for precalculus (including trig), and one for calculus. Alternatively, or together with, you could look at the topics in Khan Academy, as the previous poster suggested.

I need a book which starts from the very bottom (covering many topics) and explains why some rules are the way they are. I'm thinking to buy Finite Math and Applied Calculus by Stefan Waner and Steven Costenoble. What's your opinion about the book?

Can you recommend me other books?
Thanks for reading.

 

[Mandelbroth]

Hi, I've studied mathematics at high school and I will start university next year. I'm worried about my mathematical knowledge in general. There are many "gaps" which lead to confusion.

To give you some examples:
1) I've difficulty understanding why, for instance, 5 - (-2) = 5 + 2. The only explanation I have is: 5 - (-2) = 5 + -1*(-1*2) = 5 + (-1*-1)*2 = 5+2.
2) I can't really intuitively understand implicit differentiation.
3) I've difficulty grasping the basic counting principle, etc...

I need a book which starts from the very bottom (covering many topics) and explains why some rules are the way they are. I'm thinking to buy Finite Math and Applied Calculus by Stefan Waner and Steven Costenoble. What's your opinion about the book?

Can you recommend me other books?
Thanks for reading.

1) Imagine a number line, pretending that "right" is the positive direction and "left" is the negative direction. Consider a real number, $n$. To subtract a number $m$ from $n$, to obtain $n-m$, we start at $n$ on the number line and go to the left $m$ units. If $m$ is negative, then it is like we are moving $m$ units to the opposite of left. In other words, subtracting a negative number is like walking backward on the number line.

2) It's just an application of the chain rule. For example, if we have a circle given by $x^2+y^2=r^2$, and we want to find the slope of that circle at a point. We just apply the derivative to both sides of the equation, obtaining $\frac{d}{dx}\left[x^2+y^2\right]=2x+\frac{d}{dx}\left[y^2\right]=2x+\frac{d(y^2)}{dy}\frac{dy}{dx}= 2x+2y\frac{dy}{dx}=\frac{d}{dx}\left[r^2\right]=0$, assuming that r is a constant. Then, we solve for $\frac{dy}{dx}$, obtaining $\frac{dy}{dx}=\frac{-x}{y}$.

3) If you imagine possible events in the form of a tree, it becomes fairly intuitive.

I've heard that Spivak's Calculus constructs many principles of numbers from scratch. I can't say that I've personally read it, but fellow forumer micromass suggested it, along with Lang's Basic Mathematics, in a somewhat similar thread, and he's almost always right. I have an introduction to differential geometry by Spivak that periodically oozes masses of mathematical awesomeness whenever I open it, so I would not be surprised if his calculus book is excellent as well.

 

[Atran]

Thank you all for the answers.

Can you be more specific?

I need to know how a mathematician thinks it or to see it from a mathematician's point of view.

Does an exact number have an infinite number of significant digits?
Suppose 0.5±0.25, then 0.25≤x≤0.75: Is it correct to infer, given 0.245 and 0.755 being exact values, 0.245≤x<0.755?

 

[Mandelbroth]

Thank you all for the answers.


I need to know how a mathematician thinks it or to see it from a mathematician's point of view.

Does an exact number have an infinite number of significant digits?
Suppose 0.5±0.25, then 0.25≤x≤0.75: Is it correct to infer, given 0.245 and 0.755 being exact values, 0.245≤x<0.755?

If a number is on the interval $[\frac{1}{4},\frac{3}{4})$, then it is definitely on the interval $[0.245,0.755)$. Somehow, though, I don't think this is what you are asking.

An "exact" number doesn't involve significant digits. For example, in reality, 5 has 1 significant digit. In math, though, we actually mean that 5 is 5 and there is no error in the measurement obtaining that number. So, if you want to think about it in terms of significant digits, 5=5.000000000...

 

[Atran]

I meant that if, for example, x=2.0 is a measurement with two significant figures, then x can be any value between (1.95) and (2.05): 1.95≤x<2.05 or 1.95≤x≤2.04.
Am I thinking right?

 

[Mark44]

I meant that if, for example, x=2.0 is a measurement with two significant figures, then x can be any value between (1.95) and (2.05): 1.95≤x<2.05 or 1.95≤x≤2.04.
Am I thinking right?

Pretty close. The 2.0 measurement indicates that x could be anywhere between 1.95 and 2.05, as you wrote. That would be 1.95 ≤ x ≤ 2.05.

 

[WannabeNewton]

I have an introduction to differential geometry by Spivak that periodically oozes masses of mathematical awesomeness whenever I open it

ewwwwwww

 

[Mandelbroth]

ewwwwwww

It's the good kind that smells like strawberries.

 

[WannabeNewton]

It's the good kind that smells like strawberries.

If it oozes anything, it's ew

 

[Atran]

Pretty close. The 2.0 measurement indicates that x could be anywhere between 1.95 and 2.05, as you wrote. That would be 1.95 ≤ x ≤ 2.05.

So do 1.95 and 2.05 consist of three significant digits? Or are the two values "exact"?
Because if each has three significant figures then 1.95 is any value between 1.945 and 1.955, and for x: 1.945 ≤ x, which is wrong since 1.95 ≤ x.

 

[MarneMath]

It would be great if you could read the first chapter in Spivak's Calculus. He gives you a rather intuitive understanding of the properties of real numbers, and inequalities. Also are you aware that significant figures is a method to help maintain an significant values through out a calculation where measurements are involved.?

A number like 5 has no significant value what-so-ever. However, if I got that number 5 came from a ruler and that ruler was accurate to two more decimal places, then we would write 5.00 (units) to demonstrate the accuracy of the ruler. So note that when working with purely mathematical numbers where measurement is not involved, then the idea of significant figures does not come into play at all.

 

[Googl]

It becomes much clear when you think about number lines. Especially with negative numbers and letters. Here is an example of number lines: Number lines