Pure Mathematics for Beginners –
Accelerated and Expanded Edition
Just 19.99 on Amazon
Hi everyone! Pure Mathematics for Beginners – Accelerated and Expanded Edition is now available in paperback from Amazon. Similar to its predecessor, this book was written to provide a rigorous introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The book consists of 16 lessons. Explanations to all the problems in the book are included as a downloadable PDF file.
So, what do I mean by “accelerated and expanded” edition?
By “accelerated” I mean that the book covers most of the material from the standard edition within the first half of the book. For example, the first lesson on set theory now covers relations, functions and equinumerosity (in addition to all the basics). However, nothing is left out. Everything from the original edition is included. In fact, more exposition has been added to the original content, as well as more examples and additional clarifying remarks.
By “expanded,” I mean that a huge amount of additional content has been added to the book. In fact, most of the content in Lessons 9 through 16 consists of material that is not covered in the original edition (although some of the content can be found in my other books from this series such as Real Analysis for Beginners and Abstract Algebra for Beginners).
The paperback is now on sale on Amazon for only $19.99. Note that once the sale ends (in about 24 hours), the price of this book will go up to $64.99.
The promotion is now over. Thanks to all who participated. The book is available at Amazon here: Pure Mathematics for Beginners – Accelerated and Expanded Edition
You can get the solution guide in paperback here: Pure Mathematics for Beginners – Accelerated and Expanded Edition – Solution Guide (Note that you can download the solution guide as a PDF for free, but many readers prefer to have a physical copy of the solution guide.)
If you have any questions, feel free to contact me at the following email:
Thank you all for your continued support!
A Trick For Free Two Day Shipping
I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.
Sign Up For Amazon Prime For Free
If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.
This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.
After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.
Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.
Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:
Sign Up For Amazon Prime For Free
If you think your friends might be interested in this special offer, please share it with them on Facebook:
Thank you all for your continued support!
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Setting Up a Ratio
Today I would like to share the method I teach for solving problems involving ratios. I have a simple 4 step system.
Step 1: Identify two key words and write them down one over the other.
Step 2: Next to each of these key words write down the numbers, variables or expressions that correspond to each key word in two columns.
Step 3: Draw in 2 division symbols and an equal sign.
Step 4: Cross multiply and divide.
This procedure is best understood with a simple example:
Example: The sales tax on an $8.00 shirt is $0.60. At this rate what would be the sales tax on a $12.00 shirt?
Try to solve the problem yourself before checking the solution below.
Solution: We begin by identifying 2 key words. In this case, such a pair of key words is “shirt” and “tax.”
shirt 8 12
tax 0.60 x
Choose the words that are most helpful to you. Notice that we wrote in the shirt prices next to the word shirt, and the tax prices next to the word tax. Also notice that the tax for an $8 shirt is written under the number 8, and the (unknown) tax for a $12 shirt is written under the 12.
Now draw in the division symbols and equal sign, cross multiply and divide the corresponding ratio to find the unknown quantity x.
8/.060 = 12/x
8x = 12(0.60)
8x = 7.2
x = 0.90
So the tax on a $12 shirt is $0.90.
I’d also like to provide a quicker solution for the more advanced student.
* Mental math: If the tax on an $8.00 shirt is $0.60, then the tax on a $4.00 shirt would be $0.30 at this rate. Thus, the tax on a $12.00 shirt would be $0.90.
Here are a few more problems for you to try. Try to use my 4 step method each time. I will provide solutions to these throughout the week.
1. Running at a constant speed, a cheetah traveled 200 miles in 5 hours. At this rate, how many miles did the cheetah travel in 4 hours?
2. A copy machine makes 4800 copies per hour. At this rate, in how many minutes can the copy machine produce 920 copies?
3. The height of a solid cone is 22 centimeters and the radius of the base is 15 centimeters. A cut parallel to the circular base is made completely through the cone so that one of the two resulting solids is a smaller cone. If the radius of the base of the small cone is 5 centimeters, what is the height of the small cone, in centimeters?
More Problems with Explanations
If you are preparing for the SAT, ACT, or an SAT math subject test, you may want to take a look at the Get 800 collection of test prep books.
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Fun Math T-shirts
Hello everyone. Today I just want to share with you some of my favorite math related t-shirts. Check out my t-shirt collection by clicking the image above. Here are a couple of my favorites:
Click here to see all my favorite t-shirts
The links in this blog post are affiliate links. If you click on one of these links and decide to make a purchase I will receive a commission on the sale. This does not cost you anything additional. These small commissions help to keep the rest of the content on my blog free, so thank you!
Algebra Handbook for Gifted Middle School Students
The Algebra Handbook for Gifted Middle School Students gives you the most effective methods, tips, and strategies for solving algebra problems in both conventional and unconventional ways. The questions presented in this book were created to prepare middle school students for the MATHCOUNTS competition and include many types of number theory, algebra, probability and statistics questions that one can expect to find on the MATHCOUNTS exams.
Click the following link to get to the book’s Amazon page: Algebra Handbook
When you purchase the Algebra Handbook from Amazon, you will also get a free PDF version of 555 Geometry Problems
Take a look at my product page to see all of my books: Get 800 Product Page
A Trick For Free Two Day Shipping
I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.
Sign Up For Amazon Prime For Free
If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.
This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.
After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.
Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.
Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:
Sign Up For Amazon Prime For Free
Please share this information with your Facebook friends:
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The Contrapositive
A statement of the form “if p, then q” is known as a conditional statement. An example of such a statement is “If you are a cat, then you have fur.” Another common way to say this is “All cats have fur.”
There are 3 other statements that often come up in association with a conditional statement. Let’s use the example above to illustrate.
Conditional: If you are a cat, then you have fur.
Converse: If you have fur, then you are a cat.
Inverse: If you are not a cat, then you do not have fur.
Contrapositive: If you do not have fur, then you are not a cat.
The most important thing to know for standardized tests such as the SHSAT is that the contrapositive is logically equivalent to the original conditional statement! The converse and inverse are not.
For example, suppose the conditional statement “All cats have fur” is true. You may want to rewrite this as “If you are a cat, then you have fur.” It follows that “If you do not have fur, then you are not a cat” is also true.
In particular, if you are given the statement “Skittles does not have fur,” you can infer “Skittles is not a cat.”
Note that neither the converse nor the inverse is logically equivalent to the original conditional statement, but they are equivalent to each other.
Example 1
All of Jim’s friends can ski.
If the statement above is true, which of the following statements must also be true?
(A) If John cannot ski, then he is not Jim’s friend
(B) If Jeff can ski, then he is not Jim’s friend.
(C) If Joseph can ski, then he is Jim’s friend.
(D) If James is Jim’s friend, then he cannot ski.
(E) If Jordan is not Jim’s friend, then he cannot ski.
Solution: The given statement can be written in conditional form as “If you are Jim’s friend, then you can ski.” The contrapositive of this statement is “If you cannot ski, then you are not Jim’s friend. Replacing “you” with “John” gives the correct answer as choice (A).
Example 2
If a beverage is listed in menu A, it is also listed in menu B.
If the statement above is true, which of the following statements must also be true?
(A) If a beverage is listed in menu B, it is also in menu A.
(B) If a beverage is not listed in menu A, it is not listed in menu B.
(C) If a beverage is not listed in menu B, it is not listed in menu A.
(D) If a beverage is not listed in menu B, it is in menu A.
(E) If a beverage is listed in menu B, it is not listed in menu A.
Solution: Simply observe that the statement in choice (C) is the contrapositive of the given statement. So the answer is choice (C).
Conclusion
If you understand how to identify the contrapositive of a conditional statement, then you will be able to easily get most logic questions correct that appear on standardized tests.
If you will be taking the ACT, you may want to check out my series of ACT math prep books:
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Integers, Prime Numbers And Prime Factorizations
I would like to begin today with a few simple definitions of terms that appear in number theory problems on the ACT , GRE and SAT math subject tests.
Definitions
The integers are the counting numbers together with their negatives.
…,-4, -3, -2, -1, 0, 1, 2, 3, 4,…
The positive integers consist of the positive numbers from the above list.
1, 2, 3, 4,…
Next we have prime numbers.
A prime number is a positive integer that has exactly two factors (1 and itself). Here is a list of the first few primes:
2, 3, 5, 7, 11, 13, 17, 19, 23,…
Note that 1 is not prime. It has only one factor.
A little trick:
Here is a quick trick for determining if a large number is prime: take the square root of the integer and check if the integer is divisible by each prime up to this square root. If not, the number is prime.
For example let’s try to figure out if 3001 is a prime number. Note that when we take the square root of 3001 in our calculator we get approximately 54.8. Now with our calculators we divide 3001 by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and 53 (all the prime numbers below 54.8). Since none of these are integers, 3001 is prime.
The Fundamental Theorem of Arithmetic
The fundamental theorem of arithmetic says “every integer greater than 1 can be written “uniquely” as a product of primes.”
The word “uniquely” is written in quotes because prime factorizations are only unique if we agree to write the primes in increasing order.
For example, 6 can be written as 2 · 3 or as 3 · 2. But these two factorizations are the same except that we changed the order of the factors. To make things as simple as possible we always agree to use the canonical representation. The word “canonical” is just a fancy name for “natural,” and the most natural way to write a prime factorization is in increasing order of primes. So the canonical representation of 6 is 2 · 3. As another example, the canonical representation of 18 is 2 · 3 · 3. We can tidy this up a bit by rewriting 3 · 3 as 32. So the canonical representation of 18 is 2 · 32. If you are new to factoring, you may find it helpful to draw a factor tree. For example here is a factor tree for 18:
To draw this tree we started by writing 18 as the product 2 · 9. We put a box around 2 because 2 is prime, and does not need to be factored anymore. We then proceeded to factor 9 as 3 · 3. We put a box around each 3 because 3 is prime. We now see that we are done, and the prime factorization can be found by multiplying all of the boxed numbers together. Remember that we will usually want the canonical representation, so write the final product in increasing order of primes.
By the Fundamental Theorem of Arithmetic above it does not matter how we factor the number – we will always get the same canonical form. For example, here is a different factor tree for 18:
For practice, why don’t you try to draw a factor tree for 6137? Note that this is much more challenging than any number you will have to factor on a standardized test. I’ll have the solution for you tomorrow, so we can compare notes then. Remember, I am not looking for artistic merit – just make your factors clear in your drawing.
How many prime numbers are there?
This is going off on a bit of a tangent, but there are an infinite number of prime numbers. This was first proved by the ancient Greek mathematician Euclid.
Interestingly, the largest prime number found so far is 257,885,161 – 1. That’s a number with 17 million digits! I dare you to find the next one greater than that.
In the meantime, if you want to learn mathematical strategies to efficiently answer math questions on standardized tests, I would suggest that you take a look at the Get 800 collection of test prep books. Click on the picture below for more information about these books.
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555 Math IQ Questions
Today I would like to announce the release of 555 Math IQ Questions for Middle School Students.
The paperback is now on sale on Amazon for only $4.69.
This sale may last for only a few hours, and once the sale ends the price will go up to $34.99. Amazon will automatically raise the price once a predetermined number of copies has been sold. If you want to take advantage of the sale price I strongly recommend you purchase the book now.
The discount is ended, but I will honor all other promotions until the end of this month (see below).
Click the following link to get to the book’s Amazon page: 555 Math IQ Questions
As an additional incentive to purchase this book today, I will also give you an additional book for FREE as a downloadable PDF file. You can choose ANY of my other books. So if you would like a different book, go ahead and purchase this one (as soon as it is available), forward me your Amazon confirmation email and let me know which of my books you would like for free. You will be provided with a link to download your additional free book. This offer is available until the end of today (January 19, 2015) the month (January 31, 2015).
Take a look at my product page to see all of my books: Get 800 Product Page
I am also making the following promotion available until January 31, 2015: if you buy any 2 of my books in paperback, you get all 6 of the following SAT math prep books for free.
A Trick For Free Two Day Shipping
I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.
Sign Up For Amazon Prime For Free
If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.
This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.
After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.
Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.
Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:
Sign Up For Amazon Prime For Free
If you think your friends might be interested in this special offer, please share it with them on Facebook:
Thank you all for your continued support!
