Category: High School Algebra

Fractions with Terminating Decimals

College Algebra, College Mathematics, High School Algebra, High School Mathematics

Late last month, we have talked about fractions with terminating decimals as well fractions with non-terminating decimals. We ended up with a conjecture that a fraction is a terminating decimal if its denominator has only the following factors: 2 (or its powers), 5 (or its powers) or both.  In this post,we refine this conjecture. This conjecture is the same as saying

A rational fraction \frac{a}{b} in the lowest terms has a terminating decimal if and only if the integer b has no prime factor other than 2 and 5.

Note that we have already explained the only if part in the preceding post. It remains to show that if part which is

if \frac{a}{b} is in lowest terms and b contains at most 2 and 5 as factors, then the fraction is a terminating decimal. » Read more

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Math Misconception: Incorrect Real Number System Diagram

High School Algebra, High School Mathematics

If you search images of the real number system on the internet, you will be surprised that there are  a lot of incorrect real number system diagrams. One of the common incorrect diagrams is shown in the first figure below. If we interpret the  diagram, all the numbers inside the the oval are real numbers. That is, our universal set is the set of real numbers.

Notice that there are numbers that are outside the rational number circle and also outside the irrational number circle that are real numbers (the solid bluish part). But we know that real number is either rational or irrational. Therefore, the numbers on the bluish part of the diagram outside the rational and irrational number circles do not exist.  » Read more

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Math Trick 2: Multiplying 2 Digit Numbers by 11

High School Algebra, High School Mathematics

In the previous post, we learned a cool math trick on squaring numbers (positive integers) ending in 5. We did not only learn the math trick itself, but we also discussed why it works. In this post, we will explore another math trick which is multiplying 2 digit numbers by 11.

The Copy-Add-Copy Method

To multiply 2 digit numbers by 11, we will use the copy-add-copy method. We will copy and add the digits of the number multiplied by 11. Below are the steps of this method.

Steps in Multiplying 2-Digit Numbers by 11

  1. Copy the ones digit of the number multiplied by 11 to the ones digit of the product.
  2. Add the ones digit and the tens digit of the number and copy the sum (see *) to the tens digit of the product.
  3. Copy the tens digit of the number (see **) to the hundreds digit of the product. » Read more
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