Fractions with Terminating Decimals

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

Curve Sketching 3: Understanding Vertical and Horizontal Asymptotes

This is the third part  of the Mathematics and Multimedia Curve Sketching Series. In the first part of this series, we have learned how to sketch linear functions, while in the second part, we have learned how to sketch quadratic functions.  In this post and the next post, we will discuss about another important property of some functions that can be used in curve sketching.

asymptote

In Curve Sketching 2, we have learned the different properties of quadratic functions that can help in sketching its graphs.  This property is called the asymptote. » Read more

Vectors, Parallelograms, and Commutativity

Basics of Vectors

Scalar quantities are quantities specified by magnitude. Mass, area, density are examples of scalar quantities.  There are quantities that have both magnitude and direction.  For example, when we say 2 kilometers east, 2 kilometers is our magnitude and east is our direction. Quantities with both magnitude and direction are called vectors.

In mathematics, vectors are usually represented by a directed line segment. The arrow  of the directed segment is called its head, and the other end is called its foot.

Comparing Vectors

Two vectors are said to be equal if they have the same length and the same direction. Plainly speaking, if we look at the geometric representation of vectors,  equal vectors have  the “same slope and the same direction (of the arrow). In Figure 1, it is clear that there only two equal vectors – vector u and vector v.

Figure 1

The negative of a vector is the vector  equal in length to a certain vector, but with opposite direction. For instance, in Figure 1, – u and u are negative of each other and vector u is also the negative of vector v. » Read more

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