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 in the lowest terms has a terminating decimal if and only if the integer has no prime factor other than and .
Note that we have already explained the only if part in the preceding post. It remains to show that if part which is
if is in lowest terms and contains at most and as factors, then the fraction is a terminating decimal. » Read more
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.
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
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.
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.
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