Simple math: Obtaining a common denominator to add fractions

1/2 + 1/4

To add these two fractions, we must get a denominator that is the same. This is called the common denominator. We can get a common denominator by multiplying the two denominators, in this problem it would be 2 x 4 = 8. But that is not the lowest common denominator, which is preferred when adding and subtracting fractions. To get the lowest common denominator, let's look at all the multiples of 2 and then the multiples of 4.

Multiples of 2 are

2, 4, 6, 8, 10, 12, …...

Multiples of 4 are

4, 8, 12, 16, 20, …..

The lowest common denominator is the smallest number that is the same in both sets of multiples. Notice the 4 is bold in each set. That is the lowest common denominator.

Now that we have the common denominator, we have to make ½ into an equivalent fraction with 4 as the denominator. We learned how to do this in the previous section. If we multiply the numerator and denominator by 2, we get 2/4.. Notice ¼ already has a denominator of 4, so we don't have to change this fraction in order to add.

2/4 + 1/4 = 3/4

 

Let's now consider a little more difficult problem where we'll have to change both fractions before adding.

3/5 + 1/6

 

We need to get a common denominator, so get all the multiples of 5.

5, 10, 15, 20, 25, 30, 35, ….

Next get all the multiples of 6.

6, 12, 18, 24, 30, 36, 42, …..

Notice that 30 is the smallest of the numbers that are common to both lists, so 30 is the common denominator.

Next we get equivalent fractions with 30 in the denominator. Multiply the first fraction by 6/6 to get 18/30. Multiply the second fraction by 5/5 to get 5/30. Now we can add the fractions to get 18/30 + 5/30 = 23/30

When giving the value of a trigonometric ratio of an angle, plus the sign of another trigonometric ratio of the same angle, we can get all other trigonometric ratios of the angle.

For example.  Suppose tan(B) = -3 and we know cos(B) >0.

cot(B) = 1/tan(B) = -1/3

we know that 1 + tan^2(B) = sec^2(B), so  1+ (-1/3)^2 = sec^2(B)

1 + 1/9 = sec^2(B)

sec^2(B) = 10/9

sec(B) =+/- sqrt(10)/3

since cosine is positive and tangent is negative, we know sine must be negative. Therefore, csc, which is 1/sin, is also negative.

sec(B) = sqrt(10)/3

cos(B) = 3sqrt(10)/10

sin^2(B) + cos^2(B) = 1

sin^2(B)+ 9/10 = 1

sin^2(B) = 1/10

sin(B) = -sqrt(10)/10

csc(B) = -sqrt(10)

Just a bit about Z-scores

Remember the normal distribution is symmetrical. It's a bell shaped curve. The standard normal distribution has mean 0 and standard deviation of 1. 

A "Z-score" is a certain number of standard deviations from the mean than an data value is.  Each Z-score has a corresponding probability that an observed value is less than that many standard deviations from the mean.

For example

Z-score of 1.28..  P(observed value  is less than 1.28) = .90 (that is obtained from a standard normal chart)

Z-score of -1.28  P(observed value is less than -1.28) = .10

notice there is .10 to the right of 1.28 and .10 to the left of -1.28. That makes sense since the distribution is symmetrical.

For any sports fans out there, you know math is used all the time calculating statistics for players and teams. These fascinating numbers are used as ways to compare player and team performance on several criteria.

For those interested in knowing and understanding statistics from football, basketball, baseball, hockey or golf, let me know and I'll do my best to explain them.

When rationalizing the denominator when the denominator is a square root, you simply multiply the numerator and denominator by whatever is under the radical sign.

For example,

2/sqrt(3) , multiply by sqrt(3)/sqrt(3) to get 2sqrt(3)/3

When you have a higher root, like the cube root, 4th root and so on, the process is a little more complex.

Suppose you have   4/cuberoot(6).. we have to multiply by whatever will make a perfect cube under the radical.. so in this case we multiply by cuberoot(6 * 6) because that will give 3- 6's under the radical which makes cube root of 216 = 6

So the answer is 4cuberoot(6)/6, simplified to 2cuberoot(6)/3.

Same thing applies if there are variables in the denominator under the radical.. if you have cuberoot(x) you need to multiply by cuberoot(x^2) which makes cuberoot(x^3) = x.

Understanding vectors

The world of vectors surrounds our every move. Our world is one of pushes and pulls. Suppose you are leaning against a wall. What keeps you from sliding down the wall? Forces act against your body, including the pull of gravity. There are forces against your back, feet, shoulders, and head, if your head is touching the wall. Each of these forces has a direction in terms of an angle from the horizontal axis and amount of force exerted, called the magnitude. Such quantities that involve both direction and magnitude are called vectors.
Here's some examples of vectors. Suppose you are pulling a cart up a 20 degree incline, requiring an effort of 85 pounds. The magnitude is 85 pounds and the direction is 20 degrees.

As another example, suppose you are driving due west at 60 miles per hour. The magnitude is the speed, 60 miles per hour. The direction of the vector is due west.

Signs on a post that point in different directions and show the distance to each city are all vectors. Each sign defines a vector for each city.

Some quantities are described completely by given a magnitude, such as temperature. Suppose your body temperature is 98 degrees. Then the magnitude is 98, but there is no direction. These quantities are not vectors, they are scalars. Vectors can be multiplied by scalars.

A line segment that is given a direction is called a directed line segment. Geometrically, a directed line segment in a vector. Suppose we have line segment PQ. P is the initial point and Q is the terminal point. Vectors are often denoted by a boldface letter, such a v.

Suppose we have two vectors, v and w. There are four possible relationships between the two vectors. If v and w have the same magnitude and direction, they are equal. They can have the same magnitude and different direction. They can have the same direction and different magnitude, or they can have different magnitude and different direction.

To show that two vectors are equal, we take the initial and terminal points of both vectors and use the distance formula to get the magnitude. For example, suppose v has a starting point (0,0) and terminal point (4, 5) and w has a starting point of (-1, -2) and terminal point (3, 3). Using the distance formula learned in geometry, you see that magnitude of v = square root of 41 and the magnitude of w = square root of 41.
One way to show that v and w have the same direction is to get the slope of the lines on which they lie. Using the slope formula, we see both are 4/5 and both are pointed upward and to the right.

A vector can be multiplied by a real number. Suppose a vector is multiplied by 3. The vector will be 3 times as long as the original vector pointing in the same direction. If the vector is multiplied by a negative number, it points in the opposite direction from the original.

There are many other concepts dealing with vectors, such as vectors in the rectangular coordinate system, representing vectors in rectangular coordinates and finding its magnitude and operations of vectors. Those are topics for a more detailed discussion on vectors.

This introduction to vectors is one I use when working with students who are having difficulty understanding the basics of vectors.