Octal Representation of Numbers

Numbers is a vast subject but not a very tough topic in mathematics to understand. The vastness is because of the deductive skills of scholars and different representation of the same number to increase the security of the same. Octal Numbers is one such type of representation of number like the other representations like decimal, binary, etc. The word Octal or octa number system refers to the number eight and in Octal Number System the number is supported with the base 8. Octals are very similar to the decimal number other than in decimal we have 0 – 9 and in this we have 0 – 7.

It is not a very big task to convert decimal to octa number system or from this to decimal number. The Decimal to Octal Conversion can be understood through an example. But before going to an example let us get to know the steps to be followed while converting the decimal to octa number system.

The following are the steps used for the conversion: One should know the division and multiplication of numbers well so that he / she can solve the problem in ease. The decimal number should be written. The number should be divided by 8.

The quotient of the number should be written below. The remainder of the number after dividing has to be written on the right end. The number written below has to be further divided by 8. This division process has to be done until the number that is the remainder is less than the octa number system digits.

The process is completed once the above steps are followed. The answer has to be written now. The answer can be found by writing the numbers that is the remainder starting from the final remainder bottom to top. The answer should be written with subscript 8 to avoid confusion. Before going to an example for further explanation we should know the Octal Table which represents the octa number system in a table form with the corresponding decimal number for them.

This representation helps in the conversion of octa number system to decimal and the conversion of decimal to octa number system.

Let us consider the following example for understanding the conversion of decimal to octa number system.
Convert 9910 to octa number system: 99 / 8 = 12 with a remainder 3, 12 / 8 = 1 with a remainder 4, the answer for the question which is given will be: 9910 = 1438.

linear algebra sample final

Linear algebra is mainly use to find the solution of systems of linear equations in some unknowns. Linear algebra has a demonstration in analytic geometry and is generalize in operator theory. Linear algebra associates with the families of vectors called vector spaces, and with functions contain one input vector and output another vector, according to rules. Linear algebra has the application with the theory of systems of linear equations, determinants, and matrices. The sample problems are discussed below for final review.

Linear Algebra Sample Problems for Final:

The sample problems are solved in detail for final review as shown below.

Problem 1:

Evaluate the linear equation   -3(x - 4) / 6 - (x - 2) / 3 = -2 x

Sol:

The given equation has rational expressions. To eliminate the denominators by multiplying by the LCD
-3(x - 4) / 6 - (x - 2) / 3 = -2 x

The LCD is equal to 6*3 = 18. Multiply both sides by LCD.
18 * [-3 (x - 4) / 6 - (x - 2) / 3] = 18* [-2 x]

Simplify to eliminate the denominator.
-9(x - 4) - 6(x - 2) = -36x

Multiply factors and grouping the terms

-15x + 48 = -36x

Subtract 48 to both sides
-15x + 48 - 48 = -36x -48

Group like terms
-15x = -36x - 48

Add 36x to both sides
-13x + 36x = -36x - 48 +36x

Grouping the terms
23x = -48

Multiply both sides by `1/23 `
x = - `48/23`

Conclusion:
x = - `48/23` is the solution for the above given equation.

Sample Linear Algebra Practice Problems for Final:

1) Evaluate the linear equation   -3(x - 4) / 7 - (x - 2) / 2 = -2 x

2) Solve the linear equation -5(-x - 7) = 5x – 32.

3) Evaluate the linear equation   -3(-x +3) = x - 7

Answers:

1)   X = -38/15 is the solution for the above given equation.

2) The above equation has no solution.

3)  X=One is the solution for the above equation.

Order of Operations

Order or operations is one of the most important topics in mathematics. Order of operations is also known as PEMDAS or BODMAS. As the term suggests, order of operations refers to the steps to perform a mathematical operation. When it is a simple addition or subtraction, you can add or subtract instantly. For example: Sunny bought Nerf guns India from Fisher Price India collection. He already has Angry Birds toys India and Dartboard games.

How many variety of toys and games collection he has in total: Nerf guns + Angry Birds toys India + Dartboard = 3 variety of toys and games. This is addition operation in mathematics. John has three Nerf guns of Fisher Price India brand. He gave one to his cousin Andrew. How many Nerf guns does he have? 3 Nerf guns – 1 Nerf gun = 2 Nerf gun. This is subtraction operation in mathematics. But what do you do when a mathematical expression has more than one operation. For example: Sunny had 3 guns, his father gave him 2 new guns and he gave one to his friend John. How many he has now? Mathematically: (3+2) – 1 = 4. This is a mathematical expression.

While calculating a mathematical expression, it is essential to follow some rules or order of operation. This order of operation is referred to as BODMAS or PEMDAS as abbreviation for the functions in mathematics. The order of operations is expressed as below.
1. [B] Brackets or Parenthesis
2. [O] Orders (Exponents, roots etc)
3. [DM] Division and Multiplication
4. [AS] Addition and Subtraction

The above order of operation states that in a mathematical expression, at first the operation inside brackets needs to be calculated, thereafter the powers, roots and more, then division and multiplication and finally addition and subtraction. This should be the order of operation while simplifying or calculating a mathematical expression.
For example: There are six kids. Father bought one Nerf Guns India and two building block toys for each kid. How many toys father bought in total?
Mathematically, (1+2) x 6
Applying order of operations (1+2) = 3 x 6 = 18
Therefore, father bought 18 toys in total.

This is referred as order of operations or BODMAS in mathematics.

Estimating with Confidence

Concept of ‘estimation with confidence’ is normally used by everybody in their daily lives. For example, a person may say ‘I believe there are 90 percent chances that Congress will win in the next elections with more than 350 seats’. Here, the person’s confidence level is 90 percent and he is estimating by how much seats congress will win. Let’s move forward to the formal concept of ‘estimation with confidence’.

There are two types of tests one-tailed and two-tailed, we will discuss ‘estimation with confidence’ in each of the case separately.

Estimating with Confidence in a One Tailed Test

In the single tailed test we estimate with the given level of confidence whether the estimated value is same as the population value or either statistically greater than or less than the population value/expected value (for simplicity we are assuming normal distribution). In this we can have right or left tailed test.

Left Tailed : Let us assume the given level of confidence is 85%, from Fig-I we see yellow area is the rejection region. So, with 85% confidence we can say that our estimated value won’t b statistically smaller from the population value/expected value and there is only 15 % probability that it will be statistically smaller.

FIG-I

Right Tailed: In this case the calculation remains the same, the only difference we don’t want statistically greater value than population value/expected value rather than less than as in the previous case. We represent right tailed estimation as in Fig-II. Here yellow shaded region is the rejection area. Confidence level is 85%. So, with 85% confidence we can say that our estimated value won’t b statistically greater from the population value/expected value and there is only 15 % probability that it will be statistically greater.

FIG-II

Estimating with Confidence in Two Tailed Test

Under a two tailed estimation with confidence we do not want either too big or too small value in comparison to the population value/expected value. In other words, we have rejection area on the both tails of the curve (as shown in Fig-III). Here, the yellow shaded area is the rejection area. Here, the confidence level is of 70%. So, with 70% confidence we can say that our estimated value won’t b statistically different from the population value/expected value and there is only 30% probability that it will be statistically different.

FIG-III