Raju

Imaginary Unit and Standard Complex Form

by

Imaginary Unit and Standard Complex Form

The Imaginary Unit is defined as
i =√-1

The reason for the name “imaginary” numbers is that when these numbers were first proposed several hundred years ago, people could not “imagine” such a number.

It is said that the term “imaginary” was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we find the imaginary unit being used in mathematics and science. Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity.

Imaginary numbers occur when a quadratic equation has no roots in the set of real numbers.

imaginary-number-standard-number-1

A pure imaginary number can be written in bi form where  b  is a real number and   i   is   √-1

A complex number is any number that can be written in the  standard form  a  +  bi,  where a  and  b are real numbers and  i  is the imaginary unit. .

A complex number is a real number a, or a pure imaginary number bi, or the sum of both.

Note these examples of complex numbers written in standard a + bi form: 2 + 3i, -5 + bi .

imaginary-number-standard-number-2

Rational (Fractional) Exponents

by

Rational (Fractional) Exponents

Rational (fractional) exponents are an alternate way to express roots

Rational-Exponents-1

Notice:   The denominator of the rational exponent becomes the index of the radical, and the numerator becomes the exponent of the radicand (expression inside the radical).
Rational-Exponents-2

We have already discussed simplifying radicals such as:
Rational-Exponents-3

Let’s look at these two problems in a new light! When asked to simplify these radicals, it is often easier to rewrite the radicals using rational exponents and solve the problems by dealing with the laws of exponents.

Notice how applying the rules for dealing with the exponents makes quick work of the variable

Rational-Exponents-4

Look at these examples:
Rational-Exponents-5

When dealing with rational exponents, the Rules for Exponents are still valid!!!
Rational-Exponents-6
Rational-Exponents-7

Check out how these problems are done using rational exponents:
Rational-Exponents-8

Radicals | Simplification, Properties, Addition and Multiplication of Radicals

by

Radicals | Simplification, Properties, Addition and Multiplication of Radicals

A radical or the principal nth root of k:
k, the radicand, is a real number.
n, the index, is a positive integer greater than one.

radpic1

 Properties of Radicals:

Properties-of-radicals

Simplifying Radicals:
Radicals that are simplified have:
– no fractions left under the radical.
– no perfect power factors in the radicand , k.
– no exponents in the radicand , k, greater than the index, n.
– no radicals appearing in the denominator
of a fractional answer.

Examples:  (The following examples demonstrate various solution methods.)

Simplify:  Factor the radicand to isolate the perfect power factor(s), which will allow them to be removed from under the radical.  You will need to remember your rules for working with exponents in order to isolate the perfect powers.

perfect squares
4, 9, 16, 25, 36, …
  x2, x4, x6, x8, …
  x2y2, x2y4, 16x6y8, …
powers are “even”

perfect cubes
8, 27, 64, 125, …
  x3, x6, x9, x12, …
  x3y3, x3y6, 27x6y9, …
powers are “multiples of 3”

radicals-examples
To add radicals

simplify first if possible, and add “like” radicals.

adding-radicals

To multiply radicals 

multiplying -radicals-1
multiplying -radicals -2

What are the Physical Changes of a Substance

by

What are the Physical Changes of a Substance

Physical changes
Physical changes are the changes in which no new substances are formed. For example, the formation of ice from water kept in a freezer is a physical change because the ice when kept in the open changes back to water.

In most of the physical changes, properties such as colour, shape, size, or physical form of the substance may change. Physical changes may or may not be reversible. An example of physical change is discussed below.

Crystallization
Crystals are the purest solid form of a substance having a definite geometrical shape. The process by which an impure compound is converted into its crystals is known as crystallization.
This process is used in obtaining common salt from seawater.

Let us take common salt as an example. Chemically, common or table salt is sodium chloride (NaCl). Common salt is obtained by evaporating sea water. The salt, thus obtained, contains certain undesirable substances such as magnesium chloride, sand, etc., mixed with sodium chloride. These impurities are removed from common salt through the process of crystallization.

A crystal of sodium chloride
A crystal of sodium chloride

In crystallization, maximum amount of common salt is dissolved in boiling water. The solution is then filtered to remove the insoluble impurities. The filtered solution is left undisturbed for a few hours. Sodium chloride will aggregate and form crystals with well- defined geometrical shapes, leaving behind the undesirable impurities in the solution.

Activity
Aim: To prepare crystals of copper sulphate from an impure sample of copper sulphate
Materials needed: A beaker, distilled water, powdered copper sulphate, stirrer, and sulphuric acid
Method:
1. Take a beaker and fill it half with distilled water. Now keep on adding powdered copper sulphate into the water with constant stirring till no more of it dissolves.
2. Add half a test tube of sulphuric acid to it.

physical-changes-substance-2
3. Start heating and again add copper sulphate till a saturated solution is produced.
4. Filter the hot solution immediately to remove insoluble impurities and keep the saturated solution of copper sulphate undisturbed for a few hours.
Observation: Well-shaped crystals of copper sulphate are formed.