1.Indices and the Law of Indices
Indices in maths is used to refer to numbers that has a multiply of itself in an exponential form. In plain English, indices are a number with a power. The index of a number shows you how many times to use the number in a multiplication.
A simple example of index is am; where a is the base number and m is the power.
Indices are also regarded as powers of numbers. It defines a number that shows how many times a number multiplies by itself.
You should know about these following points before we proceed further with indices.
1. Index plural form is called an indices
2. Index can be interchanged with power or exponential
3. Indices rules apply when the base numbers are the same.
Now let’s define a real indices example below
7 x 7 = 72 = 49
The indices above show that 7 is the base number while 2 is the index of the 7.
In maths, the concept of indices can be understand and easily apply with laws of indices. There are about six laws of indices and we are going to look at each one. But before we proceed, we will have a glance at the entire laws.
Indices are a valuable way of more conveniently expressing large numbers. They as well offer us with a lot of useful properties for influence them with the use of what is known as the Law of Indices.
What are Indices?
The expression 25 is defined as follows:
25 = 2 x 2 x 2 x 2 x 2
We call “2” the base and “5” the index.
To influence expressions, we can reflect on using the Law of Indices. These laws merely apply to expressions with the same base, for instance, 34 and 32 can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 35 and 57 as their base varies (their bases are 3 and 5, correspondingly).
Six rules of the Law of Indices
a0 = 1
Any number, except 0, whose index is 0 is always equal to 1, in spite of the value of the base.
a0 = 1
a-m = 1/am
Simplify 2-2 = 1/4:
Rule 3: a-m x an = am+n
To proliferate expressions with the same base, copy the base and add the indices.
5 x 53
Simplify : (note: 5 = 51)
5 x 5 x 5 x 5 = 625
a-m ÷ an = am-n
To divide expressions with the same base, duplicate the base and subtract the indices.
An Example: 5(y9 – y5)
5(y9 – y5) = 5y4
(am)n = amn
To raise an expression to the nth index, replica the base and multiply the indices.
Simplify (y2)6: = y12
am/n = n√am
1252/3 = 25
Logical reasoning is a system of making conclusions based on a set of grounds or information. Normally, logical reasoning is subdivided into two main types known as deductive and inductive reasoning. Whereas the principles of logic can be used to form a strong argument for or against a conclusion, the system has a lot of vulnerabilities, which includes the potential for false grounds, fallacies, and intentional distortion of reason.
To make a conclusion with the use of logical reasoning, evidence or facts ought to be first presented. For example, if a grocer wants to know if he is selling more beets than turnips, he may collect evidence about the amount of the two vegetables in latest shipments, how many have been sold, and if any product loss has occurred as a result of theft or damage. If his information shows that he sold 52 turnips and 75 beets in the same month, with no loss as a result of theft or damage, he can logically conclude that he sells more beets than turnips based on the evidence.
The type of reasoning in the above instance is referred to as deductive reasoning. This type of logic takes place when the information adds up to a distinct, irrefutable conclusion. Given that the information is correct, deductive reasoning can show an absolute truth or fact. Inductive logic, by contrast, makes use of information or premises to determine a highly probable, but not total, conclusion. Whereas inductive logical reasoning can be far more difficult to comprehend than deductive reasoning, it commonly forms the bulk of the majority of logic-based arguments.
One type of inductive reasoning takes care of conclusions that have to do with the future. If the grocer from the first instance given wants to know whether he will sell more turnips or beets over the next month, an absolute answer is not possible to get hold of, for the fact that it is a game of chance. Based on his past sales, the grocer might suppose that since he sold more beets in January, he will also sell more in February; nevertheless, if an E.coli outbreak in beets at the beginning of February makes people afraid of buying any, his former conclusion may be false. Making use of his sales records and knowledge of buying trends, he may be able to deduce an inductive argument that suggests a high probability of selling more beets than turnips, but his premises cannot add up to an complete guarantee.
Logical reasoning can be a good servant but a poor master. Whereas the principles of making use of accurate information to draw a strong conclusion may be very good, they often times break down when they are made use of incorrectly. A logical fallacy happens when an erroneous or unconfirmed conclusion is drawn from premises. There are dozens of forms of logical fallacies that serve as tripwires and pitfalls to good logical reasoning . Take time to learn how to make a sound, and convincing argument.
Definition of Logical Reasoning
Logical reasoning is the process of making use of a rational, systematic series of steps based on sound mathematical procedures and given statements to come to a conclusion. Geometric proofs make use of logical reasoning and the definitions and properties of geometric figures and terms to state definitively that something is constantly true. In logical reasoning, an if-then statement (as well referred to as a conditional statement) is a statement formed when one thing means another and can be written as and read as “If P then Q.” A contra positive is the provisional statement formed when negating both sides of the implication and could be written asand read as “If not Q, then not P.” Anything that is not established is referred to as a conjecture.
Logical reasoning is the process which makes use of arguments, statements, information and proverbs to decide if a statement is true or false,. This leads to logical or illogical reasoning. In today’s logical reasoning, three various kinds of reasoning can be differentiated as deductive reasoning, inductive reasoning and abduction reasoning and are based on deduction, induction and abduction respectively.
Deductive reasoning emanates from the philosophy and mathematics and is the most observable type of reasoning. Deduction is a method for applying a common rule (major premise) in particular situations (minor premise) of which conclusions can be drawn. For instance,
Major premise: All humans are mortal
Minor premise: Socrates is human
Conclusion: Socrates is mortal
Straight away the observable and straightforwardness of the conclusion can be drawn from the premises above the example of deductive reasoning. Observe that deductive reasoning no fresh information is provided, it only rearranges information that is previously known or given into a new statement or conclusion.
The exact opposite of deductive reasoning is inductive reasoning. In this type of logical reasoning particular conclusions are generalized to general conclusions. A well-known hypothesis is ‘all swans are white’. This conclusion was derived from a large amount of observations without observing any black swan. Inductive reasoning, nevertheless is a risky form of logical reasoning since the conclusion can easily be mistaken when, looking at the swans model, a black swan is spotted. Nevertheless, in the modern day, inductive reasoning is the most frequently used type of reasoning in physics and philology.
Abductive reasoning is the third type of logical reasoning and is rather comparable to inductive reasoning, due to the fact that conclusions drawn here are based on probabilities. In abductive reasoning, it is assumed that the majority of reasonable conclusion is as well the correct one. For instance:
Major premise: – The jar is filled with yellow marbles
Minor premise: – I have a yellow marble in my hand
Conclusion: – The yellow marble was taken out of the jar
The adductive reasoning example vividly illustrates that conclusion might look obvious; however, it is solely based on the most believable reasoning. This type of logical reasoning is mainly used within the field of science and research.
Formal and Informal Logic Reasoning
In addition to these 3 types of logical reasoning, it is as well possible to make a difference between formal reasoning and informal reasoning. Formal reasoning is a type of logical reasoning based on valid premises and thus valid conclusions; therefore, it is a form of deductive reasoning. It makes available no new information, but just rearranges already known information to a new conclusion.
In addition to this formal reasoning we as well have informal reasoning. This type of logical reasoning possesses all the elements of formal reasoning, like the deduction part; nevertheless, it as well includes probabilities and truths about premises and conclusions. It can be said that informal reasoning is connected to abductive reasoning, one of the other three types of logical reasoning explained above
Linking these two forms of logical reasoning together with the three various forms results in the following differences in logical reasoning:
a. Formal deductive reasoning
b. Informal deductive reasoning
a. Formal inductive reasoning
b. Informal inductive reasoning
a. Formal abductive reasoning
b. Informal abductive reasoning
Wrong can be Right Logically
Within logical reasoning it can occasionally occur that the premises and conclusion look as if clearly wrong, but are logically speaking correct when applying one of the types of logical reasoning stated above. Notice that conclusions are drawn based on logical reasoning and not on the validity of the context of particular premises or conclusions. For instance:
Major premise: – Eating a lot makes you lose weight
Minor premise: – Craig is obese
Question: – What can we do to make Craig lose weight?
Conclusion: – We can make Craig eat a lot
By mere observation of the context of the words, you would think that this conclusion is incorrect, for the fact that you are aware that from everyday life, eating a lot does not make you lose weight at all but rather makes you gain weight. Nevertheless, based on logical reasoning this conclusion is most certainly correct, since both premises are valid, which routinely makes the conclusion a valid conclusion. What you require to understand is that the correct answer to any given logical reasoning argument needs the proper identification of relationships between assertions (normally facts and opinions), not the correctness of those assertions.
Logical Reasoning in Aptitude Tests
Logical reasoning commonly is a very significant section in aptitude tests and/or IQ tests. Logical reasoning is universal and it is made use of in every form of reasoning, in every job, in every field and every day. Therefore, if you possess good logical reasoning skills you ought to be able to apply this everywhere. Better developed logical reasoning skills make you able to comprehend, examine, and to query arguments based on statements or questions. These skills are generally made use of to identify clues that make an argument weaker, or to identify a particular assumption. Logical reasoning can be tested in quite a few different ways, nevertheless on Fibonicci you would get the most significant and most commonly accepted type of logical reasoning known as syllogisms.
There is a special notation known as functional notation that is constantly used in mathematics when one variable is described in terms of another. The notation f(x) [read f of x] is frequently used to name a second variable. Instead of writing y = 3x + 2 you may write f(x) = 3x + 2 or g(x) = 3x + 2 or possibly even y(x) = 3x + 2. You can make use of any letter. This notation showcase that f or g or y is a function of the variable x, which means that it can be articulated in terms of x. To find the value of f(2), merely replace every x with the value 2. To obtain the value of f (4), substitute each x in the given formula with the value 4. To discover the value of f(-3), substitute each x in the formula with the value -3. Observe that f(x) does NOT mean to multiply f times x.
Given f(x) = 2x + 3.
Find the values of a) f(0), b) f(7), c) f(-5).
(a) f(0) means that x = 0. Substitute the x with the value of 0.
f(x) = 2x + 3
f(0) = 2(0) + 3 = 3
(b) f(7) means that x = 7. Substitute the x with the value of 7.
f(x) = 2x + 3
f(7) = 2(7) + 3 = 14 + 3 or 17
(c) f(-5) means that x = -5. Substitute the x with the value of -5.
f(x) = 2x + 3
f(-5) = 2(-5) + 3 = -10 + 3 or -7
Surds, indices and logarithms are very much connected. They are, the majority of times learnt together. Therefore your understanding of indices and logarithms will assist towards your understanding and the use of surds. Numbers whose square roots cannot be determined in terms of rational numbers like √2, √3, √5 and so on are referred to as surds. Such numbers can constantly be found in Trigonometry when determining the ratio of angles; like Cos 30 = √3 / 2, tan 60 = √3, tan30 = 1/√3; and in coordinate geometry in the computation of distances. It is therefore essential for you to have a good understanding of surds.
At the end 0f this tutorial students ought to be able state what surd means, perform the mathematical operations of addition, subtraction, multiplication and division of surds. The students would as well be able to rationalize denominators of surds.
When you studied numbers, you were told that a number that is “square” is one that can be expressed as the square of a few other rational numbers. For instance:
9 = 32
81 = 92
(4/9) = (2/3)2
However, not all numbers are rational numbers, that is they do not possess exact square roots. Examples are:
2√3 and so on.
The square roots of numbers that do not possess exact square roots are called Surds.
Numbers like that are referred to as irrational numbers.
Even though estimated square roots of irrational numbers can be found from tables of square roots, it is normally simpler to work with SURDS themselves. You must however not that at this stage whenever you work with square roots, only positive square roots are taken into consideration.
Reduction to Basic Form
Any surd which is made up of a square number as a factor inside the radical – the square root sign is not in the basic form. For instance
√108 , are not in the basic form since they could be reduced further through simplification. The following examples will showcase this concept:
√27=√ (9X3) = √9√3 = 3√3
√50 = √ (25 X 2) =√ 25 √2 = 5√2
√108 = √ (36 X 3) =√ 36 √3 = 6√3
From the three examples above, 3√3, 5√2 and 6√3 cannot be simplified further; they have been reduced to their basic form. Surds that cannot be simplified any further are said to be in their basic form.
Addition and Subtraction of Similar Surds
Surds that are in the basic form can be added and subtracted. The example below will demonstrate the idea.
Simplify √80 +√20 -√45
First reduce all the surds to their basic forms.
= √(16X5) + √(4X5) – √(9X5) RHS(the right hand side):
=4√ 5 + 2√5 – 3√5 = 6√5 – 3√5 = 3√5
Observe that the above surds can be added or subtracted because they are in comparable form, that is, numbers under the radical signs are the same and they have the same index. Mixed surds like 2√2 + 2√7 – 2√3 are not equivalent, therefore, they cannot be added or subtracted. This means that they cannot be simplified further.
Multiplication and Division of Surds
Multiplication and division of surds are performed through two fundamental laws of surds.
i) For multiplication of surds, the rule is: √a √b = √(a.b)
= √(5X3 )
√ 2 √7
= √ (2X7)
= √ (14)
For division of Surd, the following rule applies: √a /√b =√ (a/b)
√6/√3 = & radical (6/3)
√18/√3 = √(18/3)
Rationalizing the Denominator
A surd like √3/5 cannot be simplified further; but a surd like 2/√3 can be written in a convenient form, since it is not normal to have the radical ie the square root symbol in the denominator. The process of removing the radical from the denominator is known as rationalization. In order to perform rationalization, you must have knowledge of conjugate surds.
Given a surd (a – √b), its conjugate is defined as (a + √b) and vice-versa.
When a surd is multiplied by its conjugate, their product is no more a surd.
Multiply (a+√b) by its conjugate (a-√b) to obtain (a+√b)(a-√b)
= a2 – a√b+a√b-√b√b
= a2 – √b√b
You can see that the result is no longer a surd. Now, the rule is: To rationalize a radical denominator of a surd, multiply both numerator and denominator of the surd by the conjugate of the denominator.
Points to remember about Surds
A surd is a square root which cannot be reduced to a whole number. For example,√4 = 2 is not a surd. But √5 that is not a whole number is surd. You could use a calculator to find that but, instead of this; we frequently leave our answers in the square root form, as a surd.
Surds are mathematical expressions that contain square roots. Nevertheless, it must be emphasized that the square roots are ‘irrational’ i.e. they do not result in a whole number, a terminating decimal or a recurring decimal.