An Introduction to Fundamental Algebra

When we want to know the length of a piece of cloth, we find that there is a unit, called metre

When we want to know the distance between Dhaka and Calcutta, we find out that this distance contains a smaller distance called Kilometre

When we want to find out the weight of a quantity of rice, we find out that this weight contains a smaller unit called Kilogram.

So from the above mentioned examples, it is very clear that whenever we want to find out the measurement of a certain quantity, we do so by finding out that the quantity contains smaller thing of the same kind. The smaller thing, chosen for this purpose is called unit, and the number which shows how often this unit is contained in the thing is called numerical measure.

Anything that can be denoted by a number is called a quantity. Thus time, weight, money, distance, etc are shown in the preceding article as quantities.

Quantity is also often used in the sense of number, integral or fractional.

An algebraic expression is also sometimes called a quantity

Example

Q) If a minute and a half is represented by 30 what is the unit of time?

A) A minute and a half is equivalent to 90 seconds

Now since 30 is the measure of 90 seconds, it is clear that the unit of time is contained 30 times in 90 seconds

Hence the unit of time is $\dfrac{1}{30}$ –th part of 9 seconds, and is, therefore, equal to 3 seconds.

What is Algebra?

Algebra like Arithmetic is a science of numbers, with this distinction that the numbers in algebra are generally denoted by letters instead of figure. Hence whenever concrete quantities come under the domain of Algebra, it is only their measures with which we must concern ourselves.

Symbols Signs and Substitutions:

1) Symbols:

The letters of the alphabet $a,b,c,d,...,z$ , are used to denote numbers and the signs $+, -, \times, =, \div$ , etc. are used either to denote operations to be performed upon the number to which they are attached or as abbreviations, these letters and signs are called symbols.

The letters as distinguished from the signs are called symbols of quantity.

2) The Plus Sign:

The + sign is read plus and and when placed before a number indicated that the number is to be added to what precedes it. Thus $a+b$ , which is read “ $a$ plus $b$ “, means that the number denoted by $b$ is to be added to that denoted by $a$ . Hence if $a$ denotes 5 and $b$ denotes 3 , then $a+b=8$ . Again $a+b+c$ means that the number denoted by $b$ is to be added to that of $a$ and the result should be added to the number denoted by $c$ . Hence if $a,b,c$ denote 5,3,2 respectively, $a+b+c=10$

3) The Minus Sign:

The sign ‘-‘ is read minus and when placed before a number indicated that the number is to be subtracted from what precedes it. Thus $a-b$ , which is read as “ $a$ minus $b$ , means that the number denoted by $b$ is to subtracted from the number denoted by $a$ . So if $a$ denotes 8 and $b$ denotes 3 then $a-b=5$ . Again $a-b-c$ , means that number denoted by $b$ is to to subtracted from the number denoted by $a$ and then the result is to subtracted from $c$ . Hence if $a,b,c$ denote 8,3,1 respectively, $a-b-c$ denotes $a-b-c=8-3-1=4$ .

Note:

When numbers or quantities are connected with each other with plus or minus, the order of operations is from left to right. Thus $a-b+c$ means that the number denoted by $b$ is to be subtracted from that denoted by $a$ and to the result thus obtained is to be added then number denoted by $c$ .

4) The Sign Plus or Minus:

The sign $\pm$ is read “plus or minus” and when placed before a number indicates that the number is to be either added or subtracted from what precedes it. Thus if $a$ denotes 7 and $b$ denotes 2, $a \pm b$ denotes either $a+b=9$ or, $a-b=5$

5) Sign of Difference:

The sign $\sim$ when placed between two numbers indicates that the smaller of the two is to be subtracted from the greater. Thus, if $a$ denotes 5 and $b$ denotes 8 $a \sim b=8-5=3$

6) Sign of Multiplication:

The sign ‘ $\times$ ‘ is read “into” and when placed between two numbers indicates that the number on the right is to be multiplied by that on the left, Thus $a \times b$ means that the number denoted by $b$ is to be multiplied to $a$ , hence if $a$ denoted 5 and $b$ denoted 2 then the resulting number will denote 10

Sometimes the sign ‘ $\times$ ‘ is replaced by a dot. Thus, $a.b$ and $5.2$ means the same as $a \times b$ and $5 \times 2$ .

7) Sign of Division:

The sign $\div$ is read by and when placed between two numbers indicated that the number on the left of it is to be divided by that on the right. Thus ${a}{b}$ means that the number denoted by $a$ is to be divided by that denoted by $b$ .

8) Expression and Term:

Any intelligible collection of letters, figures and signs of operation are called algebraic expression. Such a collection is also sometimes called algebraic quantity or briefly a quantity.

The parts of an algebraic expression that are connected by a plus or minus are called its terms. Thus, $5a+ab \div c \times d-8c \times f \div g$ is an algebraic expression of which the terms are $5a, ab \div c \times d, 8c \times f \div g$ .

Expressions are either simple or compound. A simple expression is one which has no parts are connected by the sign ‘+’ and ‘-‘, i.e., which has only one term and hence can also be called monomial. A compound expression is one which contains two or terms.

If it contains two terms, it is called binomial, three terms then trinomial and if more than three polynomial.

9) Functions variables:

Any expression involving a letter is called a function of that letter. Thus $x^8+5x+8$ is a function of $x$ . $a^2+ab+b^2$ is a function of $a$ and $b$ . The letters of which a function consists are called its variables. Thus $x^8+ 5xy+y^2$ is a function of which the variables are $x$ and $y$ .

10) Sign of Equality:

The sign ‘ $=$ ’ is read ‘equals to’ and when placed between two expressions indicates that they are equal to one another.

11) Coefficient:

The number expressed in figures or symbols which stands before an algebraic quantity as a multiplier is called its coefficient. Thus in 5abc, 5a is the coefficient of bc. A coefficient which is purely a numerical quantity is called a numerical coefficient.

12) Power, Index Exponent:

If a quantity is multiplied by itself any number of times, the product is called power of that quantity. For example $a \times a \times a \times a=a^4$

The small figure or letter placed above a quantity and to the right of it to express its power is called the Index or Exponent of that power.

13) Brackets:

Each of the symbols {}, [], () is called a pair of brackets. When an algebraic expression is enclosed within brackets it is to be regarded as a single quantity by itself. Thus $(a+b)x$ means that the umber denoted by $x$ is to be multiplied to the result of $a+b$

14) Roots:

That quantity whose square is equal to any given quantity a is called the square root of a and is denoted by the symbol $\sqrt{a}$

15) Like and unlike terms:

Terms or simple expressions are said to be like when they do not differ at all or differ only in their numerical coefficients, otherwise they are called unlike.

16) Factor:

If any number is equal to the product of two or more numbers, each of the latter is called a factor of the former.

Examples

1) If $a=2, b=3, c=5$ , find the value of $5a+8b+7c$

$5a= 5 \times 2= 10$

$8b= 8 \times 3 =24$

$7c= 7 \times 5 =35$

$\therefore$ , $5a+8b+7c=10+24+35=69$

2) If $a=1, b=2, c=3, d=6, e=5, f=0$ , find the value of:

$abc -d \div b \times a +def +b \div a \times c -d +bc$

$abc=1 \times 2 \times 3=6$

$d \div b \times a=\dfrac{6}{2} \times 1=3$

$def=6 \times 5 \times 0=0$

$b \div a \times c={2}{1} \times 3=6$

$d \div bc={6}{2} \times 3=1$

Hence required value $=6-3+0+6-1=8$

3) If $a=3$ find the numerical value of $a^5-5a$

$a^5 =a \times a \times a \times a \times a = 243$

$5a= 5 \times 3= 15$

Hence value $243-15=228$

4) If $a=3, b=5, c=8, d=12 and e=20$ , find the difference between the numerical values of:

$a[c+b^2- a(e-d)]$ and $a[c+( b^2-a)(e-d)]$

1st expression: $3 \times [8+5^2-3 \times (20-12)]$

$=3 \times [8+25-3 \times 8]$

$=3 \times [8+25-25]$

$=3 \times 9 =27$

2nd expression: $3 \times [8+(5^2-3) \times (20-21)]$

$= 3 \times [8+22+8]$

$= 3 \times [8+176]$

$=3 \times 184$

$=525$

Required value: $525-27=525$

Exercise

What will be the measure of 2 quintals and 20 kilograms, when a kilogram is the unit of weight?
If Rs 5.71 is the unit of money, what will be the measure of Rs. 51.39?
If the unit of weight is 7.5 kilograms, what number will represent 1.5 quintals?
If a sum of 400 rupees be represented by 16 what will be the measure of Rs. 225
If a length of 8 metres and 8 decimetre be represented by 22, what will be the measure of 4 metres and 8 decimetre
If find
1. $\: b+c \times a$
2. $\: a \div c \div 2b$
3. $\: 80 \div c \times ab +80 \div ac \times b$
If find the numerical values of:
1. $\: 8m-3p \div mn+p \times 3r+5s \div 2 \times p$
2. $\: s \times 6 \div 5m \times 8p \div 16n$
Find the value of $y^6 - 65y^4 + 66y^2- 21y +40$ , when $y=8$
Find the value of $50y^7- 51y^4+ 35y- 563y^5 -19$ when $y=3$

Example

What is Algebra?

Symbols Signs and Substitutions:

Examples

Exercise

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