A mathematical operation of dividing an algebraic symbol by another algebraic symbol is called division of algebraic symbols.

There are two types of algebraic symbols in algebraic mathematics and often appear in division form in four different styles. The process of performing the division with algebraic symbols is same in all the cases but the approach of simplification slightly varies. These four cases are very useful to study the system of multiplying algebraic symbols in the algebraic mathematics.

Like algebraic symbols are divisible symbols due to the like property. So, the quotient of division of two like algebraic symbols is either $1$ or $\u20131$ and it depends upon the signs of the like algebraic symbols which involve in the division.

1.

Division of Like Algebraic symbols

$a$ is a alphabetic letter and also an algebraic symbol and divide it by the same algebraic symbol. The numerator and denominator are having same algebraic symbol in this division. So, the quotient of them is $1$ by the successful division.

$\frac{a}{a}$

$\Rightarrow \frac{a}{a}=1$

$\u2013a$ is a negative algebraic symbol and divide it by the same symbol. Same result repeats in the case of division of negative like algebraic symbols.

$\frac{\u2013a}{\u2013a}$

$\Rightarrow \frac{\u2013a}{\u2013a}=1$

If an algebraic symbol appears in both numerator and denominator of a division, the quotient of the division is $1$.

2.

$a$ and $\u2013a$ are two like algebraic symbols which have opposite signs. The quotient of division is $\u20131$ due to the successful division and the negative sign in the quotient represents the involvement of like algebraic symbols which have opposite signs.

$\frac{a}{\u2013a}$

$\Rightarrow \frac{a}{\u2013a}=\frac{1\times a}{(\u20131)\times a}$

$\Rightarrow \frac{a}{\u2013a}=\frac{1}{(\u20131)}\times \frac{a}{a}$

$\Rightarrow \frac{a}{\u2013a}=\frac{1}{(\u20131)}\times 1$

$\Rightarrow \frac{a}{\u2013a}=\frac{1}{(\u20131)}$

$\Rightarrow \frac{a}{\u2013a}=\u20131$

Now, reverse this case but the quotient of the division is also same.

$\frac{\u2013a}{a}$

$\Rightarrow \frac{\u2013a}{a}=\frac{(\u20131)\times a}{1\times a}$

$\Rightarrow \frac{\u2013a}{a}=\frac{(\u20131)}{1}\times \frac{a}{a}$

$\Rightarrow \frac{\u2013a}{a}=\frac{(\u20131)}{1}\times 1$

$\Rightarrow \frac{\u2013a}{a}=\frac{(\u20131)}{1}$

$\Rightarrow \frac{\u2013a}{a}=\u20131$

If two like algebraic symbols with opposite signs are involved in division, the quotient of the division is $\u20131$.

Unlike algebraic symbols actively involve in division but one algebraic symbol cannot be divided by another algebraic symbol due to the unlike. So, the quotient of division of two unlike algebraic symbols is either a positive or a negative fraction but it depends upon the signs of the unlike algebraic symbols.

1.

Division of Unlike Algebraic symbols

$a$ and $b$ are two letters and unlike algebraic symbols. The alphabet in numerator $a$ cannot be divided by the alphabetic letter in denominator $b$ in the division due to the unlike. They are always in the same form when they participate in the division.

$\frac{a}{b}$

$\u2013a$ and $\u2013b$ are two unlike algebraic symbols. They are actually formed by the two unlike algebraic symbols $a$ and $b$ but their signs are negative and like. The numerator $a$ cannot be divided by the denominator $b$ in this division but their signs can be cancelled.

$\frac{\u2013a}{\u2013b}$

$\Rightarrow \frac{\u2013a}{\u2013b}=\frac{(\u20131)\times a}{(\u20131)\times b}$

$\Rightarrow \frac{\u2013a}{\u2013b}=\frac{(\u20131)}{(\u20131)}\times \frac{a}{b}$

$\Rightarrow \frac{\u2013a}{\u2013b}=1\times \frac{a}{b}$

$\Rightarrow \frac{\u2013a}{\u2013b}=\frac{a}{b}$

The two negative unlike algebraic symbols formed a fraction in algebraic form which is equal to the fraction of two positive unlike algebraic symbols.

So, if two unlike algebraic symbols with same signs participate in division, the quotient is a positive fraction of the algebraic symbols.

2.

$\u2013a$ and $b$ are two unlike algebraic symbols which have opposite signs and the quotient of the division is a negative fraction in algebraic form.

$\frac{\u2013a}{b}$

$\Rightarrow \frac{\u2013a}{b}=\frac{(\u20131)\times a}{1\times b}$

$\Rightarrow \frac{\u2013a}{b}=\frac{(\u20131)}{1}\times \frac{a}{b}$

$\Rightarrow \frac{\u2013a}{b}=(\u20131)\times \frac{a}{b}$

$\Rightarrow \frac{\u2013a}{b}=\u2013\frac{a}{b}$

Now consider $a$ and $\u2013b$ to perform division and it returns the same result.

$\frac{a}{\u2013b}$

$\Rightarrow \frac{a}{\u2013b}=\frac{1\times a}{(\u20131)\times b}$

$\Rightarrow \frac{a}{\u2013b}=\frac{1}{(\u20131)}\times \frac{a}{b}$

$\Rightarrow \frac{a}{\u2013b}=(\u20131)\times \frac{a}{b}$

$\Rightarrow \frac{a}{\u2013b}=\u2013\frac{a}{b}$

Therefore, if two unlike algebraic symbols having opposite signs involved in division, the quotient of the division is an algebraic form based negative fraction.

Division of Like Algebraic Symbols having same signs

$\frac{a}{a}=1$

$\frac{\u2013a}{\u2013a}=1$

Division of Like Algebraic Symbols having opposite signs

$\frac{a}{\u2013a}=\u20131$

$\frac{\u2013a}{a}=\u20131$

Division of Unlike Algebraic Symbols having same signs

$\frac{a}{b}=\frac{a}{b}$

$\frac{\u2013a}{\u2013b}=\frac{a}{b}$

Division of Unlike Algebraic Symbols having opposite signs

$\frac{\u2013a}{b}=\u2013\frac{a}{b}$

$\frac{a}{\u2013b}=\u2013\frac{a}{b}$

Copyright © 2012 - 2017 Math Doubts, All Rights Reserved