वास्तविक संख्याओं पर संक्रियाएँ

Here we will be learning operations on Real Numbers.

वास्तविक संख्याओं पर संक्रियाओं का नियम

• एक परिमेय संख्या और अपरिमेय संख्या का योग या अंतर अपरिमेय होता है।
• अपरिमेय संख्या के साथ एक गैर-शून्य परिमेय संख्या का गुणनफल या भागफल अपरिमेय संख्या होती है।
• जब दो अपरिमेय संख्याओं को जोड़ा, घटाया, गुणा या विभाजित किया जाता है, तो परिणाम एक परिमेय या अपरिमेय संख्या हो सकती है।

यदि $a$ और $b$ धनात्मक वास्तविक संख्याएँ हैं, तो हमारे पास है,

• ${\sqrt {ab}}={\sqrt {a}}{\sqrt {b}}$
• ${\sqrt {\frac {a}{b}}}={\frac {\sqrt {a}}{\sqrt {b}}}$
• $({\sqrt {a}}+{\sqrt {b}})({\sqrt {a}}-{\sqrt {b}})=a-b$

• $(a+{\sqrt {b}})(a-{\sqrt {b}})=a^{2}-b$
• $({\sqrt {a}}+{\sqrt {b}})({\sqrt {c}}+{\sqrt {d}})={\sqrt {ac}}+{\sqrt {ad}}+{\sqrt {bc}}+{\sqrt {bd}}$
• $({\sqrt {a}}+{\sqrt {b}})^{2}=a+2{\sqrt {ab}}+b$

What are Mathematical Operations?

The four basic Mathematical operations are addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($/$).

Operations on Two Rational Numbers

These are some of the operations:

Addition of Two Rational Numbers

When two rational numbers are added, the result is a rational number. For example, $0.24+0.68=0.92$. $0.92$ can be written as ${\frac {92}{100}}$, which is a ratio or the ${\frac {p}{q}}$ form.

Subtraction of Two Rational Numbers

When two rational numbers are subtracted, the result is a rational number. For example, $0.93-0.22=0.71$ which can be written as ${\frac {71}{100}}$.

Multiplication of Two Rational Numbers

When two rational numbers are multiplied, the result is a rational number. For example, $0.5$ multiplied by $185$ is $92.5$, which can be written as ${\frac {925}{10}}$.

Division of Two Rational Numbers

When a rational number is divided by another rational number, the result is a rational number. For example, $0.352$ divided by $0.6$ is $0.58$, which can be written as ${\frac {58}{100}}$.

Operations on two Irrational Numbers

Addition of Two Irrational Numbers

When two irrational numbers are added, the result can be an irrational or a rational number. For example, ${\sqrt {3}}$ added to ${\sqrt {3}}$ is $2{\sqrt {3}}$ which can which is a rational number. However, when $2{\sqrt {5}}$ is added to $5{\sqrt {3}}$, we get a non-terminating and non-recurring decimal, an irrational number. It is written as $2{\sqrt {5}}+5{\sqrt {3}}$.

Subtraction of Two Irrational Numbers

Similarly, when two irrational numbers are subtracted, the result can be an irrational or a rational number. ${\sqrt {2}}$ is subtracted from ${\sqrt {2}}$, the answer is $0$. When $4{\sqrt {5}}$ is subtracted from $5{\sqrt {3}}$, we get $5{\sqrt {3}}-4{\sqrt {5}}$.

Multiplication of Two Irrational Numbers

The product of two irrational numbers can be an irrational number or a rational number. For example, when ${\sqrt {2}}$ is multiplied by ${\sqrt {2}}$, we get $2$ which is a rational number.