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. However, when ${\sqrt {2}}$ is multiplied by ${\sqrt {3}}$, we get ${\sqrt {6}}$ which is an irrational number.

Division of Two Irrational Numbers

Similar to multiplication, we can get either an irrational number or a rational number as a result when an irrational number is divided by another. For example, when ${\sqrt {2}}$ is divided by ${\sqrt {2}}$, we get $1$ which is a rational number. But when ${\sqrt {2}}$ is divided by ${\sqrt {3}}$, we get ${\frac {\sqrt {2}}{\sqrt {3}}}$, which is an irrational number.

Operations on a Rational and an Irrational Number

Addition of an Irrational and a Rational Number

The sum of a rational and an irrational number is always irrational. For example, when $2$ is added to $5{\sqrt {3}}$, we get $2+5{\sqrt {3}}$, which is a rational number.

Subtraction of an Irrational and a Rational Number

The difference between a rational and an irrational number is always irrational. For example, when we subtract $5{\sqrt {3}}$ from $2$, we get  $2-5{\sqrt {3}}$, which is irrational.

Multiplication of an Irrational and a Rational Number

The product of a rational and an irrational number might be rational or irrational. For example, when $2$ is multiplied by ${\sqrt {2}}$, we get $2{\sqrt {2}}$ which is an irrational number, but when ${\sqrt {12}}$ is multiplied by ${\sqrt {3}}$, we get ${\sqrt {36}}$, or $6$, which is a rational number.

Division of an Irrational Number with a Rational Number

When a rational number is divided by an irrational number or vice versa, the quotient is always an irrational number. For example, when $8$ is divided by ${\sqrt {2}}$, we get ${\frac {8}{\sqrt {2}}}$, which is an irrational number. The answer can be further simplified to $4{\sqrt {2}}$ which is also an irrational number.

Examples

1.$({\sqrt {11}}+{\sqrt {7}})({\sqrt {11}}-{\sqrt {b}})$

$11-7=4$

2.$({\sqrt {3}}+{\sqrt {7}})^{2}$

$({\sqrt {3}})^{2}+2({\sqrt {3}})({\sqrt {7}})+({\sqrt {7}})^{2}$

$3+2({\sqrt {21}})+7$

$10+2({\sqrt {21}})$

3. $(5+{\sqrt {7}})(2+{\sqrt {5}})$

$10+5{\sqrt {5}}+2{\sqrt {7}}+{\sqrt {35}}$

4.$({\sqrt {7}}+{\sqrt {5}})({\sqrt {7}}-{\sqrt {5}})$

$({\sqrt {7}})^{2}-({\sqrt {5}})^{2}$

$7-5=2$

यहां हम वास्तविक संख्याओं पर संक्रियाओं की विधि को सीखेंगे।

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,

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

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

उदाहरण

1.$({\sqrt {11}}+{\sqrt {7}})({\sqrt {11}}-{\sqrt {b}})$

$11-7=4$

2.$({\sqrt {3}}+{\sqrt {7}})^{2}$

$({\sqrt {3}})^{2}+2({\sqrt {3}})({\sqrt {7}})+({\sqrt {7}})^{2}$

$3+2({\sqrt {21}})+7$

$10+2({\sqrt {21}})$

3. $(5+{\sqrt {7}})(2+{\sqrt {5}})$

$10+5{\sqrt {5}}+2{\sqrt {7}}+{\sqrt {35}}$

4.$({\sqrt {7}}+{\sqrt {5}})({\sqrt {7}}-{\sqrt {5}})$

$({\sqrt {7}})^{2}-({\sqrt {5}})^{2}$

$7-5=2$