Comprehensive theory, key formulas, diagrams, and memory aids for Number Systems.
The number system is the fundamental language of mathematics. From counting objects using natural numbers to understanding the profound implications of real numbers and irrational quantities, this chapter introduces you to the vast continuum of numbers. In Class 9 mathematics, Number Systems form the foundation for algebra, coordinate geometry, and polynomials.
The number line is a visual representation where every point corresponds to a unique real number, and every real number corresponds to a unique point on the line.
Between any two rational numbers, there exist infinitely many rational numbers. For instance, between $1$ and $2$, we can find $3/2$, between $1$ and $3/2$ we find $5/4$, and so on.
In the 5th century BC, the Pythagoreans in Greece discovered that there are numbers that cannot be written in the form of $p/q$. These numbers are called Irrational Numbers.
A number is called irrational if it cannot be written in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$.
When we combine the collection of all rational numbers and irrational numbers together, we get the collection of Real Numbers ($\mathbb{R}$). Thus, every real number is either rational or irrational.
One of the most effective ways to distinguish between rational and irrational numbers is through their decimal expansions.
When you convert a rational number ($p/q$) into a decimal, two things can happen: 1. Terminating Decimal Expansion: The remainder becomes zero after some steps. For example, $1/2 = 0.5$, $7/8 = 0.875$. 2. Non-Terminating Repeating (Recurring) Decimal Expansion: The remainder never becomes zero, but a repeating block of digits appears in the quotient. For example, $1/3 = 0.333\dots = 0.\overline{3}$, and $1/7 = 0.142857142857\dots = 0.\overline{142857}$.
Rule: A number whose decimal expansion is terminating or non-terminating repeating is rational.
The decimal expansion of an irrational number is non-terminating and non-repeating. For example, $\sqrt{2} = 1.4142135623\dots$ and $\pi = 3.1415926535\dots$.
Rule: A number whose decimal expansion is non-terminating non-repeating is irrational.
We can represent irrational numbers like $\sqrt{2}, \sqrt{3}, \sqrt{5}$ on the number line using the Pythagoras Theorem.
Locating $\sqrt{2}$: 1. Draw a number line with a point $O$ representing $0$ and a point $A$ representing $1$. So, $OA = 1$ unit. 2. Construct a perpendicular $AB$ of $1$ unit at $A$. 3. Join $OB$. By Pythagoras theorem in right $\triangle OAB$, $OB^2 = OA^2 + AB^2 = 1^2 + 1^2 = 2$. Thus, $OB = \sqrt{2}$. 4. Using a compass with center $O$ and radius $OB$, draw an arc intersecting the number line at $P$. The point $P$ corresponds to $\sqrt{2}$ on the number line.
We can continue this process, constructing perpendiculars on the newly formed hypotenuses to create the "Square Root Spiral" and locate $\sqrt{3}, \sqrt{4}, \sqrt{5}$, and so on.
While rational numbers satisfy commutative, associative, and distributive laws for addition and multiplication, irrational numbers also obey these laws. However, the sum, difference, quotients, and products of irrational numbers are not always irrational.
Let $a$ and $b$ be positive real numbers. 1. $\sqrt{ab} = \sqrt{a}\sqrt{b}$ 2. $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ 3. $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$ 4. $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$ 5. $(\sqrt{a} + \sqrt{b})(\sqrt{c} + \sqrt{d}) = \sqrt{ac} + \sqrt{ad} + \sqrt{bc} + \sqrt{bd}$ 6. $(\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b$
Rationalization is the process of eliminating the radical (square root or cube root) from the denominator of a fraction. We do this by multiplying both the numerator and the denominator by an appropriate expression called the rationalizing factor.
Example 1: Rationalize $\frac{1}{\sqrt{2}}$ Multiply numerator and denominator by $\sqrt{2}$: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.
Example 2: Rationalize $\frac{1}{2 + \sqrt{3}}$ The rationalizing factor is the conjugate: $(2 - \sqrt{3})$. Multiply numerator and denominator by $(2 - \sqrt{3})$: $\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$.
Let $a > 0$ be a real number, and let $m$ and $n$ be rational numbers. The laws of exponents are: 1. $a^m \cdot a^n = a^{m+n}$ 2. $(a^m)^n = a^{mn}$ 3. $\frac{a^m}{a^n} = a^{m-n}$ (where $m > n$) 4. $a^m b^m = (ab)^m$ 5. $a^0 = 1$ 6. $a^{-n} = \frac{1}{a^n}$
Definition of rational exponents: Let $a > 0$ be a real number and $n$ be a positive integer. Then, $a^{1/n} = \sqrt[n]{a}$ (the $n$-th root of $a$). Furthermore, $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.
The Number Systems chapter bridges the gap between basic arithmetic and advanced algebra. By mastering real numbers, rationalizing denominators, and applying the laws of exponents, you are well-prepared to tackle polynomial equations and advanced algebraic structures in subsequent chapters.