Visualizing Real Numbers on the Number Line: A number line is a straight line with numbers placed at equal intervals, usually represented horizontally, extending infinitely in both directions.
Successive Magnification:
Numbers with more decimal expansions (e.g., \(5.63434\dots\)) require a method called Successive Magnification.
This process visualizes the representation of numbers on the number line through a magnifying glass.
Step - 1: Locate the Range: Find the range between the nearest smallest and largest integers (e.g., \(5.63434\) is between \(5\) and \(6\)).
Step - 2: Magnify (1st Decimal): Magnify the range and divide the portion into \(10\) parts. Look for the range of the number with the first decimal point (e.g., \(5.6\) is between \(5.6\) and \(5.7\)).
Step - 3: Magnify (2nd Decimal): Again, magnify the new range and divide it into \(10\) parts. Look for the range of the number with two decimal points (e.g., \(5.63\) is between \(5.63\) and \(5.64\)).
Step - 4: Repeat: Repeat the process of magnification (dividing into \(10\) parts) until the entire decimal is located on the number line (e.g., up to \(5.63434)\).
Operations and Properties of Irrational Numbers:
- Real Numbers are either rational or irrational.
- Rational numbers are closed under addition and multiplication.
Irrational numbers are not closed under addition, subtraction, multiplication, and division. - Rational numbers satisfy the commutative, associative, and distributive properties for addition and multiplication.
Irrational numbers also satisfy the commutative, associative, and distributive properties for addition and multiplication.
Important!
The sum, difference, product, and division of two irrational numbers may or may not be irrational.
Let us study the operations of rational and irrational terms:
| Operation | Result Type | Example property |
| Rational + Irrational | Always irrational | \(\frac{1}{2} + \sqrt(2) = \) Irrational |
| Rational – Irrational | Always irrational | \(5 - \sqrt(3) = \) Irrational |
| Rational \(\times\) Irrational | Always irrational (if rational is non-zero) | \(4\times\sqrt(5) =\) Irrational |
| Rational \(\div\) Irrational | Always irrational (if rational is non-zero) | \(6\div\sqrt(7) = \) Irrational |
Rationalizing the Denominator:
Rationalizing the denominator is the process of converting an expression whose denominator contains a term with a square root (or a number under a radical sign) to an equivalent expression whose denominator is a rational number.
Procedure to rationalize the denominator of irrational terms:
- For an expression to be in a simpler form, the denominator should not have an irrational number.
- In a Single Term, to rationalize a denominator with a single irrational term, multiply and divide by the same irrational number and simplify.
Laws of Radicals and Exponents:
Laws of Exponents:
For numbers with powers like \(a^m, b^n,...\), where \(a\) and \(b\) are the bases and \(m\) and \(n\) are the exponents:
- \(a^m \cdot a^n = a^{m+n}\)
- \((a^m)^n = a^{mn}\)
- \(\frac{a^m}{a^n} = a^{m-n}\), where \(m > n\)
- \(a^m b^m = (ab)^m\)
Laws of Radicals:
The symbol \(\sqrt{ }\) is called the radical sign.
If a > 0 is a real number and n is a positive integer, then \(\sqrt[n]{a} = b\), if \(b^n = a\) and \(b > 0\).
For positive real numbers \(a\) and \(b\):
- \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
- \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
- \(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b\)
- \((\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b\).