JEE Mathematics Number System Notes Set 01

Classification of Numbers

Counting numbers 1, 2, 3, … form the set of natural numbers, written N. Adding or multiplying two natural numbers always gives another natural number, so N is closed under addition and multiplication, but not under subtraction (for example 2 − 6 = −4 is not natural). Including 0 gives the whole numbers W, and including the negative numbers as well gives the integers Z = {…, −2, −1, 0, 1, 2, …}. The number 0 is neither positive nor negative.

Rational and Irrational Numbers

A rational number is any number that can be written as \( \frac{p}{q} \) with p, q integers and q ≠ 0. Examples are \( \frac{2}{3} \), \( -\frac{5}{6} \) and 4 (= \( \frac{4}{1} \)). A number that cannot be written in this form is irrational, such as \( \sqrt{2} \) and \( \pi \) (approximately 1.414 and 3.14159).

Decimal test: every rational number has a decimal expansion that either terminates (for example \( \frac{7}{16} = 0.4375 \)) or repeats a block of digits forever (for example \( \frac{1}{3} = 0.333… \), \( \frac{1}{7} = 0.142857142857… \)). An irrational number has a decimal expansion that neither terminates nor repeats. A fraction in lowest terms terminates exactly when the denominator has no prime factor other than 2 and 5.

Real and Complex Numbers

Rational and irrational numbers together make up the real numbers R, and every real number corresponds to exactly one point on the number line. Complex numbers have the form a + ib with a, b real and \( i = \sqrt{-1} \); every real number is a complex number with b = 0. The standard chain of sets is N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C. Powers of i repeat with period 4: i, −1, −i, 1.

Absolute Value (Modulus)

The modulus of a real number a is its distance from 0: |a| = a when a ≥ 0 and |a| = −a when a < 0. Always |a| ≥ 0 and |a|² = a². To solve |x − 1| = 5 consider the two cases x − 1 = 5 and x − 1 = −5, which give x = 6 and x = −4. In general |x − a| < r means a − r < x < a + r.

Representing Numbers on the Number Line

Integers sit at equal steps either side of 0, and a rational number such as \( \frac{3}{2} \) sits between the integers 1 and 2. Irrational square roots can be located with right triangles. Take OA = 1 on the line and draw AP = 1 perpendicular to OA, so OP = \( \sqrt{2} \). Swing an arc with centre O and radius OP to meet the line at P₁. Now erect P₁Q = 1 perpendicular to the line: OQ = \( \sqrt{3} \). Repeating this spiral gives \( \sqrt{4}, \sqrt{5}, … \) one after another. Negative roots are marked on the left of O in the same way.

Useful Results with Surds

1. The sum of a rational and an irrational number is irrational, and the product of a non-zero rational and an irrational number is irrational. The sum or product of two irrationals can be rational, for example \( \sqrt{2} \cdot \sqrt{2} = 2 \).
2. To prove 1 + \( \sqrt{2} \) is irrational, assume it equals \( \frac{a}{b} \); then \( \sqrt{2} = \frac{a - b}{b} \) would be rational, which is false.
3. To rationalise, multiply by the conjugate: \( \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b} \).

Worked Examples

Example 1. Find the difference between the greatest and the least of \( \frac{5}{8}, \frac{7}{12}, \frac{9}{16}, \frac{25}{48} \).
Over the common denominator 48 these become \( \frac{30}{48}, \frac{28}{48}, \frac{27}{48}, \frac{25}{48} \). The required difference is \( \frac{30 - 25}{48} = \frac{5}{48} \).

Example 2. 500 equal fencing rods have a total length of 6.045 km. Find the length of one rod.
Length = \( \frac{6.045 \times 1000}{500} \) m = 6.045 × 2 = 12.09 m.

Example 3. For a = 2, b = 3, c = 4 evaluate \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) and \( \frac{(a^2 + 1)(b^2 + 1)}{c^2 - 1} \).
The first is \( \frac{6 + 4 + 3}{12} = \frac{13}{12} = 1\frac{1}{12} \). The second is \( \frac{5 \times 10}{15} = \frac{10}{3} \).

Example 4. Solve |x − 1| = 5.
If x − 1 ≥ 0 then x − 1 = 5, so x = 6. If x − 1 < 0 then −(x − 1) = 5, so x = −4. The solutions are x = 6 and x = −4.