World of Numbers Class 9 PPT & Notes – Ganita Manjari Ch 3

Chapter 3 of Ganita Manjari Part I follows numbers from tally marks cut into bone to the real number line. On the way it meets zero (Shunya), negative numbers, fractions, irrational numbers such as √2 and π, and the decimal expansions that tell us which kind of number we are looking at.

A herder matched each cow with one pebble: one pebble went out in the morning and one came back in the evening. An empty pot meant that the whole herd was safe. This one-to-one matching gave us the natural numbers 1, 2, 3, …

• Found in the Lebombo Mountains of southern Africa
• About 35,000 years old
• Carved with 29 uniform notches
• Believed to be a counter for the phases of the moon, or a menstrual counter
• The oldest physical proof of counting

• Found near the headwaters of the Nile, in Congo, and dated to around 20,000 BCE
• It carries three columns of notches
• One column groups the notches as 11, 13, 17 and 19, which are the primes between 10 and 20
• Another column shows doubling, that is, multiplying by 2

• Lothal and Harappa used standardised weights for trade with Mesopotamia
• The Vedas named powers of 10 up to parardha, which is 10¹²
• The Lalitavistara names numbers up to tallakshana, which is 10⁵³
• Powers of 10 in the Rigveda led to the idea of place value

Ancient Indians counted on the joints of the fingers. Each finger has 3 joints and the thumb is the pointer that touches them, so four fingers give 12 counts on one hand. This is the root of the ancient base-12 systems, and the practice is still seen in India today (Exercise Set 3.1, Q4).

For centuries the number line started at 1. The Babylonians and the Mayans used only a placeholder to mark an empty column, so “nothing” was not yet a number that you could add, subtract or multiply. Brahmagupta turned the void into a number in 628 CE.

Shunyata means emptiness. The Upanishads and Buddhist texts revered this state, and Patanjali linked it to stilling the mind. Emptiness entered art, language and literature, and Aryabhata and finally Brahmagupta carried it into mathematics.

The Bakhshali Manuscript, written on birch bark in the early centuries CE, uses a bold dot, the bindu, to mean zero. It is the oldest surviving written zero from India, and a blank space finally became a written symbol. But a symbol needs rules before it becomes a number.

• Zero is what remains: a − a = 0
• When zero is added, nothing changes: a + 0 = a
• When zero is subtracted, nothing changes: a − 0 = a
• Any number multiplied by zero becomes zero: a × 0 = 0
• Written in the Brahmasphutasiddhanta, 628 CE

Fortunes (Dhana) are the positive numbers and debts (Rina) are the negative numbers. Moving left of zero gives the negatives. Integers are the positives, the negatives and zero together.

A negative number is a debt, and multiplying by a negative number means removing that debt. Remove 4 debts of Rs 3 each and you are Rs 12 richer. Therefore (−3) × (−4) = +12.

Fractions represent parts of a whole. Every positive fraction has a negative twin, and the minus sign may sit above or below the bar. Integers and all fractions together form the rational numbers ℚ: any number p/q where p and q are integers and q is not 0.

One rational number has infinitely many forms. Divide out the common factors to simplify: 12/30 becomes 2/5 when both numbers are divided by 6. We choose the co-prime form, in which p and q share no common factor except 1, to represent all of them. So 1/2 stands for 2/4, 3/6, 6/12 and the rest.

• Equality: a/b = c/d exactly when ad = bc
• Addition and subtraction: make the denominators equal first
• Multiplication: numerator times numerator, denominator times denominator
• Division: multiply by the reciprocal of the second number
• Addition and multiplication are commutative
• The distributive law holds: p(q + r) = pq + pr

Rational numbers are closed under addition, subtraction, multiplication and division, provided that you never divide by zero. Compare the natural numbers, which are not closed under subtraction: 5 − 8 is not a natural number.

• Mark a point and call it 0, the origin. Equal steps to the right and left give the integers
• To place p/q, cut one unit into q equal parts and move p parts to the right if p/q is positive, or to the left if it is negative
• 1/2 sits exactly halfway between 0 and 1
• Improper fractions also sit on the line: 9/4 is the same as 2 and 1/4, so it lies between 2 and 3. Cut that unit into four equal parts and move one part to the right of 2
• In the same way −7/4 lies between −2 and −1

The absolute value of a number is its distance from 0. So |5/3| = 5/3 and |−5/3| = 5/3 too, and |0| = 0. An absolute value is never negative. The distance between two numbers a and b is |a − b|.

Between the integers 1 and 2 lies the rational 3/2, and between 1 and 3/2 lies another rational, 5/4. The average of a and b, which is (a + b)/2, always lies between a and b. So there are infinitely many rational numbers between any two points.

Baudhayana composed the Shulbasutra around 800 BCE. Building fire altars needed exact lengths. For a square of side 1, the diagonal d gives d² = 2, so the diagonal measures √2. Numbers that are not a ratio of integers are called irrational.

This is a proof by contradiction, credited to Hippasus (about 400 BCE).

1. Assume √2 = p/q in lowest form, with p and q co-prime
2. Square both sides and clear the denominator: 2q² = p²
3. So p² is even, which means p is even. Let p = 2k
4. Substituting gives q² = 2k², so q is even too
5. p and q are both even, so they share the factor 2
6. This contradicts the lowest form, so √2 is irrational

1. On the number line measure OA = 1 unit
2. At A draw a perpendicular and mark AB = 1 unit
3. Join O to B, so OB = √2 units
4. With O as centre and OB as radius, draw an arc
5. The arc cuts the line at P, so P represents √2

The same method extends to any √n.

Start with a right triangle whose two legs are both 1 unit; its hypotenuse is √2. Add a fresh 1 unit perpendicular to that hypotenuse. The new hypotenuses give √3, then √4, then √5, and so on, so any length √n can be constructed (Fig. 3.14).

• π is the circumference of a circle divided by its diameter
• Aryabhata (499 CE) gave the approximation 3927/1250 = 3.1416, but called it asanna, meaning only an approximation
• Lambert proved in 1761 that π is truly irrational
• Madhava of Sangamagrama, founder of the Kerala School, found the exact infinite series: π = 4(1 − 1/3 + 1/5 − 1/7 + …)

The rational numbers are dense, but they still leave gaps. The irrational numbers fill every gap. Together they form the real numbers ℝ, and every length and measurement lives on this line. The decimal expansion of a number tells us which type it is.

• A decimal terminates when the remainder becomes 0. Example: 3/8 = 0.375, and the division simply stops
• A decimal repeats when the digits loop forever. Example: 5/11 = 0.454545…, which we write as 0.45 with a bar over the 45
• Every rational number must do one or the other

Divide 1 by 7 using long division. At each step the remainder must be 1, 2, 3, 4, 5 or 6, and it can never be 0 here. Only six remainders are possible, so one of them must return. Once a remainder repeats, the whole division loops, and that is what creates a repeating decimal.

• Take p/q in its lowest form
• Find the prime factors of the denominator q
• The decimal terminates only if q has 2s, 5s or both as its prime factors
• Any other prime factor makes it repeat

Case 1: a terminating decimal. Count the digits after the decimal point and write the decimal over the matching power of 10, then cancel common factors. Example: 0.35 = 35/100 = 7/20.

Case 2: a pure repeating decimal. Let x be the decimal. Multiply by 10 raised to the number of repeating digits, subtract, and solve.

• x = 0.666…, so 10x = 6.666…. Subtracting gives 9x = 6, so x = 2/3
• x = 0.454545…, so 100x = 45.4545…. Subtracting gives 99x = 45, so x = 5/11

Case 3: a general repeating decimal. First shift the non-repeating digits, then shift one full repeating block, subtract, and solve.

• x = 0.1666…, so 10x = 1.666… and 100x = 16.666…. Subtracting gives 90x = 15, so x = 1/6
• x = 2.35777…, so 100x = 235.777… and 1000x = 2357.777…. Subtracting gives 900x = 2122, so x = 2122/900 = 1061/450

1/7 = 0.142857142857…, repeating forever. The block 142857 is a cyclic number: the very same digits simply rotate in a cycle.

• 142857 × 2 = 285714
• 142857 × 3 = 428571
• 142857 × 4 = 571428
• 142857 × 5 = 714285
• 142857 × 6 = 857142

Irrational decimals never end and never repeat: √2 = 1.4142135623730950488…, π = 3.1415926535897932384…. No repeating block exists.

Decimals are also not unique: every terminating decimal has a “nines twin”. For example, 0.999… = 1 exactly. To see this, let x = 0.999…. Then 10x = 9.999…, and subtracting gives 9x = 9, so x = 1.

1 × 1 = 1 and (−1) × (−1) = 1, so no real number squared gives −1. The square root of −1 cannot sit on the number line, so mathematicians invented a new dimension, i. These imaginary numbers are essential for electrical engineering, quantum mechanics and the chip in your phone.

• Natural numbers count the students in a classroom
• Zero shows an empty bank balance
• Negative integers record a winter temperature in Ladakh
• Rational numbers measure half a cup of ghee
• Irrational numbers give a circle’s exact circumference
• Imaginary numbers run the chip inside your phone

• q can never be zero in p/q
• Zero is a number; it is not the same as the empty set
• A negative sign belongs to the whole fraction, not just one part
• Not every square root is irrational: √81 = 9
• 0.999… equals 1 exactly; it is not slightly less
• Never call a repeating decimal irrational

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