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A Square and A Cube - A Pinch of History

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The concept of a square number originates from the geometric shape of a square. If a square has a side of length nn, its area is n×n=n2n \times n = n^2.

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A number nn is called a perfect square if it can be expressed as m2m^2 for some natural number mm.

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The concept of a cube number relates to the volume of a cube. If a cube has a side of length nn, its volume is n×n×n=n3n \times n \times n = n^3.

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Historically, Indian mathematicians like Aryabhata and Bhaskara II developed methods to find square roots and cube roots.

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The Hardy-Ramanujan Number 17291729 is a famous historical anecdote. It is the smallest number that can be expressed as the sum of two cubes in two different ways: 1729=13+1231729 = 1^3 + 12^3 and 1729=93+1031729 = 9^3 + 10^3.

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A property discovered by ancient mathematicians is that the sum of the first nn odd natural numbers is n2n^2. For example, 1+3+5=32=91 + 3 + 5 = 3^2 = 9.

📐Formulae

n2=n×nn^2 = n \times n

n3=n×n×nn^3 = n \times n \times n

1+3+5+⋯+(2n−1)=n21 + 3 + 5 + \dots + (2n-1) = n^2

1729=13+123=93+1031729 = 1^3 + 12^3 = 9^3 + 10^3

💡Examples

Problem 1:

Show that 1616 is a perfect square using the sum of consecutive odd numbers property.

Solution:

We add consecutive odd numbers starting from 11: 1+3+5+7=161 + 3 + 5 + 7 = 16. Since we added 44 consecutive odd numbers, 16=4216 = 4^2.

Explanation:

The sum of the first nn odd numbers is always n2n^2. Since 1616 is the sum of the first 44 odd numbers, it is a perfect square of 44.

Problem 2:

Calculate the value of 12212^2 using vertical multiplication.

Solution:

12×1224120144\begin{array}{r} 12 \\ \times 12 \\ \hline 24 \\ 120 \\ \hline 144 \end{array}

Explanation:

To find the square of 1212, we multiply 1212 by itself, resulting in 144144.

Problem 3:

Verify the Hardy-Ramanujan property for 17291729 using the pairs (1,12)(1, 12) and (9,10)(9, 10).

Solution:

For the first pair: 13+123=1+1728=17291^3 + 12^3 = 1 + 1728 = 1729 For the second pair: 93+103=729+1000=17299^3 + 10^3 = 729 + 1000 = 1729

Explanation:

The number 17291729 is unique in history as the smallest 'taxicab number' expressible as the sum of two cubes in two distinct ways.