Cubes and Cube Roots
Exploring the World of Three-Dimensional Numbers
Hook Activity: Ramanujan and the Number 1729
This is a story about one of India's great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number was 1729. While talking to Ramanujan, Hardy described this number as "a dull number". Ramanujan quickly pointed out that 1729 was indeed interesting. He said it is the smallest number that can be expressed as a sum of two cubes in two different ways:
1729 = 12³ + 1³ = 1728 + 1
1729 = 10³ + 9³ = 1000 + 729
1729 has since been known as the Hardy–Ramanujan Number, even though this feature of 1729 was known more than 300 years before Ramanujan.
More About Hardy-Ramanujan Numbers
1729 is the smallest Hardy-Ramanujan Number. There are infinitely many such numbers. Some examples include:
- 4104 = 2³ + 16³ = 9³ + 15³
- 13832 = 18³ + 20³ = 2³ + 24³
These numbers are also known as taxicab numbers, and they have fascinating mathematical properties.
How did Ramanujan know this? Well, he loved numbers. All through his life, he experimented with numbers. He probably found numbers that were expressed as the sum of two squares and sum of two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube roots and many other interesting facts related to them.
I Do: What Are Cube Numbers?
You know that the word 'cube' is used in geometry. A cube is a solid figure which has all its sides equal.
How many cubes of side 1 cm will make a cube of side 2 cm?
The answer is 8! Because 2×2×2=8.
Consider the numbers 1, 8, 27, ... These are called perfect cubes or cube numbers. Do you know why they are named so? Each of them is obtained when a number is multiplied by itself three times.
We note that:
- 1 = 1×1×1 = 1³
- 8 = 2×2×2 = 2³
- 27 = 3×3×3 = 3³
Since 5³ = 5×5×5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9=3×3 and there is no natural number which multiplied by itself three times gives 9. We can see also that 2×2×2=8 and 3×3×3=27. This shows that 9 is not a perfect cube.
Cubes of Numbers 1 to 10
| Number | Cube | Number | Cube |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1000 |
Cubes of Numbers 11 to 20
| Number | Cube | Number | Cube |
|---|---|---|---|
| 11 | 1331 | 16 | 4096 |
| 12 | 1728 | 17 | 4913 |
| 13 | 2197 | 18 | 5832 |
| 14 | 2744 | 19 | 6859 |
| 15 | 3375 | 20 | 8000 |
Exploring Patterns in Cube Numbers
Consider a few numbers having 1 as the one's digit (or unit's). Find the cube of each of them. What can you say about the one's digit of the cube of a number having 1 as the one's digit?
Similarly, explore the one's digit of cubes of numbers ending in 2, 3, 4, ..., etc.
Try to find the one's digit of the cube of each of the following numbers:
- 3331 → 1³=1, so the unit digit is 1
- 8888 → 8³=512, so the unit digit is 2
- 149 → 9³=729, so the unit digit is 9
You Do: Test Your Understanding
Now it's your turn! Try solving the following problems to test your understanding of cube numbers.
Exercise 1: Calculate Cube Numbers
Enter a number to calculate its cube:
Exercise 2: Check for Perfect Cube
Enter a number to check if it's a perfect cube:
Exercise 3: Find the Unit Digit
Enter a number to find the unit digit of its cube:
Challenge: Hardy-Ramanujan Numbers
Can you find another number that can be expressed as the sum of two cubes in two different ways?
Try these examples:
- 4104 = 2³ + 16³ = 9³ + 15³
- 13832 = 18³ + 20³ = 2³ + 24³
Enter a number to check if it's a Hardy-Ramanujan number:
Conclusion and Summary
Through this lesson, we've learned about the concept of cube numbers, their properties, and some interesting applications. Let's review the key points:
Definition of Cube Numbers
A number multiplied by itself three times gives its cube.
For example: n³ = n × n × n
Perfect Cube Numbers
Numbers that can be expressed as the cube of an integer are called perfect cubes.
Examples: 1, 8, 27, 64, 125, etc.
Properties of Cube Numbers
• Cube of an even number is even
• Cube of an odd number is odd
• Cube numbers have specific patterns in their unit digits
Cube Roots
Cube root is the inverse operation of cubing.
If a³ = b, then a is the cube root of b, denoted as ∛b = a
Real-World Applications
Cube numbers and cube roots have many practical applications:
- Calculating the volume of cubes
- 3D design in engineering and architecture
- Density calculations in physics
- 3D modeling in computer graphics
Hardy-Ramanujan Numbers Revisited
1729, the smallest Hardy-Ramanujan number, shows us that mathematics is full of surprises. What seems like an ordinary number can have extraordinary properties.
These numbers, also called taxicab numbers, continue to fascinate mathematicians. The search for larger taxicab numbers is an ongoing area of mathematical research.
Mathematics is not just about formulas and calculations; it contains beauty, patterns, and infinite possibilities. As Ramanujan demonstrated, even seemingly ordinary numbers can hide surprising mathematical mysteries.