Cube numbers
When a number is multiplied by itself three times, the result is called a cube number (or perfect cube).
In general, for any number \(n\), we write \(n\times n\times n\) as \(n^3\), which is read as \(n\) cube.
The following table shows the cube numbers of the first \(20\) numbers.
| Number \(n\)(\(1 - 10\)) | Cube Number \((n^3)\) | Number \(n\) (\(11 - 20\)) | Cube Number \(n^3\) |
| \(1\) | \(1^3 =1\) | \(11\) | \(11^3 = 1331\) |
| \(2\) | \(2^3 = 8\) | \(12\) | \(12^3 = 1728\) |
| \(3\) | \(3^3 = 27\) | \(13\) | \(13^3 = 2197\) |
| \(4\) | \(4^3 = 64\) | \(14\) | \(14^3 = 2744\) |
| \(5\) | \(5^3 = 125\) | \(15\) | \(15^3 = 3375\) |
| \(6\) | \(6^3 = 216\) | \(16\) | \(16^3 = 4096\) |
| \(7\) | \(7^3 = 343\) | \(17\) | \(17^3 = 4913\) |
| \(8\) | \(8^3 = 512\) | \(18\) | \(18^3 = 5832\) |
| \(9\) | \(9^3 = 729\) | \(19\) | \(19^3 = 6859\) |
| \(10\) | \(10^3 = 1000\) | \(20\) | \(20^3 = 8000\) |
Unit digit of cubes
| Unit digit of \(n\) | Unit digit of \(n^3\) |
| \(0\) | \(0\) |
| \(1\) | \(1\) |
| \(2\) | \(8\) |
| \(3\) | \(7\) |
| \(4\) | \(4\) |
| \(5\) | \(5\) |
| \(6\) | \(6\) |
| \(7\) | \(3\) |
| \(8\) | \(2\) |
| \(9\) | \(9\) |
Ending patterns in cube numbers.
1. The cube of a positive number is always positive.
Example:
\(4^3 = 4 \times 4 \times 4 = 64\)
2. The cube of a negative number is always negative.
Example:
\((-4)^3 = (-4) \times (-4) \times (-4) = -64\)
3. The cube of every even number is even.
Example:
\(2^3 = 8\), \(4^3 = 64\), \(6^3 = 216\)
4. The cube of every odd number is odd.
Example:
\(3^3 = 27\), \(5^3 = 125\), \(7^3 = 343\)
5. When cubing a fraction number, cube both the numerator and the denominator separately.
Example:
\(\left(\frac{4}{5}\right)^3 = \frac{4\times 4\times 4}{5\times 5\times 5} = \frac{64}{125}\).
6. When cubing a decimal number, the number of decimal places in the answer will be three times the number of decimal places in the original number.
Example:
\((16.05)^3 = 16.05 \times 16.05 \times 16.05= 4134.520125\)
7. The number of trailing zeros in a perfect cube is always a multiple of \(3 (0, 3, 6, 9, ...)\). Therefore, a perfect cube can never end with exactly \(1, 2, 4, 5, 7, 8,\) ... trailing zeros.
Example:
\(10^3 = 1000\), \(500^3 = 125000000\).
Taxicab Numbers
The Hardy - Ramanujan Number: 1729
Mathematician G.H. Hardy visited Ramanujan in the hospital, arriving by taxicab number \(1729\). Hardy called it a "dull number," but Ramanujan instantly replied that it was the smallest number expressible as a sum of two cubes in two different ways:
\(1729 = 1^3 + 12^3 = 9^3 + 10^3\)
Therefore, those numbers are now called Taxicab Numbers.
Perfect Cubes as the sum of consecutive Odd Numbers
Every perfect cube can be written as the sum of consecutive odd numbers.
Pattern:
\(1 = 1^3\)
\(3 + 5 = 8 = 2^3\)
\(7 + 9 + 11 = 27 = 3^3\)
1. Every perfect cube \(n^3\) can be expressed as the sum of exactly \(n\) consecutive odd numbers, beginning with \(n(n - 1) + 1\) and ending with \(n(n + 1) - 1\).
Example:
Let us find the sum of consecutive odd numbers of a perfect cube \(7^3\). Here, \(n = 7\)
Starting number \(= n(n - 1) + 1\) \(= 7(7 - 1) + 1\) \(= 7(6) + 1 = 43\)
Ending number \(= n(n + 1) - 1\) \(= 7(7 + 1) - 1\) \(= 7(8) - 1 = 55\)
Therefore, the sum of consecutive odd numbers are \(43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 = 7^3\)
2. The difference between the cubes of two consecutive natural numbers, \(n\) and \(n + 1\), is given by \((n + 1)^3 - n^3 = 1 + (n + 1)3n\).
Example:
Find the differnce between \(64^3\) and \(65^3\)
Here, \(n = 64\)
\((64 + 1)^3 - 64^3 = 1 + (64 + 1) \times 3(64)\)
\(=1 + 65 \times 64 \times 3\)
\(=1 + 12480\)
\(65^3 - 64^3 = 12481\).