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Special Numbers

All numbers can be classified as odd or even. A number is even if we can be divide it by 2, otherwise it is said to be an odd number. An odd number always gives a remainder of 1 when divided by 2.

Examples of even numbers are 2, 4, 6, 12, 24, 76, 104, etc, while 1, 3, 5, 11, 37, 91, 123 are all odd numbers.

Remember – an even number ends with 2, 4, 6, 8 or 0, while an odd number ends with 1, 3, 5, 7, or 9.

The smallest 2 digit even number is 10, and the largest 2 digit even number is 98. The smallest 4 digit odd number is 1001, and the largest 4 digit odd number is 9999.

Other than odd and even numbers, there are other special numbers like palindrome numbers, square numbers, triangular numbers, Fibonacci numbers, etc.

Now let us look at these special numbers that follow certain patterns:

 

Palindrome numbers

These numbers read the same when read from left to right (i.e. forward) or right to left (i.e. backward). 121, 2332, 14541, 2121212 are all palindrome numbers.

 

Square numbers

sequence of square numbers

Look at the pattern above, they all represent the square shape. Square numbers are generated by counting the dots forming the squares. The patterns is 1, 4, 9, 16, 25, and so on, which is actually 1×1, 2×2, 3×3, 4×4, 5×5, and so on.

We can find a square number when we multiply any number by itself. We can also give an algebraic expression for a square number as Tn = n2, where Tn is the value of the nth term, and n is the term number.

So the fourth number in the sequence of square numbers is T4 = 42 = 16.

 

Triangular numbers

Similar to the pattern of square numbers, the pattern of triangular numbers is generated by forming triangles.

sequence of triangular numbers

The numbers represented in the above patterns are 1, 3, 6, 10, 15, and so on.

Notice that the next pattern is found by adding the number which is equal to the term number. Each triangle has one more row than the previous triangle, and each row has one more dot than the previous row. The number of dots in the triangle is found by adding consecutive counting numbers, as shown below:

1 = 1

3 = 1 + 2

6 = 1 + 2 + 3

10 = 1 + 2 + 3 + 4,  and so on…

The algebraic expression for a triangular number is Tn  =  {n(n~+~1)}/2

where n is the term number and Tn is the value of the nth term.

If we want the 6th term, we get T6  =  {6(6~+~1)}/2   =  {6~*~7}/2  =  42/2  =  21

By diagram it is represented as –

triangular pattern for 6th term

 

Fibonacci numbers

Fibonacci numbers are numbers that have the following pattern:

1, 1, 2, 3, 5, 8, 13, 21, …..

To work out the pattern, add the previous numbers to find the next number. So,

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8 …..

This pattern was discovered by the Italina mathematician Fibonacci, whose real name was Leonardo of Pisa (1180-1250). Fibonacci numbers are one of the most important patterns of numbers found in real life and in nature. We can see this pattern in the centre of a daisy, where there are 2 sets of intersecting spirals that radiate in opposite directions.

We can see this series in our own hands. We all have

  • 2 hands, each of which has …
  • 5 fingers, each of which has …
  • 2 knuckles

Some other examples are seed bearing scales of a pinecone, arrangement of leaves of sneezewort, pineapple scales, starfishes, sea urchins and other marine animals.

Fibonacci numbers also appear in the description of the reproduction of a population, say of honeybees according to the rules:

  1. If an egg is laid by an unmated female, it hatches a male or drone bee
  2. If, however, an egg is fertilised by a male, it hatches a female. Hence a female bee will have two parents, but a male parent has only one parent.

If we trace the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 grandparents, 5 great grandparents, and so on…

fibonacci pattern