Explore the Landscape of Number System and Divisibility Rules

Explore the Landscape of Number System and Divisibility Rules

Number system is one of the most important topics in Mathematics. Understanding of numbers is the stepping stone to explore various dimensions and areas of Mathematics. In this post we would explore various kinds of number, and the divisibility rules of some elementary numbers.

The number system is a defined arrangement for writing various numbers and expressing them mathematically using symbols that are of consistent nature. Put differently, a number system is a collection of various symbols which are called digits. The number system comprises various nomenclatures that define a particular set of numbers like natural numbers, rational numbers, irrational numbers etc. The most commonly used number system is Decimal having ten digits – 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The number system also throws light on the place value and face value of the numbers. When a number is written in the expanded form, it represents its place value. For example, 512, in base 10, is written as 500+ 10+2 coming down as five hundred + ten + two. Therefore, 5 here represents 500 as its place value. On the other hand, face value is the actual value of the digit.

So, if you are new to the number system or brushing up your concepts, here is all you need to know about the different class of numbers such as irrational numbers, rational numbers, natural numbers etc. in the number system.

Number Tree: The number tree consists of the classification of the number system. It includes various set of numbers such as real numbers, rational and irrational numbers, integers, whole numbers, natural numbers etc. The following is a pictorial representation of Number Tree.



Real Numbers: Real Numbers in the number system are values of a continuous quantity that can represent a distance along a line or alternatively include rational numbers and irrational numbers that are not imaginary in nature. A real number represents a value that lies on the number line in the number system.

Rational Numbers: Rational Numbers in the number system are those numbers that are easily expressible as the ratio of two integers. They are different from irrational numbers.

If any number written in the form of a/b, where a and b are integers and the number ‘b’ must not be equal to zero, then it one of the rational numbers belonging to the number system. The nature of rational numbers is whether recurring or terminating. A few rational number examples are ⅕, -22/555. Terminating rational numbers are of the form ¼, 8/100 etc. while recurring rational numbers are 1/12= 0.83333, 1/3 etc.

Irrational Numbers: Numbers which cannot be expressed in the form of p/q are called Irrational Numbers. Irrational numbers are usually non-terminating (numbers which don’t stop after the decimal) and non-recurring (which do not repeat themselves again and again) in nature. Rational numbers on the other hand are not so.

Example of irrational numbers: The very famous mathematical constant π is one of the irrational numbers. The number of digits after the decimal in pi is infinite and non-terminating, thus qualifying it for irrational numbers. Other examples of irrational numbers from the number system include √2, √3 etc. which means that its decimal equivalent will go on forever, unlike rational numbers.

Integers: The Integers are the numbers that are real in nature and consist the set of natural numbers along with zero and their additive inverses. They are similar to rational numbers but unlike irrational numbers. Integers are denoted as ‘I’ or ‘J’ in the number system. For example, {-1,-2.-3,-4, 0,1,2,3,4} are all integers.

Here is an interesting fact about integers. If you add, multiply or subtract integers, the result obtained will also be an integer. However, this is not true for the division operation.

Natural Numbers: The natural numbers are often referred to as the counting numbers. The smallest natural numbers is 1 and the largest natural numbers is infinite in nature. Natural numbers are also referred to as ‘N’. Similar to integers, the sum and multiplication of any two natural numbers is also one of the natural numbers.

Whole Numbers: The numbers that start from 0 and comprise the natural numbers are called whole numbers. Therefore, whole numbers must not be confused with natural numbers.

Divisibility Rules: Now, that we are familiar with the different kind of numbers, let’s move on to find the divisibility rules of different numbers.

The divisibility rule is a method of finding out whether a given integer can be divided by another number known as the divisor, without having to actually perform the division.

Here are the divisibility rules of some of the popular integers in the number system.
  • Divisible by 2: If the last digit of a number is divisible by 2, the entire number will be divisible by 2. Examples: 122, 158, 30008 etc.
  • Divisible by 4: If the last two digits of a number are divisible by 4, the entire number will be divisible by 4. Examples: 448, 7736, 1140924 etc.
  • Divisible by 8: If the last three digits of a number are divisible by 8, the entire number will be divisible by 8. Examples: 448, 7832, 1140024 etc.
  • A number is divisible by 2n if the last n digits are divisible by 2n
  • Divisible by 3: If the sum of digits of ta number is divisible by 3, the entire number will be divisible by 3. Examples: 303, 4689, 33495 etc.
  • Divisible by 9: If the sum of digits of a number is divisible by 9, the entire number will be divisible by 9. Examples: 309, 4689, 334953 etc.
  • Divisible by 5: If the last digit of a number is divisible by 5, the entire number will be divisible by 5. Examples: 495, 3375, 4670050 etc.
  • Divisible by 25: If the last two digits of a number are divisible by 25, the entire number will be divisible by 25. Examples: 425, 3375, 4670050 etc.
  • A number is divisible by 5n if the last n digits are divisible by 5n
  • Divisible by 7: Subtract twice the unit digit from the remaining number, and keep it repeating unless you get a two-digit number. If the result is divisible by 7, the original number is also divisible by 7. Examples: 154, 70084
  • Divisible by 11: If the difference between the sum of digits at the odd place and the sum of digits at the even place is zero or divisible by 11, the given number will be divisible by 11. Examples: 495, 1375, 575700 etc.
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