The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.
A Number system or numeral system is defined as elementary system to express numbers and figures. It is the unique way of representation of numbers in arithmetic and algebraic structure.
Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system.
Numbers generally known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities.
Numbers are the mathematical values or figures used for the purpose measuring or calculating the quantities. It is represented by numerals as 2,4,7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.
Types Of Numbers
There are different types of numbers categorized into sets by the number system. The types are described below:
- Natural numbers: Natural numbers are the positive numbers that count from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = 1,2,3,4,5,6,7,……………
- Whole numbers: Whole numbers are positive numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W = 0,1,2,3,4,5,………………
- Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’. The set of integers can be represented as Z=………..,-5.-4,-3,-2,-1,0,1,2,3,4,5,………….
- Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5,0.567, etc.
- Real number: Real numbers are the set numbers that do not include any imaginary value. It includes all the positive integers, negative integers, fractions, and decimal values. It is generally denoted by ‘R’.
- Complex number: Complex numbers are a set of numbers that include imaginary numbers. It can be expressed as a+bi where “a” and “b” are real numbers. It is denoted by ‘C’.
- Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.
- Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’.
What are Integers?
Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’.
The set of integers can be represented as Z =………..,-5.-4,-3,-2,-1,0,1,2,3,4,5,………….
The number with no decimal or fractional part from the set of negative and positive numbers, including zero.
Examples of integers are: -8, -7, -5, 0, 1, 5, 8, 97, and 3,043.
Types of Integers
Two types of integers are:
Positive Integers: The integer number is positive if it is greater than zero.
Example: 1, 2, 3, 4,…
Negative Integers: The integer number is negative if it is less than zero. Example: -1, -2, -3, -4,… and here Zero is defined as neither negative nor positive integer. It is a whole number.
Z = {… -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, …}
If we divide a positive integer by another positive integer, what is the resulting number?
Solution:
Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals.
So let’s assume p and q are two positive integers, then there are two scenarios in this question:
First scenario:
Assume p is directly divisible by q with no remainder left.
Example: lets assume the value of p = 10 and q = 5
So as per the question if we divide positive integer by positive integer
i.e. p / q
= 10 /5
= 2
So here we get 2 which is a positive integer means in First scenario if we divide positive integer by positive integer we will get the result which will also be an integer
Second scenario:
Assume p is not directly divisible by q
Example: let’s assume the value of p = 15 and q = 2
So here as per the question if we divide positive integer by positive integer.
i.e. p/q
= 15/2
= 7.5 is the number we will get which is not integer as integer doesnot include fractions or decimal
So the answer to the above question is that when we divide positive integer by positive integer every time we will get different numbers sometimes integers sometimes rationals.
Sample Questions
Question 1: If we add two positive integers what will be the number?
Answer:
Let’s assume p and q are two positive integers and p = 5 and q = 6
add, p + q
= 5 + 6
= 11 also an integer as well as rational number.
Question 2: If we add one positive and one negative integer, what will be the number?
Answer:
Let’s assume p is positive integer and q is negative integer,
here p = 10 and q =5
so as per the question: add, p + q
= 10 + (-5)
= 10 – 5
= 5
the number will also be an integer as well as rational number.
Question 3: if we add two negative integers then what will be the number?
Answer:
Let’s assume p is negative integer and q is negative integer,
here p = -10 and q = – 5
so as per the question add, -p + (-q )
= -10 + (-5)
= -10 – 5
= -5
The number is negative integer