The arithmetic value which is used for representing the quantity and used in making calculations are defined as Numbers. A symbol like “4,5,6” which represents a number is known as numerals. Without numbers, we can’t do counting of things, date, time, money, etc. these numbers are also used for measurement and used for labeling.
The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words.
For example, 3 is written as three in words, 35 is written as thirty-five in words, etc. Students can write the numbers from 1 to 100 in words to learn more. There are different types of numbers, which we can learn. They are whole and natural numbers, odd and even numbers, rational and irrational numbers, etc.
What is a Number System?
A Number System is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as Number System.
For example, 156,3907, 3456, 1298, 784859 etc.
What are Integers?
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.
We can represent a set of integers as Z, which includes:
- 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, …}
We have four basic arithmetic operations associated with integers are:
- Addition of Integers
- Subtraction of Integer
- Multiplication of Integers
- Division of Integers
Before all these operations we need to remember one thing If there is no sign in front of a number which means that the number is positive. For example, 6 means +6.
Absolute value of any integer is a positive number, i.e., |−3| = 3 and |4| = 4.
Addition of Integers
While adding two integers, we will have the following cases:
Case 1: If both integers have the same signs then Add the absolute values of integers and give the same sign as that of the given integers to the result. For example:
- If two integers are -3 and -5 then the sum will be -8.
- If two integers are 3 and 5 then the sum will be 8.
Case 2: If One integer is positive and another is negative then find the difference of the absolute values of the numbers and then give the original sign of the larger of these numbers to the result. For example:
- If two integers are -3 and 5 then the sum will be 2.
- If two integers are 3 and -5 then the sum will be -2.
Subtraction of Integers
At the time of the subtraction of two integers:
First Convert the operation into an addition problem by changing the sign of the subtrahend and then Apply the same rules of addition of integers
Multiplication of Integers
At the time of the multiplication of two integers:
- First, we have to Multiply their signs and get the resultant sign.
- Then Multiply the numbers and add the resultant sign to the answer.
There are some different possible cases of multiplication of integer such as below in the table:
| PRODUCT SIGNS | RESULT | EXAMPLE |
| + × + | + | 5 × 4 = 20 |
| + × – | – | 5 × (- 4) =-20 |
| – × + | – | (-5) × 4 = -20 |
| – × – | + | (-5) × (-4) = 20 |
Division of Integers
If we carry out the division operation between two integers: First we have to Divide the signs of the two operands and get the resultant sign.
Or, divide the numbers and add the resultant sign to the quotient.
There are some cases as described in the table below:
| divisions of sign | result | example |
| + ÷ + | + | 16 ÷ 4 = 4 |
| +÷ – | – | 16 ÷ (-4) = -4 |
| – ÷ + | – | (-16) ÷ 4 = -4 |
| – ÷ – | + | (-16) ÷ (-4) = 4 |
What are Non-Integers?
A number that is not a whole number, a negative whole number, or zero is defined as Non-Integer.
It is any number that is not included in the integer set, which is expressed as { …-4, -3, -2, -1, 0, 1, 2, 3, 4… }.
Some of the examples of non-integers include decimals, fractions, and imaginary numbers. Another example is the number 3.14, which is the value for pi, is a non-integer.
Another non-integer is the mathematical constant e, known as Euler’s constant, which is equal to about 2.71.
The Golden Ratio, another non-integer mathematical constant, is equal to 1.61. In the fraction form, 1/4, equal to 0.25, is also a non-integer.
Examples of Non-Integer are:
Decimals: .00987, 5.96, 7.098, 75.980 and so on…
Fractions: 5/6, ¼, 54/3, and so on…
Mixed Units: √7, 5½, and so on…
Sample Problems
Question 1. Find two consecutive integers whose sum is equal to 135?
Solution:
Let’s assume two consecutive integers(differ by 1) are:
x and x + 1
Now as per the equation:
Sum of two consecutive integers are 135
⇒ x + (x + 1) = 135
⇒ x + x + 1 = 135
⇒ 2x + 1 = 135
⇒ 2x = 135 – 1
⇒ 2x = 134
⇒ x = 134/2
⇒ x = 67
here the value of x means one number is 67
and as per the condition second number is x + 1 = 67 + 1 = 68
So these are the two consecutive integers whose sum is 135. Here 135 is integer.
Question 2. Find the numbers whose sum of three consecutive even integers is equal to 120?
Solution:
Let’s assume three consecutive integers that differ by 2 are:
x, (x + 2) and (x + 4)
Now as per the equation:
Sum of these three consecutive integers is 120
⇒ x + (x + 2) + (x + 4) = 120
⇒ x + x + 2 + x + 4 = 120
⇒ 3x + 6 = 120
⇒ 3x = 120 – 6
⇒ 3x = 114
⇒ x = 114/3
⇒ x = 38
so the value of first even integer is 38
now as per the equation
second consecutive even integer is x + 2 ⇒ 38 + 2 ⇒ 40
and third consecutive even integer is x + 4 ⇒ 38 + 4 ⇒ 42
So the three numbers are 38, 40, 42
Question 3: Raj has overdrawn his checking account by Rs. 38. The bank debited him Rs.30 for an overdraft fee. Later, he deposited Rs.160. What will be his current balance?
Solution:
Total amount deposited = Rs. 160
Amount overdue by Raj = Rs. 38
⇒ it means Debit amount = -38 (represented as negative integer)
and the Amount charged by bank = Rs. 30
⇒ Debit amount = -30
hence , Total amount debited = −38 + −30 = -68
So, the Current balance= Total deposit +Total Debit
⇒160 + ( –68 ) = 92
Hence, the current balance of Raj is Rs. 92.