Integers are signed numbers. These numbers are useful for representing facts like the rise and fall of temperature, the height above and below sea level and opposite directions or quantities. Integers include positive numbers, zero and negative numbers also.

The number line below shows examples of integers.

Integers

Integers are subsets of real numbers. Except decimals and fractions all numbers positive and negative including zero are integers.

Properties of Integers


where a, b, c are integers
Properties of Integers
** Commutative property is not applicable in case of subtraction. Example 8-5 5-8

Subtraction is also not associative. Example 7-(5-2) (7-5)-2

(i) Zero is less than every positive integer, and greater than every negative integer. Zero is neither positive nor negative.
For example, 0<1, 0<10, etc.
Also 0>-1, 0>-5, etc.

(ii) Every positive integer is greater than every negative integer.
For example, 2>-2, 2>-1,1>-1 etc.

(iii) There is no greatest or smallest integer.

(iv) The smallest positive integer is 1 and the greatest negative integer is -1.

A sequence of Integers where each integer is one more than the previous integer in the sequence are called Consecutive Integers. The consecutive integers can be represented by n, n+1, n+2,....

Consecutive Even Integers:

The sequence of integers which start with an even number and each integer in the sequence is 2 more than the previous integer in the sequence are called Even Consecutive Integers. Example 2, 4, 6.......

Consecutive Odd Integers:

The sequence of integers which start with an odd integer and each of the following integers in the sequence are two more than the previous integer in the sequence are called as Consecutive Odd Integers. Example 35, 37, 39, .....

We must be able to represent a sequence of consecutive integers. If the first number in the sequence is n, then depending upon what sort of consecutive integers we need, we decide on what constant to add to it. For example,

n, n+1, n+2, .... Represents a sequence of consecutive integers
n, n+2, n+4,..... Represents a sequence of odd integers if n is odd and a sequence of even integers if n is even.
n, n+5, n+10,..... represents a sequence of consecutive multiples of 5 if the first integer is a multiple of 5.

We must also be able to reconstruct the required consecutive integer sequence, once we know the value of n. For example when n = 6

If we have to find three consecutive integers then they are 6, 7, 8
If we have to find four consecutive even numbers, 6, 8, 10, 12
If we have to find three consecutive multiples of three then 6, 9, 12

Examples on Consecutive Integers:

Given below are some examples on consecutive integers.

Example 1:

The sum of the greatest and least of a set of three consecutive integers is 120. What are the three integers in the sequence?

Solution:

Let the three consecutive integers in the sequence be x, x+1, x+2
The sum of the consecutive integers is 120.
x + x +1 + x+2 = 120
3x + 3 = 120
Solve for x
3x = 120 - 3
x = $\frac{117}{3}$ = 39
x+1 = 40
x+2 = 41
Hence, the three consecutive integers are 39 ,40, 41

Example 2:

If the difference between the cubes of two consecutive integers is 217, then find the integers?

Solution:

Let the two consecutive integers be n and n+1. Then the difference between their cubes is
n³ - (n+1)³ = 217
Now we have to solve for n.
n³ - [n³ +1 + 3n(n+1)] = 217 Since (n+1)³ = n³ + 1 + 3n(n+1)
n³ - n³ - 1 - 3n² - 3n = 217 Eliminating the braces
3n² + 3n - 216 = 0 Simplify
n² + n -72 = 0 multiplying by $\frac{1}{3}$
(n + 9)(n - 8) = 0 Factorizing
n = -9 or n = 8
Now we have to decide which one to take up. Let us try
n = 8 then the next integer is 9
the difference between the cubes of 8 and 9 is 9³ - 8³ = 217
Let n = -9. The next consecutive integer is n+1 = -8.
(-9)³ - (-8)³ = -217 which is not what is given.
Hence the two consecutive integers are 9, 8

Natural numbers are called as positive integers.

Thus, 1, 2, 3, 4......... are positive integers.
+1,+ 2,+ 3,+ 4,.....or 1, 2, 3, ....

Observe that these numbers start with a + sign or without any sign. These are Natural numbers. Hence Natural numbers are a subset of Integers. The positive numbers here are greater than zero.

Numbers -1, -2, -3 ... are called as negative numbers. They are just opposite in sign to the natural numbers 1, 2, 3...

Observe that,

1 + (-1) = 0
2 + (-2) = 0
3 + (-3) = 0
4 + (-4) = 0 and so on ..............

-1 is called negative of one and -1 and 1 are called the opposites of each other. Similarly, -2 is also called negative of 2 and -2 and 2 are called the opposites of each other.

Also, -3 is also called negative of 3 and -3 and 3 are called the opposites of each other and so on. Combining these new numbers with whole numbers, we obtain a new collection of numbers which is written as

...............-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.....................

These numbers are called integers.
So, the negative numbers, zero and the natural numbers together are called integers.
For example, -8,-2,0,2,8
6.2, , 3.5 these are not integers.

The negative numbers are called negative integers.
Thus, ......... -4,-3,-2,-1 are negative integers.

Natural numbers are called as positive integers.
Thus, 1,2,3,4......... are positive integers.
0 is neither positive nor negative.

The collection of numbers which contain all negative integers , positive integers and 0 are called a Set of Integers.

Before we go on to the addition of Integers let us recall that 6 and -6 are the opposites which are at equal distance from zero.

The distance of any number from zero on the number line is called as the absolute value of that number. Since the distance cannot be negative, the absolute values are always positive!

|6| = |-6| = 6

Sum is the term used for the outcome of adding two or more numbers. One way to add integers is by using the Number line. We have to start from the first number as the initial point. Move to the right to add positive numbers and to the left to add negative numbers.

Rules for addition of Integers

a) To add two positive integers, add the absolute values. The sign of the sum is positive.
4 + 5 = 9

b) To add two negative integers, add the absolute values of the two numbers and the sign of the sum is also negative.
-4 + (-5)

Step 1: |-4| + |-5| = 9
Step 2: Sign = -
Hence the sum is -9

c) To add a positive integer and a negative integer, find the difference of the absolute values and give the sum, the sign of the greater absolute value integer.
Case 1: Add -4 + 5
Step 1: The absolute values of |-4| = 4 and |5| = 5
The difference between 5 and 4 is 1
Step 2: The greater absolute value integer is 5 and its sign is positive. Hence give the difference positive sign.
The Difference is 1

Case 2: Add 4 and -5
Step 1: The absolute values
|4| = 4
|-5| = 5
The difference is 1
Step 2: The sign of the greater absolute value integer is minus.
Hence the difference is -1.
Use of shaded squares to understand the rules of addition of Integers better. Let the empty square be negative integers and the shaded squares represent the positive Integers.
When an empty square and shaded square combine, they cancel each other.

Let us consider all the four cases of addition.
Adding Integers Example
In the case of 4 + 5, both are represented by shaded squares since they are both positive numbers. Hence there are no squares that will cancel each other and so the sum is 9 shaded squares, which is +9.

In the case of -4 + (-5), both the numbers are negative and hence are represented by empty squares. There are no squares that cancel up and so all the empty squares are retained and so we get 9 empty squares which is -9.

In the case of -4 +5, -4 is represented by four empty squares and 5 by five shaded squares. Since there are empty squares as well as shaded squares, the four empty squares pair up with for shaded squares and as a result only one shaded square is left out. Hence the sum is +1

In the case of 4+(-5), 4 is represented by four shaded squares and -5 by five empty squares. The four shaded squares, pair up with four empty squares and cancel each other and as a result there is one empty square that is left out. Hence the sum is -1

Examples on adding integers:

Given below are some of the examples that explains how to add integers.

Example 1:

Add -6 + 9 using model.

Solution:

- 6 is a negative number and so we have to represent it using 6 empty squares and 9 is a positive and so represent it using 9 shaded squares.

Since there are both empty squares and also shaded squares, 6 empty squares will pair up with 6 shaded squares and will cancel each other. There are 3 shaded squares left out and so the sum is +3.
Adding Integer Example
Hence, -6 + 9 = 3

Example 2:

Find the sum of -99 and +99.

Solution:

Step1: The absolute values are
|-99| = 99 and |99| = 99
Step 2: Find the difference of the absolute values
99 - 99 = 0
Step 3: Since the difference is 0, it has no sign and so we need not attach any sign.
The sum of -99 + 99 = 0

Before we start with subtraction, let us review the terms of Subtraction.

Minuend: The first number from which the second number must be subtracted is called the minuend.

Subtrahend: The second number that is the number that has to be subtracted is called as the subtrahend.

Difference: The answer of subtraction.

To subtract Integers, all we have to do is to add the minuend and the opposite of the subtrahend. Remember that opposite of 5 is -5 and that -9 is 9!

  • Step 1: Change the sign of the subtrahend. If the subtrahend is positive, then make it negative and if it is negative change it to positive.
  • Step 2: Change the subtraction sign to addition.
  • Step 3: Follow the rules of addition of Integers.

There can be six possible cases.

Case 1: Both the minuend and subtrahend are positive. Like 9 - (+4)
Case 2: Both the minuend and subtrahend are negative. Like -9 - (-4)
Case 3: Minuend is positive and the subtrahend is negative. Like 9 - (-4)
Case 4: Minuend is negative and the subtrahend is positive. Like -9 - (+4)
Case 5: Minuend is zero. Like 0 - (4) or 0 - (-4)
Case 6: Subtrahend is zero. Like 4 - 0 or -4 - 0

Subtraction of Integers can be performed on a number line or using Models like counters or just by following the rules or steps.

Examples on Subtracting Integers

Given below are some of the examples that explains how to subtract integers.

Example 1:

Follow rules of subtraction to solve 15 - (-7)

Solution:

Step 1: Change the sign of the second number (subtrahend). The opposite of -7 is +7. So -7 is changed to +7
Step 2: Change the subtraction sign to addition. 15 - (-7) = 15 + 7
Step 3: Follow the rules of addition of Integers.

Since both the numbers are positive, we have to add them up and the answer is also positive.

15 + 7 = 22

Example 2:

Use counters to subtract 6 - 2

Solution:

Step 1: Start with six counters
Step 2: Remove two counters
Step 3: Now count the remaining counters. There are 4 positive counters and so, the answer is 4.

Subtracting Integers Example

Let us start with a quick review of the terms related to multiplication. The numbers to be multiplied are the factors and the result of multiplication is called as product.

4 × 5 = 20
4 and 5 are factors and 20 is the product.

There are four possibilities of multiplying two factors

Case 1: Both the factors are positive: When both the factors are positive, multiply the factors and the product is always positive. 3 × 5 = 15
Case 2: One factor is positive and the other is negative: When one of the factors is negative and the other is positive that is when the factors are of opposite signs, multiply the factors and the product has a negative sign. -3 × 5 = -15 or 3 × -5 = -15
Case 3: Both the factors are negative: When both the factors are negative, multiply the factors and then the product is always positive. -3 × -4 = 12
Case 4: One of the factors is zero: When one of the factors is zero, then the product is always zero. 0 × 6 = 0 or -5 × 0 = 0 Remember that zero is neither positive nor negative.

In short, when we multiply two Integers,

• If the factors are of the same sign then the product is always positive.

• If the factors are of the opposite signs then the product is always negative.

Here is an easy way to remember the sign rules for multiplication

(+) × (+) = (+)
(+) × (-) = (-)
(-) × (+) = (-)
(-) × (-) = (+)

Examples on Multiplying Integers:


Given below are some of the examples on multiplying integers.

Example 1:

The temperature in Arctic region is dropping by 4 degree centigrade each hour. Find the drop in temperature after 6 hours?

Solution:

There is a drop in temperature and so the drop in temperature after 6 hours is -4 times 6
-4 × 6 = -24 degree centigrade.
The product is negative since (-) × (+) = (-)
The answer is negative since the temperature is dropping.

When we divide a negative integer by a positive integer, we divide them as whole numbers and put a minus sign (-) before the quotient. So we get a negative integer.

$\frac{(-12)}{3}$ = (-4)

$\frac{(-32)}{4}$ = (-8)

$\frac{(-20)}{5}$ = (-4)

In general, if a and b are two integers, then
$\frac{(-a)}{b}$ = -($\frac{a}{b}$), Where b $\neq$ 0

(ii) When we divide a positive integer by a negative integer, we divide them as whole numbers and put a minus sign (-) before the quotient. So we get a negative integer.

$\frac{21}{(-7)}$ = (-3)

$\frac{24}{(-6)}$ = (-4)

$\frac{20}{(-5)}$ = (-4)

In general, if a and b are two integers, then

$\frac{a}{(-b)}$ = -($\frac{a}{b}$), Where b $\neq$ 0

(iii) When we divide a negative integer by a negative integer, we divide them as a whole numbers and put a positive sign (+). So we get a positive integer.

$\frac{(-18)}{(-6)}$ = 3

$\frac{(-12)}{(-4)}$ = 3

$\frac{(-32)}{(-8)}$ = 4

In general, if a and b are two integers, then

$\frac{(-a)}{(-b)}$ = $\frac{(a)}{(b)}$ , Where b $\neq$ 0

(iv) When we divide a positive integer by a positive integer, we divide them as a whole numbers and put a positive sign (+). So we get a positive integer.

$\frac{16}{4}$ = 4

$\frac{12}{3}$ = 4

$\frac{40}{8}$ = 5

In general, if a and b are two integers, then

$\frac{a}{b}$ = $\frac{a}{b}$, Where b $\neq$ 0

Rules for Division of Integers:

Rule 1 : The quotient of two integers with like signs is positive.
Rule 2 : The quotient of two integers with unlike signs is negative.

Examples on Dividing Integers:


Given below are some of the examples on dividing integers.

Example 1:

$\frac{(-8)}{4}$

Solution:

Here dividend is (-8) and divisor is 4.So, the single jump of (-8) can be split into 4 equal jumps of (-2).

Thus, $\frac{(-8)}{4}$ = -2.

Example 2:

$\frac{(-6)}{(-3)}$

Solution:

Here dividend is (-6) and divisor is (-3).So, the single jump of (-6) can be split into (-3) equal jumps of 2.

Thus, $\frac{(-6)}{(-3)}$ = 2.

The greatest integer function is represented by the symbol [x] and could be explained as the largest integer less or equivalent to x.
This has an infinite number of breaks or steps one at every integer value in its domain.
The greatest integer function is an integer corresponding to the variable of the function.
And as greatest integer function always rounds up the variable values to nearest integer and is better known as floor function.
The variable values are usually positive and negative real values.
Example of simple greatest integer function:
[ax] where, "a" is a constant and "x" is the variable of the function.
Without the loss missing the generality, let "a" = 1, the function values of this function could be given out as the following for the corresponding values of the variables.


Greatest Integer Function Example

f(t) = f$_{1}$ (t) + f$_{2}$ (t)
Where, f$_{1}$ (t) = 2[2t] and f$_{2}$ (t) = 5[3 cost]
Now to maintain the function of this nature, and compare the function f(t) superposed with f$_{1}$ (t) and f$_{2}$ (t).

Greatest Integers Function Example

We could also notice that f$_{1}$ (t) remains as a constant in each of the constant time interval with a time span of 0.5
The time intervals in which the function values are constants may not necessarily be constant.
It is also worth to notice that f$_{2}$ (t) to which the interval length varies from one interval to another.
Combining the two functions f$_{1}$ (t) and f$_{2}$ (t) the piecewise constant function f(t) varies piecewise-constantly with the varying time intervals in which function values are constant.

Given below are some of the integer word problems that explains us how to easily solve them.

Example 1:

Angela had some money with her. She spent it all to buy lunch worth $6 a day for 7 days. How much money did she have 7 days ago?

Solution:

Angela spent 6dollars to buy lunch for each day and so it denoted by -6.
She bought lunch for 7 days and we are to find the money she had 7 days ago so it is -7
The money she had 7 days ago = -$6 × -7 = $42.
Since the product of two negative numbers is positive.(-) × (-) = (+)
Also, the positive solution is true because it is the money she had 7 days ago!

Practice Problems on Integer Word Problems:


Given below are some of the practice problems on Integer word problems.

Practice Problem 1:

Morgan lost money through a tear in his pocket. He started with 50 cents, lost 20 cents, put another 80 cents and spent 25 cents, but lost 12 cents again and then at the end put in 30 cents. How many cents should there be in his pocket at the end of the day?

Practice Problem 2:

Natalie gained 14 pounds during the holidays. She dieted and lost 23 pounds but gained another 5 pounds when she went on vacation. By end of the year she lost another 9 pounds. How much and in what proportion the net change in weight?

Practice Problem 3:

The temperature for the first two days of the week fell through 15 degrees. It rose by 10 degrees on each consecutive day for the next 3 days and fell again by 5 degrees. If the original temperature was 41 degrees, what is the final temperature ?