Chapter 6 Integers (Concepts)

Integer and Its Representation on Number Line

In earlier classes, we have learned about whole numbers ($0, 1, 2, 3, ...$) which include natural numbers and zero. Now, we will extend our number system further to include a new type of numbers: negative numbers. This expanded collection of numbers is called Integers.

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Need for Negative Numbers

Sometimes, we need to represent quantities that are below a certain point, or in an opposite direction from a positive quantity. For example, whole numbers are not enough to represent:

• Temperature below $0^\circ\text{C}$. For instance, $5^\circ\text{C}$ below zero is represented as $-5^\circ\text{C}$.
• Losses in financial transactions. A loss of $\textsf{₹}500$ in a business can be written as $-\textsf{₹}500$.
• Depths below sea level. A location 100 meters below sea level is represented as $-100$ m.
• Money owed to someone (debt). A debt of $\textsf{₹}1000$ can be represented as $-\textsf{₹}1000$.

These situations introduce the concept of numbers that are less than zero. These numbers are called negative numbers.

Examples of negative numbers are $-1, -2, -3, -10, -100$, etc. The symbol '$-$' (minus sign) placed before a number is used to indicate that it is a negative number.

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Integers

The collection of numbers that includes whole numbers ($0, 1, 2, 3, ...$) and negative numbers ($..., -3, -2, -1$) is known as Integers.

We can represent the set of integers as: $\{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\}$

Let's categorise integers:

• The numbers $1, 2, 3, 4, ...$ are located to the right of zero on the number line. These are called positive integers. Generally, a '+' sign is not written before positive integers (e.g., $5$ is the same as $+5$).
• The numbers $..., -4, -3, -2, -1$ are located to the left of zero on the number line. These are called negative integers. The minus sign '$-$' is always written before a negative integer.
• The number 0 is a unique integer. It is neither a positive integer nor a negative integer. It is the point of origin or reference on the number line.

We can summarise the relationship between different number systems:

• Natural Numbers: $\{1, 2, 3, 4, ...\}$ (These are all positive integers).
• Whole Numbers: $\{0, 1, 2, 3, 4, ...\}$ (These include 0 and all positive integers).
• Integers: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ (These include negative numbers, zero, and positive numbers).

Thus, all natural numbers are whole numbers, and all whole numbers are integers. However, not all integers are whole numbers (e.g., $-5$ is an integer but not a whole number), and not all integers are natural numbers (e.g., $0$ and $-5$ are integers but not natural numbers).

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Representation of Integers on a Number Line

A number line is a visual tool used to represent numbers. We can easily represent integers on a number line following these steps:

1. Draw a straight line and mark a point exactly in the middle. Label this point as 0. This point is called the origin.
2. Choose a unit distance. Starting from 0, mark points to the right of 0 at equal intervals corresponding to this unit distance. Label these points sequentially as $1, 2, 3, 4, ...$. These represent the positive integers.
3. Starting from 0, mark points to the left of 0 at the same equal intervals as used on the right side. Label these points sequentially as $-1, -2, -3, -4, ...$. These represent the negative integers.
4. Draw arrows on both ends of the line. These arrows indicate that the number line extends infinitely in both the positive (right) and negative (left) directions.

Here is what an integer number line looks like:

Key observations from the number line:

• Numbers on the number line increase as you move from left to right.
• Numbers on the number line decrease as you move from right to left.
• Every positive integer is greater than zero and every negative integer.
• Every negative integer is less than zero and every positive integer.
• Zero is greater than every negative integer and less than every positive integer.
• For any two integers, the integer to the right on the number line is always greater than the integer to the left. For example, $2 > -3$ because 2 is to the right of -3. Similarly, $-1 > -4$ because -1 is to the right of -4.

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Answer:

Absolute Value & Comparison of Integers

Integers include positive numbers, negative numbers, and zero. To work effectively with them, it's crucial to understand their magnitude (size) and how to order them correctly. This involves the concepts of absolute value and the rules for comparing integers on a number line.

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Absolute Value

The absolute value of an integer is its numerical value without regard to its sign. Geometrically, it represents the distance of that integer from zero on the number line. Since distance is always a positive quantity or zero, the absolute value of any integer is always non-negative.

The symbol for the absolute value of an integer 'a' is $|a|$.

• The distance of $5$ from $0$ is $5$ units. So, $|5| = 5$.
• The distance of $-5$ from $0$ is also $5$ units. So, $|-5| = 5$.
• The distance of $0$ from itself is $0$ units. So, $|0| = 0$.

Formal Definition

Mathematically, the absolute value of an integer $x$ is defined as follows:

$|x| = \begin{cases} x & , & \text{if } x \ge 0 \\ -x & , & \text{if } x < 0 \end{cases}$

This definition means:

• If the number is positive or zero, its absolute value is the number itself. (e.g., $|7| = 7$)
• If the number is negative, its absolute value is its opposite (the negative of the negative number), which makes it positive. (e.g., $|-7| = -(-7) = 7$)

Let's look at some more examples:

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Comparison of Integers

Comparing integers means determining whether one is greater than ($>$), less than ($<$), or equal to ($=$) another. The number line provides a clear visual aid for this.

Golden Rule: On a horizontal number line, numbers increase as you move from left to right. Therefore, any integer to the right of another integer is greater.

Rules for Comparing Integers

From the golden rule, we can derive the following specific rules:

1. Positive vs. Negative Integers: Every positive integer lies to the right of every negative integer. Therefore, any positive integer is always greater than any negative integer.

e.g., $1 > -1000$ and $-50 < 2$.

2. Integers and Zero: Zero is to the left of all positive integers and to the right of all negative integers.

• Every positive integer is greater than 0. (e.g., $5 > 0$)
• Every negative integer is less than 0. (e.g., $-5 < 0$)

3. Comparing Two Negative Integers: This is the trickiest case. As you move further to the left from zero, the numbers get smaller. This means the negative number with the smaller absolute value (the one closer to zero) is the greater number.

e.g., To compare $-3$ and $-8$: $|-3| = 3$ and $|-8| = 8$. Since $3 < 8$, we have $-3 > -8$. On the number line, $-3$ is to the right of $-8$.

Case	Comparison Rule	Example
One positive, one negative	The positive integer is always greater.	$3 > -300$
Two positive integers	The one with the larger value is greater.	$25 > 12$
Two negative integers	The one with the smaller absolute value is greater.	$-10 > -20$
Any integer and $0$	Positive integers are $> 0$. Negative integers are $< 0$.	$9 > 0, -9 < 0$

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Answer:

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Answer:

Addition of Integers

We have learned how to compare integers and represent them on a number line. Now, let's learn how to perform addition with integers. We can add integers using the number line or by following specific rules based on their signs.

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Addition on the Number Line

Adding integers on the number line is like taking steps. We start at the first integer and then move a certain number of steps to the right or left depending on the second integer:

• To add a positive integer, move that many units to the right on the number line.
• To add a negative integer, move the absolute value of that integer's units to the left on the number line.

Let's look at some examples:

Example: Add $3 + (-5)$.

To find the sum of $3$ and $-5$ using the number line:

1. Start at the point representing $3$ on the number line.
2. We are adding $-5$, which is a negative integer. So, we need to move $5$ units to the left from our starting point ($3$).
3. Count $5$ steps to the left from $3$ (i.e., $3 \to 2 \to 1 \to 0 \to -1 \to -2$).
4. We land on the point representing $-2$.

The number line representation would look like this:

So, $3 + (-5) = -2$.

Example: Add $-2 + (-4)$.

To find the sum of $-2$ and $-4$ using the number line:

1. Start at the point representing $-2$ on the number line.
2. We are adding $-4$, which is a negative integer. So, we need to move $4$ units to the left from our starting point ($-2$).
3. Count $4$ steps to the left from $-2$ (i.e., $-2 \to -3 \to -4 \to -5 \to -6$).
4. We land on the point representing $-6$.

The number line representation would look like this:

So, $-2 + (-4) = -6$.

Example: Add $-1 + 6$.

To find the sum of $-1$ and $6$ using the number line:

1. Start at the point representing $-1$ on the number line.
2. We are adding $6$, which is a positive integer. So, we need to move $6$ units to the right from our starting point ($-1$).
3. Count $6$ steps to the right from $-1$ (i.e., $-1 \to 0 \to 1 \to 2 \to 3 \to 4 \to 5$).
4. We land on the point representing $5$.

The number line representation would look like this:

So, $-1 + 6 = 5$.

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Rules for Addition of Integers

Adding integers using a number line can be time-consuming, especially for large numbers. We can use simple rules based on the signs of the integers:

Rule 1: Adding two positive integers:

When you add two positive integers, the sum is always positive. Simply add their absolute values.

Positive Integer + Positive Integer = Positive Integer

Example:

Example:

Rule 2: Adding two negative integers:

When you add two negative integers, the sum is always negative. Add their absolute values and then put a minus sign before the result.

Negative Integer + Negative Integer = Negative Integer

Example:

Absolute values are $|-5|=5$ and $|-7|=7$.

Add the absolute values: $5 + 7 = 12$.

Since both original integers were negative, the sum is negative.

Example:

Absolute values are $|-10|=10$ and $|-3|=3$.

Add the absolute values: $10 + 3 = 13$.

The sum is negative: $-13$.

Rule 3: Adding a positive integer and a negative integer:

When you add a positive integer and a negative integer, the sign of the sum depends on which integer has the larger absolute value. To find the sum:

1. Find the absolute value of each integer.
2. Subtract the smaller absolute value from the larger absolute value.
3. The sign of the result is the same as the sign of the integer with the larger absolute value.

Example:

Absolute values are $|10|=10$ and $|-3|=3$.

Subtract the smaller absolute value from the larger: $10 - 3 = 7$.

The integer with the larger absolute value is $10$, which is positive. So, the result is positive.

Example:

Absolute values are $|-10|=10$ and $|3|=3$.

Subtract the smaller absolute value from the larger: $10 - 3 = 7$.

The integer with the larger absolute value is $-10$, which is negative. So, the result is negative.

Example:

Absolute values are $|5|=5$ and $|-12|=12$.

Subtract the smaller absolute value from the larger: $12 - 5 = 7$.

The integer with the larger absolute value is $-12$, which is negative. So, the result is negative.

Example:

Absolute values are $|-5|=5$ and $|12|=12$.

Subtract the smaller absolute value from the larger: $12 - 5 = 7$.

The integer with the larger absolute value is $12$, which is positive. So, the result is positive.

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Properties of Addition of Integers

Addition of integers follows certain properties, which are helpful in calculations:

• Closure Property: When you add any two integers, the result is always an integer. We say that integers are closed under addition.

  If $a$ and $b$ are any two integers, then $a+b$ is also an integer.

  Example: $3 + (-8) = -5$. Here, $3$, $-8$, and $-5$ are all integers.

  Example: $-10 + (-15) = -25$. Here, $-10$, $-15$, and $-25$ are all integers.

• Commutative Property: The order in which you add two integers does not affect the sum.

  If $a$ and $b$ are any two integers, then $a + b = b + a$.

  Example: $5 + (-2)$. Using the rule, $|5|=5, |-2|=2$. Difference is $5-2=3$. Sign of $5$ (larger absolute value) is positive. So, $5+(-2) = 3$.

  Now, $(-2) + 5$. Using the rule, $|-2|=2, |5|=5$. Difference is $5-2=3$. Sign of $5$ (larger absolute value) is positive. So, $(-2)+5 = 3$.

  Thus, $5 + (-2) = (-2) + 5 = 3$.

• Associative Property: When adding three or more integers, the way you group them does not affect the sum.

  If $a$, $b$, and $c$ are any three integers, then $(a + b) + c = a + (b + c)$.

  Example: Let's find the sum of $2 + (-3) + 5$.

  Group 1: $(2 + (-3)) + 5$. First, $2 + (-3) = -1$. Then, $-1 + 5 = 4$.

  Group 2: $2 + ((-3) + 5)$. First, $(-3) + 5 = 2$. Then, $2 + 2 = 4$.

  Both groupings give the same result: $(2 + (-3)) + 5 = 2 + ((-3) + 5) = 4$.

• Additive Identity: Adding the integer $0$ to any integer does not change the value of the integer. The integer $0$ is called the additive identity for integers.

  If $a$ is any integer, then $a + 0 = 0 + a = a$.

  Example: $-7 + 0 = -7$.

  Example: $15 + 0 = 15$.

• Additive Inverse: For every integer $a$, there exists another integer, denoted by $-a$, such that their sum is $0$. The integer $-a$ is called the additive inverse (or opposite) of $a$.

  If $a$ is any integer, then $a + (-a) = (-a) + a = 0$.

  Example: The additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.

  Example: The additive inverse of $-10$ is $-(-10) = 10$, because $-10 + 10 = 0$.

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Answer:

Subtraction of Integers

Now that we know how to add integers, we can easily understand how to subtract them. The main concept behind subtracting an integer is to convert the subtraction problem into an addition problem by adding the additive inverse of the integer being subtracted.

Remember from the previous section that the additive inverse of an integer is the integer with the opposite sign (e.g., the additive inverse of $5$ is $-5$, and the additive inverse of $-8$ is $-(-8) = 8$). When an integer is added to its additive inverse, the result is $0$.

The rule for subtraction is:

To subtract an integer 'b' from an integer 'a', add the additive inverse of 'b' to 'a'.

Since the additive inverse of $b$ is $-b$, the rule can be written as:

Let's apply this rule to different cases of subtraction.

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Rules for Subtraction (Derived from Addition Rules)

Using the rule $a - b = a + (-b)$, we can perform subtraction by following the rules of addition:

Case 1: Subtracting a positive integer from a positive integer (where the second is larger).

Example: Subtract $8$ from $5$. This is $5 - 8$.

Using the rule, $5 - 8 = 5 + (\text{additive inverse of } 8)$. The additive inverse of $8$ is $-8$.

So, $5 - 8 = 5 + (-8)$.

Now, we follow the rule for adding a positive integer ($5$) and a negative integer ($-8$).

• Absolute values are $|5|=5$ and $|-8|=8$.
• Difference between absolute values is $8 - 5 = 3$.
• The integer with the larger absolute value is $-8$, which is negative.

So, the result is negative.

Case 2: Subtracting a negative integer from a negative integer.

Example: Subtract $-4$ from $-10$. This is $-10 - (-4)$.

Using the rule, $-10 - (-4) = -10 + (\text{additive inverse of } -4)$. The additive inverse of $-4$ is $4$.

So, $-10 - (-4) = -10 + 4$.

Now, we follow the rule for adding a negative integer ($-10$) and a positive integer ($4$).

• Absolute values are $|-10|=10$ and $|4|=4$.
• Difference between absolute values is $10 - 4 = 6$.
• The integer with the larger absolute value is $-10$, which is negative.

So, the result is negative.

Notice that subtracting a negative number is equivalent to adding the corresponding positive number: $-10 - (-4) = -10 + 4$.

Case 3: Subtracting a positive integer from a negative integer.

Example: Subtract $5$ from $-2$. This is $-2 - 5$.

Using the rule, $-2 - 5 = -2 + (\text{additive inverse of } 5)$. The additive inverse of $5$ is $-5$.

So, $-2 - 5 = -2 + (-5)$.

Now, we follow the rule for adding two negative integers ($-2$ and $-5$).

• Absolute values are $|-2|=2$ and $|-5|=5$.
• Add the absolute values: $2 + 5 = 7$.
• Since both integers are negative, the sum is negative.

So, the result is negative.

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Subtraction on the Number Line

We can also visualise subtraction using the number line:

• Subtracting a positive integer: To subtract a positive integer, move that many units to the left on the number line (same as adding a negative integer).
• Subtracting a negative integer: To subtract a negative integer, move the absolute value of that integer's units to the right on the number line (same as adding a positive integer).

Example: Calculate $5 - 8$ using the number line.

To find the result of $5 - 8$ using the number line:

1. Start at the point representing $5$ on the number line.
2. We are subtracting positive $8$. This means moving $8$ units to the left from our starting point ($5$).
3. Count $8$ steps to the left from $5$ (i.e., $5 \to 4 \to 3 \to 2 \to 1 \to 0 \to -1 \to -2 \to -3$).
4. We land on the point representing $-3$.

The number line representation would look like this:

So, $5 - 8 = -3$. This matches the result we got using the additive inverse rule ($5 + (-8) = -3$).

Example: Calculate $-10 - (-4)$ using the number line.

To find the result of $-10 - (-4)$ using the number line:

1. Start at the point representing $-10$ on the number line.
2. We are subtracting negative $4$. This means moving the absolute value of $-4$ (which is $4$) units to the right from our starting point ($-10$).
3. Count $4$ steps to the right from $-10$ (i.e., $-10 \to -9 \to -8 \to -7 \to -6$).
4. We land on the point representing $-6$.

The number line representation would look like this:

So, $-10 - (-4) = -6$. This matches the result we got using the additive inverse rule ($-10 + 4 = -6$).

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Answer:

Addition or Subtraction of Three or More Integers

When you have a mathematical expression involving the addition and subtraction of three or more integers, you need a systematic approach to find the correct result. You can either solve the operations step-by-step from left to right or group the positive and negative integers together before adding.

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Method 1: Solve Step-by-Step (Left to Right)

This method involves performing the operations strictly in the order they appear from the left side of the expression to the right side. Remember to apply the rules for addition and subtraction of two integers at each step.

Example: Solve $5 - (-3) + (-7)$.

We will perform the operations one by one from left to right.

First, calculate $5 - (-3)$. Subtracting $-3$ is the same as adding its additive inverse, which is $3$.

$5 - (-3) = 5 + 3 = 8$

Now, substitute this result back into the original expression:

$5 - (-3) + (-7) = 8 + (-7)$

Next, calculate $8 + (-7)$. This is the addition of a positive integer ($8$) and a negative integer ($-7$).

Find the difference in absolute values: $|8|=8$, $|-7|=7$. Difference is $8 - 7 = 1$.

The integer with the larger absolute value is $8$, which is positive. So, the result is positive.

$8 + (-7) = 1$

Thus, the final result is:

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Method 2: Grouping Positive and Negative Integers

This method often simplifies calculations, especially when dealing with many integers. It involves separating the positive and negative integers and then combining them.

1. Rewrite the entire expression by converting all subtractions into additions of the additive inverse. Remember that $a - b = a + (-b)$ and $a - (-b) = a + b$.
2. Identify all the positive integers and all the negative integers in the rewritten expression.
3. Group all the positive integers together and find their sum (using the rule for adding positive integers). The sum will be positive.
4. Group all the negative integers together and find their sum (using the rule for adding negative integers). The sum will be negative.
5. Finally, add the sum of the positive integers and the sum of the negative integers (using the rule for adding integers with different signs).

Example: Solve $5 - (-3) + (-7)$ using the grouping method.

Rewrite the expression by converting subtraction to addition:

Identify positive integers: $5, 3$.

Identify negative integers: $-7$.

Group positive integers and find their sum:

Group negative integers and find their sum:

Now, add the sum of positive integers and the sum of negative integers:

Using the rule for adding integers with different signs ($|8|=8, |-7|=7$, difference $8-7=1$, sign of $8$ is positive):

So, $5 - (-3) + (-7) = 1$. This matches the result from Method 1.

Example: Solve $-12 + 8 - (-5) - 3 + 10$.

Rewrite the expression by converting subtractions to additions:

Identify positive integers: $8, 5, 10$.

Identify negative integers: $-12, -3$.

Group positive integers and find their sum:

Group negative integers and find their sum:

Now, add the sum of positive integers and the sum of negative integers:

Using the rule for adding integers with different signs ($|23|=23, |-15|=15$, difference $23-15=8$, sign of $23$ is positive):

So, $-12 + 8 - (-5) - 3 + 10 = 8$.

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Answer: