Introduction to Integers

Taking the Positives with the Negatives!

Because integers are fun and really really helpful Calvin!

Till now all mathematical operations have been performed on whole or natural numbers. Working with numbers beginning from zero and progressing towards the right of the number line. What exists on the left of the number line?

What are integers?

Up to this point, children could answer questions like which number is smaller than 2 (ans: 0 or 1). However, what if they were asked what is smaller than zero? Therefore, the need arises to introduce a completely new group of numbers called negative numbers.

An integer is a whole number that can be either positive or negative. Which means integers cannot be fractions and decimals (which you will learn later). Positive integers are basically the everyday numbers that we use to count things, for example ‘3 apples’, where 3 is a positive integer. As a result, positive integers are also sometimes called ‘counting numbers’. Negative integers are numbers that have a minus sign as a prefix, for example, -3 is a negative integer. Don’t worry they didn’t do anything wrong! Negative integers aren’t bad numbers! They just happen to be on the other side of the number line from positive integers!

Integers are numbers that are whole numbers and negative numbers. i.e., Integers are the set of all natural numbers, along with their negatives, and 0.

All integers are represented by the alphabet Z and are written as;

Z = {…, –4, –3, –2, –1, 0, 0, 1, 2, 3, 4, …}

✍Note: The notation \(\mathbb{Z}\) for the set of integers comes from the German word Zahlen, which means "numbers". Integers strictly larger than zero are positive integers and integers strictly less than zero are negative integers.

We see that going from \({\text{W}}\) to \(\mathbb{Z}\) has further increased the solvability of equations. We also note another very important point. If we plot the numbers in \(\mathbb{Z}\) on a line, we get a plot like the one below. Let us call this the Integer Line.

Positive and Negative Integers

Numbers that are on the left of zero are denoted by a negative () sign, thus making them negative integers. 
Numbers that are on the right of zero are positive integers.
Numbers to the left represent a decrease and numbers to the right represent an increase.
Integer numbers cannot have decimals and cannot be fractions

Numbers that are integers

\(-20, 45, -998, 26, 0\)

Numbers that are not integers

\(-1.23, \) \({6 \over 8},\) \(89.75\)

What about something like \({30 \over 6}\)? Well, at first it may look at bit odd, but if you remember the principles of simplification of fractions, it comes out to be 5, which is an integer!

All mathematical operations such as addition, subtraction, division and multiplication can be performed on integers. All mathematical operations on integers can be solved by following a few simple guidelines.

Solved Example:

Example 1: List natural numbers, whole numbers, and integers from the following:

\[\frac{6}{2}, - 8,0.4,(12 - 5),\frac{5}{2}\]

Solution: Here, \(\frac{6}{2} = 3\) and \((12 - 5) = 7\).

Now, let us think about the set of natural numberswhole numbers, and integers. We can list as follow:

Natural numbers: \(\frac{6}{2},(12 - 5)\)

Whole numbers: \(\frac{6}{2},(12 - 5)\)

Integers: \(\frac{6}{2}, - 8,(12 - 5)\)

Topics Closely Related To Integers

The image shown below shows how different topics are connected to integers. What topics are required as a pre-requisite to master this topic? It also tells that mastery over integers can help build proficiency in connected topics like rational numbers.  

Flow chart for integers

 

Why is it important?

What is a Number line?

A number line is a picture of a straight line graduated with smaller lines to denote a number. At the centre of the number line is the number, zero. Since zero does not have a value associated with it, it is neither positive nor is it negative. The right-hand side of zero on the number line represents positive integers, whereas the left-hand side of zero represents the negative integers. 

Number line showing positive and negative integers

Using the above number line, we can now familiarize ourselves with some basic operations of integers. Integers are important even in day to day life. You will often hear people say that the temperature has dropped by 5 degrees or that the prices of vegetables have gone up by 10 Rupees. The number line is useful for these situations because you can use it to visualize basic mathematical problems! 

Integer Line

We note that on the Integer Line, for every number, we can talk about the next number and the previous number. For example, for number 2, we can say that 3 is the next number and 1 is the previous number. For the number \(- 7\), we can say that is \(- 6\) the next number and is \(- 8\) the previous number. That is, for every number on the Integer Line, we can talk about the closest number(s) to that number. However, this concept of the closest number will not apply in the next higher number system, the set of Rational Numbers.

Also, we note that \(\mathbb{Z}\) is also limited in terms of the solvability of equations. For example, consider the equation \(2x = 3\). This equation has a non-integer solution and is therefore not solvable in \(\mathbb{Z}\).

enlightenedThink: My brother laughs at a problem in his math homework that we are trying to figure out. It asks what the error is in the following statement: Jeff says that every whole number is an integer and that every integer is a whole number. Explain the error.

Tip: Every pig is a mammal but not every mammal is a pig. wink

Exercises with the Number Line

Before we dive into the world of integers, let’s familiarize ourselves with the number line. If you can visualize the number line, you’ll breeze through the chapters of integers in no time! Given below is a sample of a Cueamth worksheet from an integers book. Solve questions that are given. Draw a number line in your book to solve these questions. Take your time, there’s no hurry! Don’t let the wording of the problems confuse you! Before approaching each question, break it down to three steps:

Step 1: The starting point
Step 2: The change in the number line
Step 3: The ending point

The concept of a number line

All mathematical operations on integers are easier to understand once there is a strong understanding of the number line.

Example:

6 – 3 = 3

  1. Step 1: we go three places to the right of 0, landing on to 6.
  2. Step 2: since the operation calls for a subtraction of 3, from the 6 we move 3 steps further to the left.
  3. Step 3: now we are on 3. Hence proving the addition and the utilization of the number line. 

Zeros and fractions

When it comes to integers, the number 0 is neither positive or negative. It is a counting number which symbolises that there are no objects.

Children often make errors in judgement when it comes to fractions. According to the rules, an integer cannot be a fraction. But the fraction \({25 \over 5}\) is, in fact, an integer. 5 x 5 = 25.  25 ÷ 5 = 5. Reducing it to a whole number, making it an integer. With the rules of the division, children will discover that when fractions are further reduced to a number without any remainders, they are integers.

Integers and the real world

Integers and their properties are not explicit only to the classrooms. They are everywhere around us, in our everyday activities. Anything that is increasing is positive, and anything with a decreasing factor is negative.

  • In the bank account, every time money is deposited, is positive. Nut, when money is withdrawn the balance, decreases making with withdrawal a negative.
  • With the rise in temperature during summers, it is denoted with a positive sign. But in the cold of the winters, temperature decreases and is denoted by a negative sign.
  • In business, with profit, the numbers are in positive, and losses are negative. With exercise, calories gained are positive and calories lost are negative

Having a core understanding of the number line and a strong handle on the rules of operation on integers simplifies mathematics in general.

However, we at Cuemath, are driven to put the Why before the what and ensure that in our classes, the concepts come first.

Important concepts related to Integers

The Absolute Value of an Integer

The absolute value of an integer is the numerical value of the integer without the negative or positive sign. On the number line, it is calculated as the distance between the number and zero.

For eg: the absolute value of -5 is 5. The absolute value of +5 is also 5. 

Absolute value of an Integer

Addition and Subtraction of integers

Now that we have the basics of integers completed, we can now move on to a few operations on integers. The easiest operations on integers are addition and subtraction. Before we dive into the complex problems, let’s solve some integer puzzles! These are some of the questions from the Cuemath workbook on integers. 

Pretty simple right? When you add red cubes the temperature of the water increases by (1°C x number of red cubes). When you add blue cubes the temperature of the water decreases by (1°C x number of blue cubes). Okay, that was simple! For the next couple of problems, we are going to have the temperature of the water given to us. The trick to solving the problem is finding the difference between the original temperature of the water (given to us) and the final temperature of the water. Once you have done that subtraction, you can figure out which cubes were used red or blue, depending on the increase or decrease in temperature.

Okay let’s take this puzzle to the next level, shall we? You must fill in the blanks with the appropriate integers in the dotted boxes. Remember that red cubes increase the temperature so they can be denoted by a '+' sign, and blue cubes decrease the temperature so they are denoted by a '-' sign. Here we go!

At CueMath we believe that it is important for everyone to visualize their mathematical calculations! So again, we will use the number line to visualize addition and subtraction of integers. Remember, the more you can visualize integers the faster you’ll be able to add and subtract.

Multiplication and Division of integers 

Most adults don’t understand why signs change when you multiply a positive number with a negative number. Despite it being such an important lesson! So, let’s visualize this first and solve some problems!

Okay, now that we’ve learned how signs change during multiplication here are a few exercises for you to solve!

The division of integers is the inverse process of their multiplication. Solve the following problems and write your answers on a piece of paper!

Addition of integers

If both numbers have the same sign, add both numbers and include the same sign as both.

Example:

  1. 5 + 6 = 11
  2. –6 + (–9) = –15

If both numbers have different signs, add or subtract the number and include the sign of the largest absolute integer.

Example:

  1. 9 + (–5)
    a. 9 –
    5
    b. 4
  2. –9 + 5
    a. –4 
    b. –9 is larger than 5

Addition of integers

Subtraction of integers

If both numbers have positive signs, the difference will be positive.

Example:

10 – 5 = 2

If both numbers have a negative sign, subtract the two numbers and include the sign of the largest absolute value.

Example:

  1. –10 – (–25)
    a. –10 + 25 according to the rules of multiplication 2 negatives make a positive.
    b. 15
        25 is larger than 10
  2. –25 – (–10)
    a. –25 + 10
        2 negatives make a positive
    b. –15
        25 is larger than 15

If both numbers have different signs, subtract the two and denote the difference with the sign of the largest absolute value.

Example:

  1. 18 – 5 = 13
    18 is larger than 3 and has a positive value
  2. –3 – 18 = –21
    2 negative signs call for addition
  3. 3 – 18 = –15
    18 is larger than 3

Subtraction of integers 

 

 

Multiplication of integers

If both integers have the same sign, then the product is positive.

Example:

  1. 2 x 5 = 10
  2. –2 x –5 = 10

If both integers have different signs, the product is always negative.

Example:

  1. 2 x –5 = –10
  2. –2 x 9 = –18

Multiplication of integers


Division of integers

If both integers have the same sign, then the quotient is positive.

Example:

  1. 16 ÷ 8 = 2
  2. –16 ÷ –8 = 2

If both integers have different signs, the quotient is always negative.

Example:

  1. –16 ÷ 8 = –2
  2. 10 ÷ –2 = –5

Division of integers 

 

Some properties of Integers:

  • Integers are closed under the operation of addition: if \(a,b \in \mathbb{Z}\) then \(a + b \in \mathbb{Z}\)
  • Integers are closed under the operation of subtraction: if \(a,b \in \mathbb{Z}\) then \(a - b \in \mathbb{Z}\)
  • Integers are closed under the operation of multiplication: if \(a,b \in \mathbb{Z}\) then \(ab \in \mathbb{Z}\)
  • For any integer \(a\), the additive inverse \( - a\) is an integer.
  • If \(a\) and \(b\) are integers such that \(ab = 0\) then \(a = 0\) or \(b = 0\).
  • No smallest and largest integer exists.

yesChallenge: Is the set of integers closed under the operation of division?

Tip: Take any two integers \(a\) and \(b\) and actually check if \(a,b \in \mathbb{Z}\) then \(\frac{a}{b}\) belongs to \(\mathbb{Z}\) or not.

Important Topics

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