Understanding Integers
For solving integer word problems students need the right base of knowledge on integers. Proper practice is required for solving integer questions correctly. In this article, we will help students develop the base required for answering word problems on integers for Class 7.
Let’s start with the fact that an arithmetic operation is an elementary branch of mathematics. Arithmetic operations are subtraction, addition, division, and multiplication. These arithmetic operations are used for solving integer problems. There are also other types of numbers that can be solved with the help of arithmetic operations.
It should also be noted that integers are a special group that does not contain any decimal or fractional part. Integers include positive numbers, negative numbers, and zero. Also, arithmetic operations on integers are similar to whole numbers.
It also makes it a little confusing to solve word problems on integers for class 6 and word problems on integers for class 7 pdf because there are both positive and negative numbers. This is also why integers are different from whole numbers.
Integers can also be plotted on a number line. A number line might also be used by students when learning how to solve integers questions. These types of questions are more common when it comes to integers word problems Class 7. If you have never seen a number line, then an image of a number line is attached below.
Rules of Integers
There are several rules that students need for learning how to solve integer word problems. Some of those rules are mentioned below.
The sum of any two positive integers will result in an integer.
The sum of any two negative integers is an integer.
The product of two positive integers will give an integer.
The product of two negative integers will be given an integer.
The sum of any integer and its inverse will be equal to zero.
The product of an integer and its reciprocal will be equal to 1.
Now, let’s look at addition, multiplication, subtraction, and division of signed integer numbers. This will help students to work on story problems with integers answer key.
The Addition of Signed Integer Numbers
As mentioned above, if we add two integers with the same sign, then we have to add the absolute value along with the sign that was provided with the number. For example, (+4) + (+7) = +11 and (-6) + (-4) = -10.
Also, if we add two integers with different signs, then we have to subtract the absolute values and write down the difference. This should be done with the sign of the number that has the largest absolute value. For example, (-4) + (+2) = -2 and (+6) + (-4) = +2.
Subtraction of Signed Integer Numbers
If a student wants to solve integer example problems, then he or she needs to know that while subtracting two integers, we have to change the sign of the second number. The second number should be subtracted and the rules of addition should also be followed. For example, (-7) - (+4) = (-7) + (-4) = -11 and (+8) - (+3) = (+8) + (-3) = +5.
Multiplication and Division of Signed Integer Numbers
When it comes to working on integer word problems with solutions of multiplying and dividing two integer numbers, then the rules are quite straightforward. If both the integers have the same sign, then the final results are positive. If the integers have different signs, then the final result is negative. For example, (+2) x (+3) = +6, (+3) x (-4) = -12, (+6) / (+2) = +3, and (-16) / (+4) = -4.
Properties of Integers
Students should also be familiar with the properties of integers if they want to work on integer word problems grade 6 with solutions. Some of those properties of integers are:
Closure property
Associative property
Commutative property
Distributive property
Additive inverse property
Multiplicative inverse property
Identity property
We also need to look at these properties in detail for solving integer problems in 6th grade. Let’s move on to that discussion.
Closure Property:
According to the closure property of integers, if two integers are added or multiplied together, the final result will be an integer only. This means that if a and b are integers, then:
a + b = integer
a x b = integer
For example, 2 + 5 = 7, which is an integer, and 2 x 5 = 10, which is also an integer.
Commutative Property:
According to this property, if a and b are two integers, then a + b = b + a and a x b = b x a. For example, 3 + 8 = 8 x 3 = 24 and 3 + 8 = 8 + 3 = 11. It should be noted that this property is not followed in the case of subtraction and division.
According to the associative property, if a, b, and c are integers, then:
a + (b + c) = (a + b) + c
a x (b x c) = (a x b) x c
For example, 2 + (3 + 4) = (2 + 3) + 4 = 9 and 2 x (3 x 4) = (2 x 3) x 4 = 24.
This property is only valid when it comes to addition and multiplication.
Distributive Property:
The distributive property states that if a, b, and c are integers, then a x (b + c) = a x b + a x c. For example, if we have to prove that 3 x (5 +1) = 3 x 5 + 3 x 1, then we should be start by finding:
LHS = 3 x (5 + 1) = 3 x 6 = 18
RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18
Since, LHS = RHS
This proves our example.
Additive Inverse Property:
This additive inverse property states that if a is an integer, then a + (-a) = 0. This means that-a is the additive inverse of integer a.
Multiplicative Inverse Property:
The multiplicative inverse property states that if a is an integer, then a x (1 / a) = 1. This means that 1 / a is the multiplicative inverse of integer a.
Identity Property of Integers:
The identity property of integers states that a + 0 = a and a x 1 = a. For example, 4 + 0 = 4 and 4 x 1 = 4.
Types of Integers
Earlier, we mentioned that there are three types of integers. In this section, we will look at these types of integers in more depth. The list of those types of integers is mentioned below.
Zero:
Zero can be characterized as neither a positive nor a negative integer. It can be best defined as a neutral number. This also refers to the fact that zero has no sign (+ or -).
Positive Integers:
As the name indicates, positive integers are those numbers that are positive in their nature. These numbers are represented by a positive or plus (+) sign. The positive integers lie on the right side of the zero on the number line. This also means that all positive integers are greater than zero. For example, 122, 54, and 9087268292.
Negative Integers:
Negative integers, on the other hand, are numbers that are represented by a minus (-) or negative sign. These numbers are present on the left side of the zero on a number line. For example, -182, -8292, and -2927225.
Fun Facts About Integers
The word integer comes from the Latin word “integer” which literally means whole.
You might find it interesting to note that integers are not just simple numbers on paper. Instead, these numbers have real-life applications! Both positive and negative integers are used to symbolize two contradicting situations in the real world. For example, if the temperature is above zero, then positive integers are used for denoting the temperature. But if the temperature is less than zero, then negative integers are used for denoting the temperature.
Integers can also help an individual in comparing and measuring two things like how small or big or few or more things are. These integers help in quantifying things. For example, in games like cricket and soccer, integers are used for keeping a track of scores. Movies and songs can also be rated by using integers!
FAQs on Word Problems on Integers
1. What are integers and what is their importance?
An integer is usually defined as a number that can be written without any type of fractional component. These numbers tend to consist of zero, positive natural numbers or whole numbers, and their additive inverses. For example, 2, -56, 98, -302, etc. A set of integers is always denoted by the letter “Z”.
The concept of integers is quite an important one to learn in mathematics. This is mainly because integers tend to help us compute the efficiency in both negative as well as positive numbers in a wide range of fields. They also help us to facilitate a number of calculations that are imperative to our daily lives. For example, we use integers to describe the temperature above or below the freezing point, to debit or credit money, etc.
2. What are the rules for adding and subtracting integers?
There are some rules to be noted while adding two or more integers. They are as follows:
The sum of an integer and its additive inverse is always zero.
When you add two positive integers, the result will always be a positive number which will be greater than the two integers.
When you add two negative integers, the result will always be a negative number which will be smaller than the two integers you added.
By finding the difference between the absolute value of a positive integer and a negative one, you can add both of them. And the sign of the greater number out of the two will be attached to the end product.
When you add an integer with zero, you will get the same number as the answer.
There are some rules to be noted while subtracting two or more integers. They are as follows:
When you subtract any integer from zero, the answer will be either the additive inverse or the opposite of the integer. And when you subtract zero from any integer, then the result will always be the integer itself.
If you want to subtract two integers that have the same sign, you ought to perform a subtraction operation on the absolute values of those numbers.
If you want to subtract two integers that have different signs, you ought to add the absolute values.
3. State the principles that the addition and subtraction of integers on a number line is based on.
The principles upon which the addition of integers on a number line is based are as follows:
Moving towards the right or the positive side of the number line will lead to the addition of a positive integer.
Moving towards the left or the negative side of the number line will lead to the addition of a negative integer.
Any of the integers can be taken as the base point (the point from where you start to move on the number line).
The principles upon which the subtraction of integers on a number line is based are as follows:
Every subtraction fact can also be written as an addition fact.
Moving towards the left or the negative side of the number line will lead to the subtraction of a positive integer.
Moving towards the right or the positive side of the number line will lead to the subtraction of a negative integer.
4. Is multiplying rational numbers just like multiplying integers? If so, how?
To a certain extent, multiplying rational numbers is just like multiplying integers because the rules that are applicable to the latter, are also the same for the former. Rational numbers are just numbers that can be written in the fraction form of two given integers. So, if both the divisor and the dividend have the same signs, then the quotient will be positive. And if the divisor and the dividend have different signs, then the quotient will be negative.
5. Can integers be decimals?
No. Just like whole numbers, integers can neither be fractions, nor can they be decimals. All integers can be expressed as a decimal, however, most of the numbers that are decimals cannot be expressed as integers. If there are any digits after the decimal point, and all of them are zeroes, only then can the number be identified as an integer.