RS Aggarwal Solutions Class 7 Chapter-1 Integers (Ex 1A) Exercise 1.1

In Mathematics, Integers are the sum of positive and negative numbers. Integers do not have the fractional part. Integers are numbers that can be positive, negative, or zero but cannot be fractions. On Integers, we can perform all arithmetic operations such as addition, subtraction, multiplication, and division. Integers are 1, 2, 5, 8, -9, -12, and so on. Integers are represented by the symbol "Z." Let us now go over the definition of Integers, symbols, types, operations on Integers, rules, and properties associated with Integers, and how to represent Integers on a number line in detail with many solved Examples.

About Integers.

The term Integer is derived from the Latin word "Integer," which means "whole" or "complete." Integers are a type of number that consists of zero, positive numbers, and negative numbers.

Integer Examples: – 1, -12, 6, 15.

Symbol

Integers are denoted by the symbol 'Z.'

Z= {....-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,....}

Integer Varieties

There are three kinds of Integers:

The number 0 (0)

Integers that are positive (Natural numbers)

Integers that are negative (Additive inverse of Natural Numbers)

Zero

The Integer zero is neither positive nor negative. It is a neutral number, which means it has no sign (+ or -).

Integers That Are Positive

Positive Integers are natural numbers, also known as counting numbers. These Integers are sometimes denoted by the symbol Z+. On a number line, positive Integers are on the right side of 0.

Integers That Are Negative

Negative Integers are the inverses of natural numbers. They are denoted by the symbol Z–. On a number line, negative Integers are on the left side of 0.

Integers' Rules

The following rules apply to Integers:

• If you multiply two positive Integers the result is an Integer.

• If you multiply two negative Integers the result is an Integer.

• If you add an Integer and its inverse it is equal to zero. The product of an Integer and its reciprocal equals one.

Integer Arithmetic Operations

The following are the basic Math operations performed on Integers:

• The addition of Integers

• Integer subtraction

• Integer multiplication

• Dividing Integers

Integer Addition

While adding two Integers with the same sign, add the absolute values and write the sum with the sign provided with the numbers.

As an Example,

(+4) +  (+7) = +11

-10 = (-6) + (-4)

While adding two Integers with different signs, subtract the absolute values and record the difference with the sign of the number with the highest absolute value.

As an Example,

-2 = (-4) + (+2)

(+6) - (-4)  = +2.

Integer Subtraction

When subtracting two Integers, change the sign of the second number being subtracted and follow the addition rules.

As an Example,

(-7) – (+4) = (-7) + (-4) = -11

(+8) – (+3) = (+8) + (-3) = +5.

Integer Multiplication

The rule for multiplying two Integer numbers is straightforward.

If both Integers have the same sign, the outcome is positive.

If the Integers have opposite signs, the result is negative.

As an Example,

(+2) x (+3) = +6

(-4) x (+3) = – 12

Division of Integers

The rule for dividing Integers is similar to the rule for multiplying Integers.

• If both Integers have the same sign, the outcome is positive.

• If the Integers have opposite signs, the result is negative.

Integers' Properties

The following are the most important properties of Integers:

• Closure Property

• Associative Property

• Commutative Property

• Distributive Property

• Additive Inverse Property

• Multiplicative Inverse Property

• Identity Property

Closure Property

When two Integers are added or multiplied together, the result is only an Integer, according to the closure property of Integers. Let  a and b are Integers, then:

a + b = Integer

a x b = Integer

Commutative Property

If a and b are two Integers, then the commutative property of Integers states that:

a + b = b + a

a x b = b x a

Associative Property

If a, b, and c are Integers, then the associative property states that:

a+(b+c) = (a+b)+c

(ax(bxc) = (axb)xc

Distributive Property

If a, b, and c are Integers, then the distributive property of Integers states that:

a x (b + c) = a x b + a x c

Inverse Additive Property

If an is an Integer, then the additive inverse property of Integers states that

(-a) + a = 0

As a result, -a is the additive inverse of the Integer a.

Multiplicative Inverse Property

If an is an Integer, then the multiplicative inverse property of Integers says that

1 = an x (1/a)

As a result, 1/a is the multiplicative inverse of the Integer a.

Identity Properties of Integers

Integers' identity elements are:

a + 0 = a x 1 = a

Integer Applications

Integers are more than just numbers on paper; they have a wide range of real-world applications. Positive and negative numbers produce different effects in real life. They are primarily used to represent two opposing situations.

When the temperature is above zero, for Example, positive numbers are used to denote the temperature, whereas negative numbers denote the temperature below zero. They allow one to compare and measure two things, such as how big or small, how many or how few things there are, and thus can quantify things.