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Number Properties

What is number?

We employ a mathematical object called number to count a given quantity. Numerals are symbols that represent each number. The number sixteen, for example, is expressed by the numeral 16. It is possible for a number to become either positive or negative. Marked numbers are what they’re called. If there is no sign next to the number, it is assumed to be positive. The digits to the right of 0 on the number line are positive, while those the left of 0 are negatives. A signed number’s absolute value is its magnitude. The magnitude is always positive, regardless of whether the number is negative or positive. In our daily lives, we use numbers. They are frequently referred to as numbers. We can’t count stuff, date, moment, money, or anything else without number system. These numerals are sometimes used for calculation and other times for identification. Numbers have features that allow them to conduct arithmetic on them. These figures are expressed both numerically and in words. For example, two is formed of two, thirty is written as 30, six is written as 6 and so on. To learn further, students might practice writing the numbers from 1 to 100 in words. 

There are various types of numbers that we learn in Math. Natural numbers = {1 , 2 , 3 , … }  and whole {0 , 1 , 2 , 3 , … } numbers, odd { 1, 3 , 5 , 7, 9 , … }  and even numbers {2, 4, 6, 8 , … }   Rational { p/q, q ≠ 0  and p, q ∈Z} and irrational numbers, and so on are all examples. In this article, we’ll go through all of the different varieties. Aside from that, the numbers are utilized in a variety of applications, including number series, arithmetic tables, and so on. The numerical system is as follows:

  • Specifies a set of numbers that is beneficial.
  • Providing standard representation of a number’s mathematical and algebraic structure

The fundamental features of numbers are based on simple concepts. Since no computational mathematics is necessary, you could even call it “common sense” math.

  1. Commutative,
  2. Associative, 
  3. Distributive, and
  4. Identity    Are the four basic properties of real numbers-

Commutative Property 

What does it mean to have a commutative property? 

Definition

The commutative property says that the numbers we work with is changed or replaced from their original positions without affecting the result. The characteristic applies to addition and multiplication, but just not subtracting or dividing.

 Example

, 2 + 3 = 3 + 2 = 5 

5 + 7 = 7 + 5 = 12 

But 7 – 5 = 2 

5 – 7 = -2   

Hence 7 – 5 5 – 7 

The commutative property can be applied to addition and multiplication in the instances above. Subtraction and divisions, on the other hand, are not subject to the commutative property. When we change the state of the numbers in subtraction or division, the complete problem transforms. Let’s take a quick look at the commutative property of addition and multiplication.

Addition’s Commutative Property

According to the commutative property of addition, modifying the order of the addends has no effect on the sum’s value. There are times when more than two numbers must be added. Even when more than two numbers are now being added, the commutative principle holds valid. For example, 50 + 10 + 10 + 30 equals 100, as does 30 + 10 + 10 + 50. Even when the sequence of the integers is modified, the sum equals 100 in both circumstances.

Hence, in general 

When A , and B are two numbers.

A + B = B + A   

For example, 

11 + 5 = 16 ———- (1)

5 + 11 = 16 ——— (2)

From equation (1) and (2) it is clearly proved that,   11 + 5 = 5 + 11 

Further, 

77 + 2 = 79 = 2 + 77 

3 + 5 = 5 + 3 = 8 

Multiplication’s Commutative Property

Definition

The commutative property of multiplying states that the final product is unaffected by the sequence in which the numbers are multiplied. When there are more than two numbers to be multiplied, the commutative property holds true, much as the commutative property of addition. 6 x 7 =  42, for example. Whenever we multiply 7 x 6 = 42, we have the same result. For both circumstances, the product is 42. As a result, the product is unaffected by the order or position of the numbers being multiplied.

Examples 

 2 x 3 = 6 ——— (1)

3 x 2 = 6 ——— (2) 

Hence from equation (1) and (2) show that 

2 x 3 = 3 x 2 = 6 

Further, we can see some more examples

  • 4 x 3 = 12 = 3 x 4 
  • 10 x 15 = 15 x 10 = 150 
  • 40 x 1000 = 1000 x 40 = 40000

Commutative property is not hold under subtraction and division.

For example,  

3 – 15 = – 12   ———– (1

15 – 3 = 12 ———— (2 

From equation (1) and (2) it is concluded that commutative property is not hold under subtraction. 

Also, it is not hold in division. 

3 ÷ 5 ≠ 5 ÷ 3 

100 ÷ 10 ≠ 10 ÷ 100 

Associative property 

Definition

Regardless of how you organize them, the total of two or more real numbers is always the same. When you add real numbers, the order in which they are added has no effect on the total. Whenever three or even more numbers are added (or multiplication), this characteristic indicates that the sum (or product) will be the same irrespective of how the addends are grouped (or the multiplicands). The term “associative property” derived from the term “associate,” which relates to the combination of numbers.

Use of parenthesis or brackets to sets of numbers is known as grouping.

Three or more numbers are involved in the associative property.

The numbers included in a parenthesis or bracket is treated as a single unit.

Just addition and multiplication, not subtraction or division, can be employed with the associative attribute. Let A , B and C  are numbers then according to this property 

A + (B + C) = (A + B) + C 

A X (B X C) = (A X B) X C 

EXAMPLES

2 + (3 + 5) 

= 2 + 8 

= 10 ———- (1

( 2 + 3) + 5 

= 5 + 5 

= 10 ———— (2 

From equation (1) and (2), 

2 + (3 + 5) = (2 + 3) + 5 

Further, we see one more example 

10 + (5 + 12) 

= 10 + 17 

= 27 

(10 + 5) + 12 

= 15 + 12 

= 27 

FOR MULTIPLICATION:- 

Definition 

When three or more numbers are combined, the result is the same regardless of how and why the three numbers are arranged, according to the associative feature of multiplication. The way the brackets are put in the provided multiplication phrase is referred to as grouping. To grasp the concept of the associative property of multiplication, consider the following example. The moved expression demonstrates that 6 and 5 are grouped together, but the right-hand phrase shows that 5 and 7 are grouped around each other. Nevertheless, when we multiply all of the numbers together, the end result is much the same.

Hence, in general 

A X (B X C) = (A XB) X C 

Examples

  2 x (3 x 5)

= 2 x 15 

= 30 ——- (1

 = (2 x 3) x 5 

= 6 x 5 

= 30 ———- (2

From (1) and (2) it is to be included that associative property hold under multiplication.

Let’s look at the formula in terms of numbers. Let’s multiply 2 x 3 x 4 and observe how well the formula for the associative property of multiplication is demonstrated using the steps below:

Steps for writing formula 

Step 1: Combine the numbers 2 and 3 to form (2 3) 4. When we calculate the multiplication of this expression, we obtain 6 x 4 = 24.

Step 2: Next combine the numbers 3 and 4 to become 2 x (3 x 4). When we multiply this expression by two, we get 2 x 12 and the product is 24.

Step 3: That means that the product of a multiplication expression remains the same regardless of how the numbers are grouped.

Here are some key aspects to remember about multiplication’s associative property:

When there are three or more numbers, the associative property is always true.

Subtraction and division are not affected by the associative property, which only applies to addition and multiplication.

Example 

Let’s solve one more example for the sake of better understanding

5 x (4 x 3) 

= 5 x 12

= 60 

(5 x 4) x 3 

= 20 x 3 

= 60 

Hence, it is hold for any real number.

Identity property 

Definition

An identity is a number which permits n to remain the same whether added, subtracted, multiplied, or divided with any other number (let’s name this number n). Based on the operation we’re using, the identity will be 0 or 1. The identity is 0 in addition and subtraction. The identity in multiplication and division is 1. That is, if 0 is added to or taken from n, the value of n does not change. Also, if n is split or multiplied by 1, n stays the very same.

The identical property asserts whenever an operation is used to associate an identity with a number (n), the result is n:

Examples

  • Additive Identity (0) = n +
  • Subtractive Identity (0) = n-
  • Multiplicative Identity (1) = n * 

We don’t know which identity attribute we’re using since we don’t know whichever one we’re using. That was simple! All we have to do is look at the sign for each mathematical equation. The + sign is used in the additive identity. The – sign is used in the subtractive identity. The * sign is used for multiplicative identities, while the / sign is used for divisive identities.

Identity property of addition 

Definition

The identity property of addition states that when a number n is given to zero, the output is the same as the original number, i.e. n + 0 = n.

Additive identity refers to the fact that zero can be added to any actual figure with affecting its value. Here are a few examples of addition’s uniqueness property:

Example

3 Plus 0 equals 3 (Positive Integers)

-3 + 0 equals -3 (Negative Integers)

0 + 4/5 Equals 4/5 (Fractions)

0.5 + 0 equals 0.5 (Decimals)

x + 0 equals x (Algebraic notation)

Offsetting 0 from any number represents the number itself, hence this property remains true for subtraction as well. As a result, 0 is also known as a subtractive identity.

Multiplication’s Identity Property

Definition

Whenever a number n is divided by one, the outcome is the number itself, which is known as the identity property of division.

N x 1 = N 

The multiplicative identity is one that may be repeated by any real integer with losing its value. But here are some examples of multiplication’s identity property:

Examples

  • 1 + 3 = 3 (Positive Integers)
  • -3 x 1 equals -3 (Negative Integers)
  • 1 x 4/5 Equals 4/5 (Fractions)
  • 1 x 0.5 = 1 
  • x x 1 = 1 x  x = x

Although dividing any integer by 1 equals the number itself, this condition holds true for division too though. As a result, 1 is also known as divisive identity.

Why do the Identities always have the numbers 0 and 1?

The identity always is 0 for addition and subtraction and 1 for multiplication and division, as previously stated. You might wonder why this is the case. Let’s take a closer look at how the operations function. The process of adding something to something else is known as addition. As a result, when you add 7 to 0, you get 7. Subtraction is the process of subtracting one number from another. As a result, if you remove 0 from 7, you’ll receive 7. Multiplication is the method of assigning something more than once. When you multiply two numbers, you are repeatedly adding one number by the number of times the second number is spoken. As a result, if you have 7 * 8, you are adding the number 7 x8 times. Multiplication’s identification is 1, thus if you have 7 * 1, you’re adding 7 one more time. This gives us a total of 7.

The process of dividing a number into parts is known as division. As a result, if we have 7/1, we are dividing the number 7 into two halves. Isn’t that number seven? Yes!

The math would not work if any other number were substituted for 0 or 1 in the aforementioned circumstances, which is why the identities are always 0 or 1.

Example 

3 x 1 = 3

4 /1 = 4 

5 + 0 = 5 

5 – 0 = 5 

0 + 5 = 5 

Distributive property

Definition

The distributive law of multiplication over addition and subtraction is another name for the distributive property. The investigation’s name implies that it involves dividing or dispersing something. The distribution property of addition over multiply is another name for this expression. Let’s look at several examples of the distributive property that have been solved.

Any equation containing three numbers A, B, and C of the form A (B + C) is resolved as A (B + C) = AB + AC or A (B – C) = AB – AC, according with distributive property. This indicates that operand A is shared amongst the other two operands. Multiplication distributive over simple addition is another name for this feature.

Multiplication over Addition Has a Distributive Property

Definition

When multiplying a given number by the sum of two numbers, the distribution property of multiplying over adding is used. Let us just multiply 7 by the sum of 20 + 3 as an instance. This can be expressed mathematically as 7 x (20+3). First solve the sum within the parenthesis first, and then multiply the value by 7 according to the laws of order of operations.

Example

7 x (20 + 3) = 7 x (23), which equals 161. If we use the distributive property to solve the formula, we can multiply each addend by 7 first. This is known as spreading the number 7 across the two addends, after which the products can be added.

The addition will be preceded by the multiplying of 7 x (20) and 7 x (3).

7 x (20) plus 7 x (3) equals 140 + 21 = 161

We can see that the produced outcome is the same before and after in both circumstances.

Multiplication over subtraction has a distributive feature.

The distributive property with subtraction follows the same rules as the operations above, only you’re looking for the differential rather than the total.

AX (B-C) = (A X B) – (A X C) 

5 X (4 – 1) 

= 5 X 3 

= 15 

( 5 X 4 ) – (5 X 1) 

= 20 – 5 

= 15 

EXAMPLE

 4 X (5 – 3 ) 

= 4 X 2 

= 8 ——— (1

(4 X 5) – ( 4 X 3) 

= 20 – 12 

= 8 ———- (2

Hence, it satisfies the distributive property.

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