Whole Numbers and also Its Properties

WHOLE NUMBERSNow if we add zero (0) in the set of natural numbers, we obtain a brand-new set that numbers dubbed the whole numbers. Therefore the set of entirety numbers consists of zero and the collection of organic numbers. The is denoted by W. I.e., W = 0, 1, 2, 3, . . .. Smallest whole number is zero.

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Properties of entirety numbers

All the nature of number satisfied by natural numbers are also satisfied by entirety numbers. Currently we shall find out some an essential properties of number satisfied by whole numbers.

Properties that Addition

(a) Closure Property: The amount of two whole numbers is constantly a entirety number. Permit a and b be two entirety numbers, climate a + b = c is additionally a totality number.This property is referred to as the closure property of addition

Example: 1 + 5 = 6 is a whole number.

(b) Commutative Property: The sum of two whole numbers stays the exact same if the order of number is changed. Permit a and also b it is in two whole numbers, thena + b = b + aThis property is called the commutative property of addition.

(c) Associative Property: The sum of three whole numbers stays the same even if the grouping is changed. Let a, b, and c be three whole numbers, then(a + b) + c = a + (b + c)This residential property is called the associative property of addition.

(d) identification Element: If zero is added to any type of whole number, the sum stays the number itself. As we can see the 0+a=a=a+0 wherein a is a totality number.Therefore, the number zero is called the additive identity, as it does not change the value of the number when addition is perform on the number.

Properties the Subtraction

(a) Closure Property: The difference of two whole numbers will certainly not always be a totality number. Allow a and also b be two totality numbers, then a – b will be a entirety number if a > b or a = b. If a Examples17 – 5 = 12 is a totality number.5 – 17 = – 12 is no a whole number.

(b) Commutative Property: If a and b space two whole numbers, then a – b ≠ b – a. It mirrors that subtraction of two whole numbers is no commutative. Hence, commutative building does not hold great for individually of entirety numbers, i.e.,a – b ≠ b – a.Example: 3 – 4 = – 1 and 4 – 3 = 1∴ 3 – 4 ≠ 4 – 3

(c) Associative Property: If a, b, and c are whole numbers, then (a – b) – c ≠ a – (b – c). It shows that subtraction of entirety numbers is no associative. Hence, associative property does not hold great for individually of entirety numbers.Example: (40 – 25) – 10 = 15 – 10 = 540 – (25 – 10) =40- 15 = 25∴ (40 – 25) – 10 ≠ 40 – (25 – 10)

(d) residential or commercial property of Zero: If we subtract zero from any whole number, the result remains the number itself.Example: 7 – 0 = 75 – 0 = 5

Properties of Multiplication

(a) Closure Property: If a and also b room two entirety numbers, climate a × b = c will constantly be a totality number. Hence, closure residential property holds an excellent for multiplication of totality numbers.Example: 5 × 7 = 35 (a whole number)6 × 1 = 6 (a whole number)

(b) Commutative Property: If a and b space two whole numbers, climate the product that two whole numbers remains unchanged if the order the the numbers is interchanged, i.e.,a × b = b × a.Example: 6 × 5 = 5 × 630 = 30i. E., 6 rows of 5 or 5 rows of 6 offer the very same results.so, 6 × 5 = 30 = 5 × 6

(c) Associative Property: If a, b, and c space whole numbers, climate the product the three entirety numbers remains unchanged also if they are multiplied in any type of order. Hence, associative home does hold an excellent for multiplication of totality numbers, i.e.,(a × b) × c = a × (b × c)Example:(4 × 5) × 8 = 4 × (5 × 8)20 × 8 = 4 × 40160 = 160

(d) Multiplicative Identity: If any kind of whole number is multiplied by 1, the product continues to be the number itself. Allow a whole number it is in a, thena × 1 = a = 1 × a.3 × 1 = 3 = 1 × 3Examples75 × 1 = 75 = 1 × 753 × 1 = 3 = 1 × 3Hence, 1 is dubbed the multiplicative identity.

(e) Multiplicative residential property of Zero: any whole number multiply by zero gives the product together zero.If a is any kind of whole number, then 0 × a = a × 0 = 0.Example: 3 × 0 = 0 × 3 = 0

Properties the Division

(a) Closure Property: If a and b are whole numbers, climate a ÷ b is not constantly a whole number. Hence, closure home does not hold good for department of entirety numbers.Example: 7 ÷ 5 = \(\frac 7 5 \) is no a totality number.7 ÷ 7 = 1 is a entirety number.

(b) Commutative Property: If a and b are whole numbers, climate a ÷ b ≠ b ÷ a. Hence, commutative residential property does not hold good for division of entirety numbers.Example: 18 ÷ 3 = 6 is a whole number.3 ÷ 18 = \(\frac 3 18 \) = \(\frac 1 6 \) is not a entirety number.∴ 3 ÷ 18 ≠ 18 ÷ 3

(c) Associative Property: If a, b, and c space whole numbers climate (a ÷ b) ÷ c ≠ a ÷ (b ÷ c). Hence, associative building does no hold an excellent for division of whole numbers.

Example: (15 ÷ 3) ÷ 5 = 5 ÷ 5 = 115 ÷ (3 ÷ 5) = 15 ÷ 3/5 = 15 × 5/3= 25∴ (15 ÷ 3) ÷ 5 ≠ 15 ÷ (3 ÷ 5)

(d) building of Zero: If a is a entirety number climate 0 ÷ a = 0 however a ÷ 0 is undefined.Example: 6 ÷ 0 is undefined.

Note:

Product of zero and a whole number provides zero.a × 0 = 0Zero separated by any whole number provides zero.0 ÷ a = 0a ÷ 0 = undefinedAny number split by 1 is the number itself.a ÷ 1 = a

DISTRIBUTIVE PROPERTY

You space distributing something together you separate or rest it into parts.Example: Raj distributes 4 crate of sweets. Each box comprises 6 chocolates and also 10 candies. How plenty of sweets space there in these 4 boxes?∴ Chocolates in 1 box = 6Chocolates in 4 boxes = 4 × 6 = 24Candies in 1 box =10Candies in 4 boxes = 4 × 10 = 40Total variety of sweets in 4 boxes= 4 × 6 + 4 × 10 = 4 × (6 + 10)= 4 × 16 = 64

Hence, us conclude the following:(a) Multiplication distributes end addition, i.e., a(b + c) = abdominal muscle + ac, wherein a, b, c are entirety numbers.Example: 10 × (6 + 5) = 10 × 6 + 10 × 510 × 11 = 60 + 50110 = 110This residential property is referred to as the distributive building of multiplication end addition.(b) Similarly, multiplication distributes end subtraction, i.e., a × (b – c) = abdominal muscle – ac where a, b, c are totality numbers and also b > c.Example: 10 × (6 – 5) = 10 × 6 – 10 × 510 × 1 = 60 – 5010 = 10This building is dubbed the distributive residential property of multiplication end subtraction.

Example 1: determine the following by an ideal arrangement.2 × 17 × 5Solution: 2 × 17 × 5 = (2 × 5) × 17= 10 × 17 = 170

Example 2: fix the following using distributive property.97 × 101Solution: 97 × 101 = 97 × (100 + 1)= 9700 + 97 = 9797

Example 3: Tina gets 78 clues in math in the half-yearly Examination and 92 marks in the final Examination. Reena gets 92 point out in the half- yearly Examination and 78 point out in the last Examination in Mathematics. Who has obtained the higher total marks?Solution: Tina gets the following marks = 78 + 92 = 170 complete marksReena gets the following marks = 92 + 78 = 170 complete marksSo, both that them obtained equal marks.

Example 4: A fruit seller placed 12 bananas, 10 oranges, and also 6 apologize in a fruit basket. Tarun buys 3 fruit baskets for a function. What is the total number of fruits in these 3 baskets?Solution: number of bananas in 3 baskets = 12 × 3 = 36 bananasNumber that oranges in 3 baskets = 10 × 3 = 30 orangesNumber of to apologize in 3 baskets = 6×3 = 18 applesTotal variety of fruits = 36 + 30+ 18 = 84Alternative MethodTotal number of fruits in 3 baskets= 3 × < 12 +(10 + 6)>= 3 × < 12 + 16>= 3 × 28 = 84

Representation Of entirety Numbers on A Number Line

We can represent whole numbers-on a straight line. To stand for a set of entirety numbers on a number line, let’s an initial draw a straight line and also mark a point O on it. After that, note points A, B, C, D, E, F ~ above the line at equal distance, top top the best side of allude O.

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Now, OA = ab = BC = CD and also so onLet OA = 1 unitOB = OA + ab = 1 + 1 = 2 unitsOC = OB + BC = 2 + 1 = 3 unitsOD = OC + CD = 3 + 1 = 4 units and also so on.Let the point O correspond to the totality number 0, climate points A, B, C, D, E, ….. Exchange mail to the entirety numbers 1, 2, 3, 4, 5,…. In this means every totality number have the right to be stood for on the number line.