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Are whole numbers closed under subtraction?

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Numerals are the mathematical figures used in financial, professional as well as a social fields in the social world. The digits and place value in the number and the base of the number system determine the value of a number. Numbers are used in various mathematical operations as summation, subtraction, multiplication, division, percentage, etc which are used in our daily businesses and trading activities.

Numbers

Numbers are the mathematical figures or values applicable for counting, measuring, and other arithmetic calculations. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc. The number system is a standardized method of expressing numbers into different forms being figures as well as words. It includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form on the basis of the number system used.  

The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

The elementary system to express numbers is called a number system. It is the standardized method for the representation of numerals in which numbers are represented in arithmetic and algebraic structure.

Types Of Numbers

There are different types of numbers categorized into sets by the number system. The types are described below:

  1. Natural numbers: Natural numbers counts from 1 to infinity. They are the positive counting numbers that are represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = {1, 2, 3, 4, 5, 6, 7, …}
  2. Whole numbers: Whole numbers count from zero to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W = {0, 1, 2, 3, 4, 5, …}
  3. Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals and is represented by ‘Q’.
  4. Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. They are represented by ‘P’.
  5. Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is represented by ‘Z’. Example: Z = {.., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}
  6. Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5, 0.567, etc.
  7. Real number: The set of numbers that do not include any imaginary value and are constituent of all the positive integers, negative integers, fractions, and decimal values are real numbers. It is generally denoted by ‘R’.
  8. Complex numbers: They are a set of numbers that include imaginary numbers are complex numbers. It can be expressed as a + bi where “a” and “b” are real numbers. It is denoted by ‘C’. 

Are whole numbers closed under subtraction?

Answer: 

The subset of the number system that consists of all positive integers including 0 is defined as a whole number. The whole number counts from zero to positive infinity. These numbers are mostly used for counting, measurement of fundamental quantities, and daily calculations.

Whole numbers are the only constituents of natural numbers including zero. The subset is given by {0, 1, 2, 3, 4, 5, …}, the set does not include fractions, decimals, and negative integers. The set of whole numbers is represented by ‘W’. The set can be represented as W = {0, 1, 2, 3, 4, 5, …}

Hence, the standard form of whole number is 0, 1, 2, 3, 4, 90,…

Examples of Whole Numbers

Positive integers are also known as counting numbers including zero are part of whole numbers, such as 0, 1, 2, 3, 4, 5, etc, excluding negative integers, fractions, and decimals.12, 120, 1200, etc all are examples of whole numbers.

Whole numbers are not closed under subtraction operation because when assume any two numbers, and if  subtracted one number from the other number. it is not compulsory  that the result is a whole number. Recall the definition of the whole number set W, take any two whole numbers a, b ∈  W and then add, subtract, multiply them to check whether the result is also a whole number or not.

Sample Problems

Question 1: Are whole numbers closed under subtraction in closure property?

Answer:

In this property of whole numbers, add or multiply any two whole numbers that will also result in a whole number, but when two whole number are subtracted that does not result into whole number. Example, 

  1. 2 + 5 = 7 and 80 + 40 = 120 for addition
  2. 6 × 5 = 30, and 5 × 6 = 30 for multiplication

Now for subtraction:

1 – 0 = 0  i.e a whole number, and 0 – 1 = -1 i.e not a whole number

So in this property, whole number are not closed under subtraction .

Question 2: Are whole numbers closed under subtraction in commutative property?

Answer:

It states that the operation of addition or multiplication on the number does not matter what is the order, it will give the same result even after swapping or reversing their position or, the placement of adding or multiplying numbers can be changed but it will give the same results. This property is valid for addition and multiplication not for subtraction.

  1. x + y = y + x
  2. x × y = y × x

Example:

  • If add 6 in 2 or add 2 in 6 results will be same,

7 + 2 = 9 = 2 + 7                   

  • Multiply both the real numbers,

6 × 7 = 42 = 7 × 6

  • Now for subtraction:

x – y = y – x 

7 – 2 = 5 i.e a whole number, or 2 – 5 = -7 is not a whole number. 

So once again it proved through commutative property that whole number are not closed under subtraction .

Question 3: Are whole numbers closed under subtraction in associative property?

Answer:

This property states that when three or more numbers are added (or multiplied) or the sum (or product) is the same regardless of the grouping of the addends (or multiplicands). The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property. That is, rearranging the numbers in such a manner that will not change their value.

  1. (x + y) + z = x + (y + z)
  2. (x × y) × z = x × (y × z)

Example: 

  • (8 + 5) + 6 = 8 + (5 + 6)

19 = 19  

  • (8 × 5) × 6 = 8 × (5 × 6)

240 = 240

As it can be seen even after changing their order, it gives the same result in both the operations adding as well as multiplication.

Now for subtraction:

Example: 

(x – y) – z = x – (y-z)

(6 – 5 ) – 2 =  6 – ( 5 – 2 )

– 1 = 3 

So it’s proved that whole number is not closed under subtraction in associative property

Question 4: Are whole numbers closed under subtraction in distributive property?

Answer:

This property helps simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.      

x × (y + z) = x × y + x × z                                     

x × (y – z) = x × y – x × z  

Example:

Simplify 20 × (5 + 6)                                                              

= 20 × 5 + 20 × 6

= 100 + 120

= 220

Now for subtraction:

x × (y – z) = x × y – x × z  

20 × ( 6 – 5 ) =  20 × 6 – 20 × 5

20 × 1 = 120 – 100

20 = 20

It applies same for the subtraction also. Hence, from this property it shows that whole number is closed under subtraction in this property.



Last Updated : 03 Jan, 2024
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