The properties of division of whole numbers are as follows :
Property 1:
If a and b (b not equal to zero) are whole numbers, then a ÷ b (expressed as a/b) is not necessarily a whole number.
In other words, whole numbers are not closed for division.
Verification: We know that dividing a whole number a by a nonzero whole number b means finding a whole numbers c such that a = bc.
Consider the division of 14 by 3. We find that there is no whole number which when multiplied by 3 gives us 14. So, 14 ÷ 3 is not a whole number. Similarly, 12, 5, 9, 4, 37, 6 etc. are not whole numbers.
Property 2:
If a is any whole number, then a ÷ 1 = a.
In other words, any whole number divided by 1 gives the quotient as the number itself.
Verification: We know that
(i) 1 × 5 = 5
Therefore, 5 ÷ 1 = 5
(ii) 1 × 11 = 11
Therefore, 11 ÷ 1 = 11
(iii) 1 × 29 = 29
Therefore, 29 ÷ 1 = 29
(iv) 1 × 116 = 116
Therefore, 116 ÷ 1 = 116
(v) 1 × 101 = 101
Therefore, 101 ÷ 1 = 101
(vi) 1 × 1 = 1
Therefore, 1 ÷ 1 = 1
Property 3:
If a is any whole number other than zero, then a ÷ a = 1.
In other words, any whole number (other than zero) divided by itself gives 1 as the quotient.
Verification: We have,
(i) 13 = 13 × 1
Therefore, 13 ÷ 13 = 1
(ii) 9 = 9 × 1
Therefore, 9 ÷ 9 = 1
(iii) 17 = 17 × 1
Therefore, 17 ÷ 17 = 1
(iv) 123 = 123 × 1
Therefore, 123 ÷ 123 = 1
(v) 21 = 21 × 1
Therefore, 21 ÷ 21 = 1
(vi) 1 = 1 × 1
Therefore, 1 ÷ 1 = 1
Property 4:
Zero divided by any whole number (other than zero) gives the quotient as zero. In other words, if a is a whole numbers other than zero, then 0 ÷ a = 0
Verification : We have,
(i) 0 × 7 = 0
Therefore, 0 ÷ 7 = 0
(ii) 0 × 11 = 0
Therefore, 0 ÷ 11 = 0
(iii) 0 × 17 = 0
Therefore, 0 ÷ 17 = 0
(iv) 0 × 132 = 0
Therefore, 0 ÷ 132 = 0
(v) 0 × 164 = 0
Therefore, 0 ÷ 164 = 0
Note:
In order to divide 6 by 0, we must find a whole number which when multiplied by 0 gives us 6. Clearly, no such number can be obtained. We, therefore, say that division by 0 is not defined.
Property 5:
Let a, b and c is the whole numbers and b ≠ 0, c ≠ 0. If a ÷ b = c, then b × c = a.
Verification: We have,
(i) 15 ÷ 3 = 5
Therefore, 5 × 3 = 15
(ii) 27 ÷ 9 = 3
Therefore, 9 × 3 = 27
(iii) 56 ÷ 7 = 8
Therefore, 7 × 8 = 56
(iv) 99 ÷ 11 = 9
Therefore, 11 × 9 = 99
(i) 75 ÷ 15 = 5
Therefore, 15 × 5 = 75
Property 6:
Let a, b and c be whole numbers and b ≠ 0, c ≠ 0. If b × c = a, then a ÷ c = b and a ÷ b = c.
Verification: We have,
(i) 18 = 3 × 6
Therefore, 16 ÷ 3 = 8 and 16 ÷ 8 = 3
(ii) 42 = 6 × 7
Therefore, 42 ÷ 6 = 7 and 42 ÷ 7 = 6
(iii) 72 = 8 × 9
Therefore, 72 ÷ 9 = 8 and 72 ÷ 8 = 9
(iv) 48 = 8 × 6
Therefore, 48 ÷ 6 = 8 and 48 ÷ 8 = 6
(v) 24 = 12 × 2
Therefore, 24 ÷ 2 = 12 and 24 ÷ 12 = 2
Property 7:
(Division Algorithm) If a whole number a is divided by a nonzero whole number b, then there exists whole numbers q and r such that a = bq + r, where either r = 0 or, r < b.
This can also be expressed as:
Related to this we have the following definitions:
DIVIDEND: The number which is to be divided is called dividend.In this case, a is the dividend.
DIVISOR: Divisor is the number by which the dividend is divided.Here, b is the divisor.
QUOTIENT: The number of times the divisor divides the dividend is called the quotient.
Here, q is the quotient.
REMAINDER: The number which is left over after division is called the remainder.
Here, r is the remainder. Clearly r = a – bq
Using these terms, the division algorithm can be restated as:
Dividend = Divisor × Quotient + Remainder.
Verification: Let a = 159 and b = 8.
By actual division, we find that
Quotient (q) = 19
And, Remainder (r) = 7
Clearly, 159 = 19 × 8 + 7 i.e. a = bq + r.
Representation of Whole Numbers on Number Line
Division as The Inverse of Multiplication
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