Division in General and With Decimals
Division - How times does one object (measure) fit into another.
When we have a whole number of objects, we may decide to form or try to form
equi-sized or equipollent groups. The division of a whole number N by
another whole number d counts the maximum number of times q that groups of size
d can be formed and yields a remained less the divisor d. What is left-over may
be zero or a group of count r < M. Division in the first instance
may be appear as a physical operation. We say d divides N when and only
when the remainder after division is zero. Here N = qd+r
The foregoing discussion of division may be repeated or recalled later when
needed. The concept of multiplication is needed next.
When fractions are known and allowed as multipliers, the division of a whole number N by
another whole number d is given by the proper or improper fraction N/d as is or
expressed in lowest terms or expressed as a mixed number. In the
latter case, the number of times that d goes into N is N/d
exactly, with no remainder. That being said if N/d is not a whole
number, it equals (qd+r) = q +r/d where r = remainder for the
number of wholes times q that d goes in N.
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Decimal Methods for Division
Motivating Questions.
-
How many times must 6 be added to itself to obtain
48?
-
How many times must 2 be added to itself to give
6?
-
How many times must 2 be added to itself to give
48 ?
Example 1. What number added to itself 3 times,
gives 963 = 9 hundred + 6 tens + 3 ones.
The answer is 3 hundred + 2 tens + 1 ones =321.
(Related question: How many times must 3 be added to
itself to give 963?)
Example 2. How many times does 7 go into 963 = 9
hundred + 6 tens + 3 ones?
The Euclidean Division Method gives the
answer.
9 = 7 x 1 + 2.
Thus 9 hundred = (7 x 1 + 2) hundred
Thus 963 = (7x1+2) hundred + 63 =7 x 1 hundred + 263
Thus 963 = 7 x (1 hundred) + 263.
--------------------------
2 is smaller than 7.
But 7 x 3 = 21 and 7 x 4 =28 > 26.
and 26 = 7 x 3 + 5. Therefore
263 = (7x3 +5) tens +3 = 7 x 3 tens +53
Thus 263 = 7 x (3 tens) + 53
-------------------------------
5 is smaller than 7. But 7 x 7 = 49 and 7 x 8 = 56 > 53.
Therefore 53 = 7 x 7 ones + 4 ones.
Thus 53 = 7 x (7 ones) + 4 leftover.
------------------------------------
4 is "too small to be divided by 7"
without the use of fractions.
Now
963 = 7 x (1 hundred) + 263
= 7 x (1 hundred) + 7 x (3 tens) + 53
= 7 x (1 hundred) + 7 x (3 tens) + 7 x (7 ones) + 4 leftover
= 7 x (1 hundred + 3 tens + 7 ones) + 4 leftover
= 7 x 137 + 4.
= 959 + 4
Our conclusion is that 7 goes into 963, 137 times completely, with
4 leftover = remainder.
In shorthand notation, we write the above calculation
more compactly and briefly as follows.
137
----
7 | 963
-700
---
263
-210
---
53
49
--
4 <--- the remainder.
Conclusion: The Euclidean Division Method justifies the
long division algorithm. (Algorithm is just another word
for method.)
4
If you know about fractions: 963 = 7 x ( 137 + --- )
7
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Example 3. How many times can 23 go into 478155?
We will apply the Euclidean Division Method
47 = 23 x 2 + 1 leftover
Thus 470000= 23 x 20000 + 10000.
Adding 8155 gives
478155 = 23 x 20000 + 18155
---------------------------
18 is smaller than 23
but 7 x 23 = 161 and 181 - 171 = 20 is
smaller than 23. Thus
181 = 23 x 7 + 20 leftover
Thus 18100 = 23 x 700 + 2000 leftover
Adding 55 gives
18155 = 23 x 700 + 2055 leftover
------------------------
20 is smaller than 23. But 8 x 23 = 160 +24 = 184
Thus 200 = 23 x 8 + 16 leftover
2000 = 23 x 80 + 160 leftover.
So adding 55 gives
2055 = 23 x 80 + 215 leftover
---------------------
21 is smaller than 23. But
9 x 23 = (10-1)x23 = 230 - 23 =207.
Thus
210 = 9 x 23 + 7 or
215 = 23 x 9 + 8 leftover
-----------------
Here the remainder 12 is less than 23.
Now we substitute:
478155 = 23 x 20000 + 18155
= 23 x 20000 + 23 x 700 + 2055
= 23 x 20000 + 23 x 700 + 23 x 80 + 215
= 23 x 20000 + 23 x 700 + 23 x 80 + 23 x 9 + 12 leftover
= 23 x (20000 + 700+ 80 + 9) + 12 leftover.
We conclude 23 goes into 478115,
20789 times completely with 12 leftover
The foregoing calculations can be written more compactly
in the left column. Read it first. Consider it as summary of the above.
_________________________________
|
20789 | 2078
--------- | --------
23 | 478155 | 23 | 478155
- 455000 as 23 x 2 = 46 --> 20000 | 46 (23 x 2 -> 2)
------ | - --
18155 |* 18
-16100 23 x 7 = 161 --> 700 | 00 (23 x 0 -> 0)
----- | ---
2055 |* 181
-1840 23 x 8 = 184 --> 80 | 161 (23 x 7 -> 7)
----- | ---
215 23 x 9 = 207 --> 9 | 205
-207 | 184 (23 x 8 -> 8)
---- | ---
8 Last Leftover or | 215
remainder is 8. | 207 (23 x 9 -> 9)
| ---
| 8 (less than 23).
| Stop.
|_________________________________
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The rightmost column shows the usual long division
algorithm, one that was taught in elementary school in the 1960s.
Variations of it may be found. You should be able to see how the steps
on the right correspond to those on the left. In the rightmost column,
you will see rows with * in them. The row in-between in them is usually
omitted to lessen the amount of writing. The row is included here to
help in the comparison of the Euclidean Division Method and that which I
met in elementary school in the 1960s. The right hand column is a more
cryptic implementation and variation of the Euclidean Division Method.
8
Conclusion 478155 = 20789 x 23 + 8 = (20789 + -- ) x 23.
23
Continuing the Division Process
The remainder 8 is smaller than 23, but 3 x 23 = 69. So
80 = 23 x 3 + 11 and therefore, dividing by 10, yields
8 = 23 x 0.3 + 1.1
--------------------
This gives
478155 = 20789 x 23 + 8
= 20789 x 23 + 23 x 0.3 + 1.1
= 20789.3 x 23 + 1.1
The remainder has become 1.1 instead of 8. It is much smaller.
We can do this again, and again. For example: 4 x 23 = 92. So
110 = 23 x 4 + 18
and therefore division by 100 gives
1.1 = 23 x .04 + .018
This yields again
478155 = 20789 x 23 + 8
= 20789.3 x 23 + 1.1
= 20789.34 x 23 + .018
The remainder has become smaller. This division process can be recorded
in the shorthand form described above.
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Number Theory
A. Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
B. Number Theory
Continued
Decimal Place Value
Place Value Reinforcement
Addition Method
Comparison Method
Subtraction Methods
Multiplication Methods
Division Methods
Long Division Continued
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Examples
Counting Whole No. Factors
Prime Factorization Aids
Square Roots & Primes
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Folders
Euclidean-Geometry/Complex
No.s
Complex
Numbers More 2
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