The Modulative property Is the one that allows to carry out operations with the numbers without altering the result of the equality. This is particularly useful later in algebra, since multiplying or adding up by factors that do not alter the result, allows the simplification of some equations.
For addition and subtraction, adding zero does not alter the result. In the case of multiplication and division, multiplying or dividing by one does not alter the result either.
The zero factors for addition and one for multiplication are modular for such operations. Arithmetic operations possess several properties in addition to modulative property, which contribute to the solution of mathematical problems.
Arithmetic operations and modulative property
Arithmetic operations are addition, subtraction multiplication and division. Let's work with the set of natural numbers.
Sum
The property called neutral element allows us to add a summing without altering the result. This tells us that zero is the neutral element of the sum.
As such, it is said to be the modulus of addition and hence the name of modulative property.
For example:
(3 + 5) + 9 + 4 + 0 = 21
4 + 5 + 9 + 3 + 0 = 21
2 + 3 + 0 = 5
1000 + 8 + 0 = 1008
500 + 0 = 500
233 + 1 + 0 = 234
25000 + 0 = 25000
1623 + 2 + 0 = 1625
400 + 0 = 400
869 + 3 + 1 + 0 = 873
78 + 0 = 78
542 + 0 = 542
36750 + 0 = 36750
789 + 0 = 789
560 + 3 + 0 = 563
1500000 + 0 = 1500000
7500 + 0 = 7500
658 + 0 = 658
345 + 0 = 345
13562000 + 0 = 13562000
500000 + 0 = 500000
322 + 0 = 322
14600 + 0 = 14600
900000 + 0 = 900000
The modulative property is also true for integers:
(-3) +4 + (-5) = (-3) +4 + (-5) +0
(-33) + (- 1) = (-33) + (- 1) +0
-1 + 35 = -1 + 35 + 0
260000 + (- 12) = 260000 + (- 12) +0
(-500) +32 + (- 1) = (-500) +32 + (-1) +0
1750000 + (- 250) = 1750000 + (- 250) +0
350000 + (- 580) + (- 2) = 350000 + (- 580) + (- 2) +0
(-78) + (- 56809) = (-78) + (- 56809) +0
8 + 5 + (- 58) = 8 + 5 + (- 58) +0
689 + 854 + (- 78900) = 689 + 854 + (- 78900) +0
1 + 2 + (- 6) + 7 = 1 + 2 + (- 6) + 7 + 0
And, similarly, for rational numbers:
2/5 + 3/4 = 2/5 + 3/4 + 0
5/8 + 4/7 = 5/8 + 4/7 + 0
½ + 1/4 + 2/5 = ½ + 1/4 + 2/5 + 0
1/3 + 1/2 = 1/3 + 1/2 + 0
7/8 + 1 = 7/8 + 1 + 0
3/8 + 5/8 = 3/8 + 5/8 + 0
7/9 + 2/5 + 1/2 = 7/9 + 2/5 + 1/2 + 0
3/7 + 12/133 = 3/7 + 12/133 + 0
6 + 8 + 2 + 3 = 6/8 + 2 + 3 + 0
233/135 + 85/9 = 233/135 + 85/9 + 0
9/8 + 1/3 + 7/2 = 9/8 + 1/3 + 9/8 + 0
1236/122 + 45/89 = 1236/122 + 45/89 + 0
24362/745 + 12000 = 24635/745 + 12000 + 0
Also for the irrational:
E + √2 = e + √2 + 0
√78 + 1 = √78 + 1 + 0
√9 + √7 + √3 = √9 + √7 + √3 + 0
√7120 + e = √7120 + e + 0
√6 + √200 = √6 + √200 + 0
√56 + 1/4 = √56 + 1/4 + 0
√8 + √35 + √7 = √8 + √35 + √7 + 0
√742 + √3 + 800 = √742 + √3 + 800 + 0
V18 / 4 + √7 / 6 = √18 / 4 + √7 / 6 + 0
√3200 + √3 + √8 + √35 = √3200 + √3 + √8 + √35 + 0
√12 + e + √5 = √12 + e + √5 + 0
√30 / 12 + e / 2 = √30 / 12 + e / 2
√2500 + √365000 = √2500 + √365000 + 0
√170 + √13 + and + √79 = √170 + √13 + and + √79 + 0
And so for all real.
2.15 + 3 = 2.15 + 3 + 0
144,12 + 19 + √3 = 144,12 + 19 + √3 + 0
788500 + 13.52 + 18.70 + 1/4 = 788500 + 13.52 + 18.70 + 1/4 + 0
3.14 + 200 + 1 = 3.14 + 200 + 1 + 0
2.4 + 1.2 + 300 = 2.4 + 1.2 + 300 + 0
√35 + 1/4 = √35 + 1/4 + 0
And + 1 = e + 1 + 0
7.32 + 12 + 1/2 = 7.32 + 12 + 1/2 + 0
200 + 500 + 25.12 = 200 + 500 + 25.12 + 0
1000000 + 540,32 + 1/3 = 1000000 + 540,32 + 1/3 +0
400 + 325.48 + 1.5 = 400 + 325 + 1.5 + 0
1200 + 3.5 = 1200 + 3.5 + 0
Subtraction
Applying the modulative property, as in addition, the zero does not alter the result of the subtraction:
4-3 = 4-3-0
8-0-5 = 8-5-0
800-1 = 800-1-0
1500-250-9 = 1500-250-9-0
It is true for integers:
-4-7 = -4-7-0
78-1 = 78-1-0
4500000-650000 = 4500000-650000-0
-45-60-6 = -45-60-6-0
-760-500 = -760-500-0
4750-877 = 4750-877-0
-356-200-4 = 356-200-4-0
45-40 = 45-40-0
58-879 = 58-879-0
360-60 = 360-60-0
1250000-1 = 1250000-1-0
3-2-98 = 3-2-98-0
10000-1000 = 10000-1000-0
745-232 = 745-232-0
3800-850-47-0
For the rational:
3 / 4-2 / 4 = 3 / 4-2 / 4-0
120 / 89-1 / 2 = 120 / 89-1 / 2-0
1 / 32-1 / 7-1 / 2 = 1 / 32-1 / 7-1 / 2-0
20 / 87-5 / 8-0
132 / 36-1 / 4-1 / 8 = 132 / 36-1 / 4-1 / 8
2 / 3-5 / 8-0 = 2 / 3-5 / 8-0
1 / 56-1 / 7-1 / 3 = 1 / 56-1 / 7-1 / 3-0
25 / 8-45 / 89 = 25 / 8-45 / 89 -0
3 / 4-5 / 8-6 / 74 = 3 / 4-5 / 8-6 / 74-0
5 / 8-1 / 8-2 / 3 = 5 / 8-1 / 8-2 / 3-0
1 / 120-1 / 200 = 1 / 120-1 / 200-0
1 / 5000-9 / 600-1 / 2 = 1 / 5000-9 / 600-1 / 2-0
3 / 7-3 / 4-0 = 3 / 7-3 / 4-0
Also for the irrational:
Π-1 = Π-1-0
E-√2 = e-√2-0
√3-1 = √-1-0
√250-√9-√3 = √250-√9-√3-0
√85-√32 = √85-√32-0
√5-√92-√2500 = √5-√92-√2500
√180-12 = √180-12-0
√2-√3-√5-√120 = √2-√3-√5-120
15-√7-√32 = 15-√7-√32-0
V2 / √5-√2-1 = √2 / √5-√2-1-0
√18-3-√8-√52 = √18-3-√8-√52-0
√7-√12-√5-0 = √7-√12-√5-0
√5-e / 2 = √5-e / 2-0
√15-1 = √15-1-0
√2-√14-e = √2-√14-e-0
And, in general, for the real ones:
Π -e = π-e-0
-12-1.5 = -12-1.5-0
100000-1 / 3-14.50 = 100000-1 / 3-14.50-0
300-25-1.3 = 300-25-1.3-0
4.5-2 = 4.5-2-0
-145-20 = -145-20-0
3,16-10-12 = 3,16-10-12-0
Π-3 = π-3-0
Π / 2 π / 4 = π / 2 π / 4-0
325.19-80 = 329.19-80-0
-54.32-10-78 = -54.32-10-78-0
-10000-120 = -10000-120-0
-58.4-6.52-1 = -58.4-6.52-1-0
-312,14-√2 = -312,14-√2-0
Multiplication
This mathematical operation also has its neutral element or modulative property:
3x7x1 = 3 × 7
(5 × 4) x3 = (5 × 4) x3 × 1
Which is the number 1, since it does not alter the result of the multiplication.
This is also true for integers:
2 × 3 = -2x3 × 1
14000 × 2 = 14000x2x1
256x12x33 = 256x14x33x1
1450x4x65 = 1450x4x65x1
12 × 3 = 12 × 3 × 1
500 × 2 = 500x2x1
652x65x32 = 652x65x32x1
100x2x32 = 100x2x32x1
10000 × 2 = 10000x2x1
4x5x3200 = 4x5x3200x1
50000x3x14 = 50000x3x14x1
25 × 2 = 25 × 2 × 1
250 × 36 = 250x36x1
1500000 × 2 = 1500000x2x1
478 × 5 = 478x5x1
For the rational:
(2/3) x1 = 2/3
(1/4) x (2/3) = (1/4) x (2/3) x1
(3/8) x (5/8) = (3/8) x (5/8) x1
(12/89) x (1/2) = (12/89) x (1/2) x1
(3/8) x (7/8) x (6/7) = (3/8) x (7/8) x (6/7) x 1
(1/2) x (5/8) = (1/2) x (5/8) x 1
1x (15/8) = 15/8
(4/96) x (1/5) x (1/7) = (4/96) x (1/5) x (1/7) x1
(1/8) x (1/79) = (1/8) x (1/79) x 1
(200/560) x (2/3) = (200/560) x 1
(9/8) x (5/6) = (9/8) x (5/6) x 1
For the irrational:
And x 1 = e
√2 x √6 = √2 x √6 x1
√500 x 1 = √500
√12 x √32 x √3 = V√12 x √32 x √3 x 1
√8 x 1/2 = √8 x 1/2 x1
√320 x √5 x √9 x √23 = √320 x √5 √9 x √23 x1
√2 x 5/8 = √2 x5 / 8 x1
√32 x √5 / 2 = √32 + √5 / 2 x1
And x √2 = e x √2 x 1
(Π / 2) x (3/4) = (π / 2) x (34) x 1
Π x √3 = π x √3 x 1
And finally for the real ones:
2.718 × 1 = 2.718
-325 x (-2) = -325 x (-2) x1
10000 x (25.21) = 10000 x (25.21) x 1
-2012 x (-45.52) = -2012 x (-45.52) x 1
-13.50 x (-π / 2) = 13.50 x (-π / 2) x 1
-π x √250 = -π x √250 x 1
-√250x (1/3) x (190) = -√250x (1/3) x (190) x 1
- (√3 / 2) x (√7) = - (√3 / 2) x (√7) x 1
-12.50 x (400.53) = 12.50 x (400.53) x 1
1x (-5638.12) = -5638.12
210.69 x 15.10 = 210.69 x 15.10 x 1
Division
The neutral element of the division is as in multiplication, the number 1. A given quantity divided by 1 will give the same result:
34 ÷ 1 = 34
7 ÷ 1 = 7
200000 ÷ 1 = 200000
Or what is the same:
200000/1 = 200000
This is true for each integer:
8/1 = 8
250/1 = 250
1000000/1 = 1000000
36/1 = 36
50000/1 = 50000
1/1 = 1
360/1 = 360
24/1 = 24
2500000/1 = 250000
365/1 = 365
And also for each rational:
(3/4) ÷ 1 = 3/4
(3/8) ÷ 1 = 3/8
(1/2) ÷ 1 = 1/2
(47/12) ÷ 1 = 47/12
(5/4) ÷ 1 = 5/4
(700/12) ÷ 1 = 700/12
(1/4) ÷ 1 = 1/4
(7/8) ÷ 1 = 7/8
For each irrational number:
Π / 1 = π
(Π / 2) / 1 = π / 2
(√3 / 2) / 1 = √3 / 2
√120 / 1 = √120
√8500 / 1 = √8500
√ 12/1 = √ 12
(Π / 4) / 1 = π / 4
And, in general, for every real number:
3.14159 / 1 = 3.14159
-18/1 = -18
16.32 ÷ 1 = 16.32
-185,000,23 ÷ 1 = -185,000,23
-10000.40 ÷ 1 = -10000.40
156.30 ÷ 1 = 156.30
900000, 10 ÷ 1 = 900000.10
1.325 ÷ 1 = 1.325
Modulative property is essential in algebraic operations, since the artifice of multiplying or dividing by an algebraic element whose value is 1 does not alter the equation.
However, if you can simplify the operations with the variables so as to obtain a simpler expression and achieve solving equations in an easier way.
In general, all mathematical properties are necessary for the study and development of scientific hypotheses and theories.
Our world is full of phenomena that are constantly observed and studied by scientists.
These phenomena are expressed with mathematical models to facilitate their analysis and subsequent understanding.
In this way you can predict future behaviors, among other things, which brings great benefits that improve the way of life of people.
References
- Definition of natural numbers. Recovered from: definicion.de.
- Division of integers. Recovered from: vitutor.com.
- Example of modulative property. Recovered from: ejemplode.com.
- The natural numbers. Recovered from: gcfaprendelibre.org.
- Mathematics 6. Retrieved from: colombiaaprende.edu.co.
- Math properties. Retrieved from: wikis.engrade.com.
- Properties of multiplication: associative, commutative and distributive. Retrieved from: portaleducativo.net.
- Properties of the sum. Retrieved from: gcfacprendelibre.org.