Math Examples of Positive and Negative Numbers

On simplifying the complex fraction as
(4 / 9) / (-8) = (4 / 9) * (- 1/ 8),
(4 / 9) / (-8) = - 1 / 18,
Here the negative sign shows the opposite signs of the given Rational Numbers.

The division can be simplified as follows
(3 / 7) / (-2 / 21) = (3 / 7) * (-21 / 2),
(3 / 7) / (-2 / 21) = (3 * 21) / (-7 * 2),
(3 / 7) / (-2 / 21) = - 9 / 2,
(3 / 7) / (-2 / 21) = - 4.5,
Here the negative sign is due to the opposite signs of the Rational Numbers. The multiplication result of two rational numbers having opposite signs is always negative.

The division is as follows where the number 18 is considered to be 18 / 1.
(6 / 5) / 18 = (6 / 5) / (18 / 1),
(6 / 5) / 18 = (6 / 5) * (1 / 18),
(6 / 5) / 18 = (6 * 1) / (5 * 18),
(6 / 5) / 18 = 1 / 15.
Both the Rational Numbers have the positive sign so the answer is positive.

(40 / 20) * (120 / 20) = (40 / 20) * (20 / 120),
(40 / 20) * (120 / 20) = (40 * 20) / (20 * 120),
(40 / 20) * (120 / 20) = 1 / 3,
Both the Numbers have the positive sign. So the answer is positive.

To divide two Rational Numbers is not a difficult task. When we divide two rational numbers, first we need to find out the reciprocal of the second number and then the division would be the multiplication of the first number and that reciprocal number.
Here the rational numbers are 12 / 5 and 18 / 5. The reciprocal of 18 / 5 is 5 / 18. So the division of these rational numbers is
(12 / 5) / (18/5) = (12 / 5) * (5 / 18),
(12 / 5) / (18/5) = (12 * 5) / (5 * 18),
(12 / 5) / (18/5) = 2 / 3.

The divisions of the Numbers are as follows
1. 5 / 8 to 25 / 12,
The reciprocal of 25 / 12 is 12 / 25.
(5 / 8) / (25 / 12) = (5 / 8) * (12 / 25),
(5 / 8) / (25 / 12) = (5 * 12) / (8 * 25),
(5 / 8) / (25 / 12) = 3 / 10 = 0.3,
 7 / 12 to - 21 / 9,
(7 / 12) / (- 21 / 9) = (7 / 12) * (- 9 / 21),
(7 / 12) / (- 21 / 9) = (7 * -9) / 12 * 21,
(7 / 12) / (- 21 / 9) = -1/4,
Here the negative sign is due to negative rational number – 21 / 9.

The equation can be simplified as
(-2x + 15 y) / 5 = (-2x / 10) + (15y / 10),
(-2x + 15 y) / 5 = (- x / 5) + 1.5 y,
or
(-2x + 15 y) / 5 = (- x / 5) + (3/2) y,
This is the simplified equation. Here the division of the Rational Numbers concept is used.

The division of the numbers is as follows:
 1. 14 / 3 to (-2 / 3)
If two Rational Numbers of opposite signs are divided then the result is always negative.
When we divide two rational numbers, first is the need to find out the reciprocal of the second number and then the division would be the multiplication of the first number and that reciprocal number.
Here the rational numbers are 14 / 3 and (-2 / 3). The reciprocal of (-2 / 3) is (-3 / 2).
So the division is
(14 / 3) / (-2 / 3) = (14 / 3) * (-3 / 2),
(14 / 3) / (-3 / 2) = 14 * (-3) / 3 * 2,
(14 / 3) / (-3 / 2) = -7,
- 56 / 18 to – 7 / 9.
The reciprocal of -7 / 9 is – 9 /7.
So, 
(- 56 / 18) / (– 7 / 9) = (- 56 / 18) * (-9 / 7),
(- 56 / 18) / (– 7 / 9) = (-56) * (-9) / 18 * 7,
(- 56 / 18) / (– 7 / 9) = 8 / 2 = 4.
The solution of the multiplication of two negative numbers is always positive.
 - 25 / 14 to 5 / 7,
The reciprocal of 5 / 7 is 7 / 5. Thus the division is:
(- 25 / 14) * (5 / 7) = (- 25 / 14) * (7 / 5),
(- 25 / 14) * (5 / 7) = -5 / 2.
Here the solution is negative because one number has the negative sign in the given two numbers and the numbers are multiplied having opposite signs then the result will obvious be negative.

By the property of power of zero, we Mean that  if we have any rational number say p1/q1, such that p1 and q1 are integers and q1 <> 0, then we say that there exist another rational number 0, such that if the rational number is multiplied by the number 0, the product itself becomes zero. It does not make a difference how long or small the rational number is.

For applying the product of the Rational Numbers, the following properties holds true for the rational numbers:
1.      Closure property of multiplication
2.       Commutative property of multiplication
3.      Associative property of multiplication
4.      Multiplicative Inverse property of the rational numbers
5.      Multiplicative identity of the rational numbers.
6.       Power of zero property of multiplication.

In order to find the multiplicative inverse of the rational number 12 / 15, we will simply divide 1 by this number 12 / 15. Now we observe that this statement can be framed as the mathematical statement: 1 ÷ 12 / 15.
 Here as we change the sign of division to the sign of multiplication, we write the reciprocal of 12 / 15 as 15 / 12.
So we get the statement as 1 * (15/12) = 15/12 

According to commutative property of multiplication of the Rational Numbers, we Mean that the product of a * b = b * a.
So here we first find the value of a * b = (2 / 5) * 4 = 8 / 5. Similarly now we find the value of b * a = (4) * (2 / 5) = 8 / 5.
Thus we observe that in both the cases we get the same result. So we say that the rational numbers product is commutative.

When we have to find the product of the  two Rational Numbers, we Mean that we will multiply the numerator with the numerator and the denominator with the denominator, so here we write  (5/7) * (4 / 6) = ( 5 * 4) / ( 7 * 6) = 20 / 42.
Here in the above example, we multiply the numerator 5 and 4; similarly we multiply the denominators 4 and 6.
So we get the product 20/42.
Now we will convert it to the lowest form and get 10/21.

We say that if we have two Rational Numbers say p₁/q₁ and p₂/q₂, then we say according to the Commutative Property of Rational Numbers we Mean that the product (p₁/q₁) * (p₂/q₂) = (p₂/q₂) * (p₁/q₁). If we have p1/q1 = 3/5 and p2/q2 = 7/9, then we say (p₁ / q₁) * (p₂ / q₂) = (3 / 5) * (7 / 9) = (3 * 7)/ (5 * 9) = 21 / 45. Also we say p₂/q₂ * p₁/q₁ = (7 / 9) * (3 / 5) = (7 * 3) / (9 * 5) = 21 / 45.
Thus we observe that the product in both the cases is same so the commutative property of multiplication holds true for the rational numbers.

By multiplicative inverse of any number, we Mean that if we have any rational number say p₁/q₁, then its multiplicative inverse will be q₁/p₁. Thus we say that the multiplicative inverse of the number 5/7 is 7/5. Also we say that the product of the rational number and its multiplicative inverse is the multiplicative identity, which is always 1. Thus here we try to find the product of 5/7 and its multiplicative inverse 7/5 and check what we get as the product.
=> 5/7 * 7/5 = (5*7)/ (7 * 5),
= 35 / 35 = 1.