# Representing Rational Numbers using Scientific Notations

Rational numbers are terminating or recurring decimal numbers written in the form of fraction $\frac{a}{b}$ in which 'a' and 'b' are integers and the denominator 'b' not equal to zero. Rational numbers can also be represented using Scientific Notations.

e.g, 00000000567 can be written as 5.67 * 10-9

Addition of Rational Numbers with Scientific Notation
For adding two rational numbers represented with scientific notation, we need to have equal power value. If the power values are different then we have to equalize the power values before adding the rational numbers.

e.g, 4.1234 * 10-4 + 6.203 * 10-3 = 0.41234  * 10-3 + 6.203 * 10-3 = 6.6153 * 10-3

Subtraction of Rational Numbers with Scientific Notation
Subtracting two rational numbers represented with scientific notation is the same as adding two rational numbers represented with scientific notation. First we have to equalize the power values and then subtract the rational numbers.

e.g, 4.1234 * 10-3 - 6.203 * 10-4 = 4.1234  * 10-3 - 0.6203 * 10-3 = 3.5031 * 10-3

Multiplication of Rational Numbers with Scientific Notation
For multiplying two rational numbers represented with scientific notation, we do not have to equalize the power value. We can simply multiply the Decimals and add the power values.

e.g, 4.1234 * 10-4 x 6.203 * 10-3 = 25.5775 * 10-7

Division of Rational Numbers with Scientific Notation
For dividing two rational numbers represented with scientific notation, we do not have to equalize the power value. We can simply divide the decimals and subtract the power values.

e.g, 4.1234 * 10-4 ÷ 2.203 * 10-3 = 1.8717 * 10-1

Note:

Rational numbers are the result of two integers or they can be represented in simple fraction form with the numerator and the denominator. They are the ratio of two integers like a/b. Rational number are countable numbers. In addition and subtraction we have to equalize the denominator but in multiplication and division there is no need to equalize the denominator. Let’s take some example of addition and subtraction:-

Note:-For adding and subtracting rational expressions we have to find LCMs of polynomials.

First take an example for addition when the denominator is equal:

A/C +      B/C   =   (A + B)/c

Now take an example of polynomials expression for adding.

= 5a/4a+1 + (11a + 4)/4a +1

In the first step we will combine numerators together and in second step we will take the LCM of denominators if both denominator are same than LCM will also same.

= 5a/4a+1 + (11a + 4)/4a +1

= (5a + 11a + 4)/4a + 1, in this we have combined numerators and common denominator

= 16a + 4/4a +1

Now we will reduce the above expression to the lowest term

16a + 4/4a +1

Common factor out 4

4(4a + 1)/4a +1

Divide out the common factor of 4a + 1

= 4

Let’s talk about subtraction of rational number:-

A/C -  B/C = (A – B)/C

Subtraction of polynomials expression:-

(4a2 - 3a + 1)/a -   (3a2 - 4a + 11)/a

First step will be same as I mention above, combine numerators together and in the next step put the common denominator over step 1

(4a2 - 3a + 1- 3a2 - 4a + 11)/a

Subtract every term of second num

(a2 – 7a + 12)/ a

Then reduce the lowest term

a2- 7a +12

a now no common factor, so,

(a – 3) (a – 4).

Thus we can say, for adding and subtracting rational expressions we need to be able to find LCMs of polynomials.

## 0.0035 times 10 to the power of negative 2 in scientific notation?

Scientific Notations are generally used to represent the Numbers in mathematics that seem to be very insignificant or very big. This type of number will possess just 2 factors out which first one appears in between numbers 1 and 10. Second factor is usually written in form of an exponent of 10. For example, 1012 means exponent of 10 is 12. This is what we call as scientific representation of any number. We are calling two numbers as factors because numbers that are multiplied to each other to get a product are called as factors only.

Let us consider an example to understand the concept of converting any number into its scientific form: We have to represent 0 Point 0035 times 10 to the power of negative 2 in scientific notation. Remember first factor has to lie between 1 and 10. This makes necessary for us to keep on shifting the decimal point until we get a digit lying between 1 and 10. Here, in our example the number is 0.0035. We can see that digit that lies between 1 and 10 is lying to the right of decimal at thousands place. So, at least 3 times the decimal point has to be shifted to get desired factor. That is we have to multiply 0.0035 by 103 to get 3 in the units place to the left side of decimal. We get:
0.0035 * 103 / 103 = 3.5 * 10-3,
Also number has to be multiplied by 10-2 and then final scientific notation will be decided. On multiplying 10-2 by 3.5 * 10-3, we get following result:
3.5 * 10- (3 + 2) = 3.5 * 10-5,
So, second factor is 10-5 and scientific notation for desired number is given as:
3.5 * 10-5.