What is the standard form? Explained with examples and solutions

Standard means a level of quality. The term standard is very important in practical life. Similarly, the standard form is very important in physics and mathematics to make the numbers and equitation easy to read and understand.

In this article, we will discuss the standard form of different types of data and also deal with some examples for better understanding.

What is the standard form of numbers?

A number can be written in standard form to make it easier to read. It is frequently used for extremely high or extremely low figures. Science and engineering are the main fields that employ standard form, which is similar to scientific notation.

When a number is expressed as a decimal number times a power of 10, it is written in standard form.

Example:

 if we have a number like 19870000000000. It is difficult to read with the eye. So we let the change in the standard form (scientific notation).

Step 1: Replace the decimal after the 1^st non-zero digit. We have

1.987

Step 2: Count the number of digits we jump the decimal. In this case, we move 13 digits.  Write this number in the power of 10.

Step 3: If we jump the decimal from left to right the power will be negative and if we jump the decimal from right to left the power will be positive. In this case, we jump left to right so the power will be negative.

1.987 * 10^-13

What is Engineering Notation of numbers?

The exponent of ten must be divisible by three in engineering notation, also known as engineering form, which is a kind of scientific notation. 

The number 38700 has the engineering notation as 38.700 * 10³. The exponent must be divisible by 3.

What is the standard form of Fractions?

When the numerator and denominator of a fraction are both co-prime integers, the fraction is said to be in standard form. When there is just one common factor or divisor between two integers, that pair is said to be co-prime. Examples are 5 and 3, 6 and 7, 13 and 15, etc. Two prime numbers are always co-prime by definition.

For example, the fraction of the form 6/7 is a standard form because the numerator is 6 and the denominator is 7 do have only one common divisor. The fraction 8/16 is not in standard form because the denominator and numerator have common divisors other than 1 i.e. 2, 4, and 8.

Example:

Covert the fraction 99/30 into the standard form.

Solution:

Step 1: Find all the common divisors of 99 and 33 which are nominator and denominator respectively. In this case, 99 and 30 have common divisors: 1 and 3.

Step 2: Divide the numerator and denominator by 3 we have

= (99)/3/ (30)/3

= 33/10 

The standard form of the fraction 99/30 is 33/10.

The standard form of a linear equation:

The linear equation is one form of equation in which the power (exponent) of the variable is 1.

The standard form is Ax + By = C with two variables x and y where A, B, and C are known as the real numbers.

Example:

Change the standard form to the equation 6x = 9y -99

Solution:

Step 1: We will take the variables on the left side and the constant term on the right side. 

6x – 9y = -99

Step 2: If there is a common divisor then we divide that into both sides. In this case, 3 is the common divisor. So we have the standard equation as

2x – 3y = -33 

How to calculate the problems of standard form?

The examples or problems of the standard from can be done easily either with the help of a standard form calculator or by using a manual method. Here are a few examples to solve the standard form problems manually. 

Example 1: For larger number

Find the standard form of 345.82 * 10⁺². Also, write their cross-ponding e-notation and engineering notation.

Solution:   

Step 1: We have to write them in the form of a real number.

= 345.82 *10⁺²

= 34582

Step 2: Now we will change to the standard form by replacing the decimal after 1^st non-zero number.

= 3.4582

Also, count the number of digits we move the decimal. Here we move 4 digits from right to left so the power of ten must be positive. Hence finally the standard form is

= 3.4582 * 10⁺⁴

Step 3: We will also change it into an e-notation by replacing the 10 with e.

= 3.4582 * e⁺⁴

Step 4: We will also change it into the engineering notation

= 345.782 * 10³

Example 2: For smaller number

Convert 0.0000000780 in standard form step-by-step.

Solution:

Step 1: We have to jump the decimal point from its original position to 1^st non-zero digit.

7.80

Step 2: Now count the number of digits we have jumped left to right. In this case, we have to move 8 digits. We write this number in the power of 10 while writing our decimal number in standard form.

Step 3: If we move the decimal from left to right the power of 10 will be negative. If we made the jumps from right to left, then the power of 10 will be positive.

Step 4: So, the standard form of 0.0000000780 is 7.8 * 10^-8.

0.0000000780 = 7.8 * 10^-8

Example 3:

Convert the equation 2x² = 6x – 8 in the standard form. 

Solution:

Step 1: We can change any equation with one variable in the standard form by 

(Expression) = 0

Step 2: In this case, we bring all terms right to left.

2x² – 6x + 8 = 0

Step 3: If it is common in the expression we take and change it in the simplest form

2 (x² – 3x + 4) = 0  

Divided by 2 into both sides we have

x² – 3x + 4 = 0  

Wrap up

In this post, we have learned the standard form of numbers, fractions, and equations with the help of examples. Now you will be solving any problem regarding the standard form easily after reading the whole post.