A canonical, normal, or standard form of a mathematical object is a conventional way of presenting that item as a mathematical expression in mathematics and computer science. It is frequently the one that provides the simplest representation of an object and allows it to be identified uniquely.
In this article, we will learn about the standard form, examples, and real-life applications.
Table of Contents
What is the standard form?
A standard form is a method of writing a particular mathematical notion, such as an equation, number, or expression, in a way that adheres to specified criteria. The large number which is hard to read can be converted into a readable form.
Notation of standard form:
A number is said to be in its standard form if it is “any number between 1 and 10 multiplied by some power of 10.” It is expressed as a*10n, where n is an integer and 0 < a < 10.
The usual form is frequently referred to as “scientific notation.” Alternatively, we can substitute e, or the exponential standard form, for 10. The number is also used in technical notation to describe the large value in the small.
The standard form of an equation:
The standard form of an equation is represented as
(some expression) = 0
For example, 2x + 5x^2 = 6 is not in standard form
5x^2 + 2x – 6 = 0
On the left-hand side is an expression that is also arranged in the standard form in descending exponent.
The standard form of a Fraction:
In the standard form of a fraction, we need to make the numerator and denominator co-prime numbers. It means that the common divisor is just 1 of both numerator and denominator.
If we have a fraction like 88/42 which is not in standard form because the fraction has a common divisor 2. After division by 2, we have 44/21. Now the fraction 44/21 has no common divisor. So this is in the standard form.
How Should I Write a Number in Standard Form?
The following steps are used to convert any number into standard form.
Step 1: First we note whether there is a decimal or not in the given number.
Step 2: We change the input in the decimal form if there is no decimal existing we replace the decimal with the last of the number.
Step 3: Next, count how many digits there are in the provided number after the first one and express that number as a power of 10. The exponent is negative if we move the decimal from left to right and positive if we move the decimal from right to left.
For instance, the number is 51220000000. Thus, the following is how a number is represented in standard form:
Step I: The initial number is 5
Step II: Input the decimal point after the first non-zero digit which is 5.
Step III: There are 10 digits after the number 1.
Consequently, 5.122 * 1010 is the conventional form of 512200000.
Engineering Notation:
Engineering notation is a method in which numbers are expressed as the product of 10^a where a must be divisible with 3. For example, 2.313 * 10^15, 32.121* 10^3, and 0.212 * 10^-3 are all in engineering notation where all exponents are divisible with 3.
The numbers 3.12 * 10^-2 and 9.0921 * 10^-8 are not in engineering notation because the exponents are not divisible with 3.
Examples:
We elaborate on the concept of standard form with the help of some examples.
Example 1:
Change the expression 354.32 * 10^2 into the standard form.
Solution:
Step 1: We first write the given expression
= 354.32 * 10^2
Step 2: Now move the decimal from right to left and replace it with 1st non-zero digit.
= 3.5432
Step 3: Count the number of digits’ decimal moves and put that number in the exponent of 10.
= 3.5432 * 10^2
Step 4: Now replace the given expression
= 3.5432 * 10^2 * 10^2
= 3.5432 * 10^4
Example 2:
Change the expression 0.0000000000909 into the standard form.
Solution:
Step 1: We first write the given expression
= 0.0000000000909
Step 2: Now move the decimal from left to right and replace it with 1st non-zero digit.
= 9.09
Step 3: Count the number of digits’ decimal moves and put that number in the exponent of 10.
= 9.09 * 10^10
Step 4: When we move from left to right, we put a negative sign in the exponent.
= 9.09 * 10^ (-10)
The above problem can also be converted easily with the help of a standard form calculator. You will get the step-by-step conversion of a number into standard form by using this calculator
Example 3:
Change the expression 9099.21 into the engineering Notation.
Solution:
Step 1: We first write the given expression
= 9099.21
Step 2: In engineering notation, the exponent of 10 must be divisible by 3.
= 9.00921 *10^3
Example 4:
Write in the standard form of a linear equation 2 + 5x = 9y – 6 +2x.
Solution:
We convert it into the standard form step by step.
Step 1: Make the left-hand side equal to zero.
2 + 5x – 9y + 6 – 2x = 0
Step 2: Simplify the right-hand side
8 + 3x – 9y =0
Step 3: Arrange the given equation with respect to variable x.
3x -9y + 8 = 0
Example 5:
Change the expression 0.000786* 10^2 into the standard form.
Solution:
Step 1: We first write the given expression
= 0.000786* 10^2
Step 2: Now move the decimal from right to left and replace it with 1st non-zero digit.
= 7.89
Step 3: Count the number of digits’ decimal moves and put that number in the exponent of 10.
= 7.89 * 10^-4
Step 4: Now replace the given expression
= 7.89 * 10^-4 * 10^2
= 7.89 * 10^-2
Summary:
In this post, we have learned about the standard form of numbers, equations, fractions, with examples, and applications in real life. Now you can convert any number into standard form by learning the above procedure.