Standard Form

Dhristi JEE 2022-24

What is Standard Form?

A standard form is a method of writing mathematical concepts like an equation, expressions, or numbers in standard form.


Example:

2.5 billion years is written as 2,500,000,000 years.


As you can see, reading or writing a large number like 2.5 billion is not only difficult but also time-consuming and there are chances of writing more or less 0’s while writing the larger numbers in a general form.


Hence, to write large or small numbers precisely, we use the standard form.


Standard Form Formula

The method of writing mathematical concepts like an equation, expressions, or numbers in standard form follows certain rules or formulas.


The standard form formulas vary for different Mathematics concepts.


For example, the standard form of 2,500,000,000 is 2.5 x 10⁹


What about the fractions 12/24 and 15/7? Are these standard forms of fractions?


In the case of fractions, we need to ensure that in the standard form of decimals, both numerator and denominators must be coprime.


It implies that they should not have common factors other than 1.


So, the standard form of fraction 12/24 is ½


The fraction 15/7 is already written in a standard form, as both 15 and 7 are co-prime.


Polynomial

A polynomial is defined as a mathematical expression that includes variables, coefficients, and operations of addition, subtraction, multiplication, and non-negative integers.


The standard of a polynomial of degree n is anxn………+ a1x+a0

Examples: x + 2, 3y2 - 2y + 5, -6, 1/2y2 - 2/3 y + 3/4


What does Standard form of a Polynomial Means

The standard form of a polynomial is a method of writing polynomials with the exponents in decreasing order.


Polynomials are expressed in standard form to make complex calculation easier.


A polynomial is considered to be written in standard form, if it is expressed in such a way that the term with the highest degree is written first, followed by the term which has the next highest degree, and so on.


Example : 14y4 - 5y3 - 11y2 - 11y + 8


You can find that in the above-given standard form of a polynomial, the exponents are placed in decreasing order.


How to Write Polynomials in Standard Form?

The like terms in the standard form of a polynomial are grouped, added, subtracted, and rearranged with the exponents of terms in decreasing order.


Following are the steps to write a polynomial in standard form:

  1. Write the terms.

  2. Arrange all the like terms.

  3. Find the exponent.

  4. Write the term with the highest exponent first.

  5. Write the remaining terms with lower exponents in decreasing order

  6. Write the constant term ( a number without variable) in the end.

Example: 8y⁴ + 11y³ - 6y⁴ - 8y²

 = 8y⁴ - 6y⁴ + 11y³ + 8y²

= 2y³ + 11y³ + 8y²

In the above example, the polynomial with the highest degree is 4 and that became the exponent with the first term.


Standard Form of an Equation

The standard form of an equation is written as  Ax + By = C, where A, B, and C are integers.


This form of the equation is particularly useful for determining both the x - and y-intercepts. We can find the x-intercept of an equation by substituting 0 for y and solving for the x. Similarly, we can find the y-intercept of an equation by substituting 0 for x and solving for y.


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Example: Put y² = 6 in standard form.


Answer: The equation y² = 6 in standard form can be written as” y² - 6 = 0


Standard Form of a Decimal Number

The standard form of a decimal number in Britain is known as Scientific notation, where the  number is written in the following way:


4527.7 =  4.5277 x 10³

A Number In Scientific Notation

In this example, 4527.7 can be written as 4.5277 × 10³ in scientific notation

Because, 4527.7 =  4.5277 × 1000 = 4.5277 × 10³

The standard form of decimal numbers in the United States, and in other countries using the US conventions is written in expanded form.

Example: 4.327 in expanded form can be written as:

4.327=4×1+3×110+2×1100+7×11000

4.327=4×1+3×110+2×1100+7×11000

4.327 = 4 + 0.3 + 0.03 + 0.007

Therefore,

The expanded form of 4.327 is 4 + 0.3 + 0.03 + 0.007.


Standard Form Examples

Here are some standard form examples for different mathematical concepts.

1. Write the polynomial y² - 10y + 16 - y² + y⁵ - 3y⁴ + 3y²

Solution:

To write the given polynomial equation in standard form, two rules should be followed.

  1. Write the terms in decreasing order of their powers.

  2. All the terms should be different.

Let us first arrange the given term in decreasing order:

y² - 10y + 16 - y² + y⁵ - 3y⁴ + 3y² = y⁵ - 3y⁴ + y² - y² + 3y² -10y + 16

After adding like terms, we get

y⁵ - 3y⁴  + 3y² -10y + 16

Hence, the standard form is y⁵ - 3y⁴  + 3y² - 10y + 16


2. How to write the distance between the Sun and Mars in standard form?

Solution:


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As we know, the distance between the Sun and Mars is 228,000,000 km or 141,700,000 miles.


Hence, the distance between the Sun and Mars in standard form can be written as 2.28 × 108 km or 1.417 × 108 miles.


3. Write 3253 in standard form.

Solution:

3253 can be written as 3.253 × 1000


Therefore, the standard form of 3253 is 3.253 × 10³


Standard Form

Mathematics is a subject involving numbers and formulas which scares many students and hence, they tend to feel anxious before every maths exam especially board examinations. 


Here are a few tips, every student must follow two months before their maths board examination or at the times when they are preparing for any other tests to ace their learnings. 


Let's discuss some of the DOs and DON'Ts for the same. 


Do’s of Preparation

  • Preparing a Separate Notebook

A separate notebook must be prepared by every student for jotting down theorems, formulas, rules and methods. Initially, it might sound a bit time consuming but it saves time at the eleventh hour as you do not have to go through 10 notebooks just for one formula. 

  • Revising Formulas Every Day

Revising formulas every day is easier than thinking about mugging up before an exam as you have a lot to do then, which makes your brain feel pressurised and hence, you start feeling nervous before going for your exam. Revising every day on the other hand enhances your confidence as you have been familiar with the formulas and decreases anxiety.

  • Creating a Schedule/timetable 

Managing your time is very important and it becomes a priority when exams are near. Creating a schedule enables the management of time and energy. Your schedule must be created wisely, it should neither be too hectic for you to follow nor leave you too idle or lazy. 

  • Try to Understand, not Just Cram

One needs to understand the concept as it stays in the mind for a longer duration than just cramming without knowing the logic behind it.

  •  Solving Sample Test Papers

Solving at least 2 sample tesṭ papers per week shall be considered a good habit as it assists you to go through a variety of questions, both easy and difficult. Attempting sample tests in one go with a time limit of 2.5 hours, especially between 10.30 am and 1 pm (timings of examination) prepares your body and mind for the final exam.

  • Follow the 90/10 Rule 

Students shall note that the major part of your preparation can be completed if you are thorough with the NCERT textbook because it consists of 90% stuff. The rest 10% can be covered when you try your hands on sample papers and additional study material. 


Don'ts of Preparation

  • Do not Study New Topics one Month Prior

Acquiring new topics often the helps in the preparation of difficult questions but it may confuse your mind and hence make you feel even more nervous. It’s better to practice topics that you are already familiar with than studying the new ones.

  • Do not Study one Night Before Exam

Studying everything one night before an exam won’t help you. Rather it would confuse you even more. Since it is natural to be stressed at those times, you might not be able to get the right answers and this might discourage you. This anxiousness would make you nervous and it might become more difficult for you to perform the other day. 

  • Practice more, Better Score

Practical subjects are all about practice. It means that the more you practice, the more you get. When you try your hands on different kinds of questions, you gain a better understanding of how questions can be asked and this way, you can easily prepare your mind for the final exams. 

  • Don’t Stress yourself at the end Moment

Students often get anxious right before starting the exam. Whereas they shall understand that as important as it is to revise for the exam, it is also important for the students to attempt the paper calmly so that they can avoid unnecessary errors.

  • Don’t give a Messy Answer Sheet

We usually hurry up so much that we forget about presenting our work neatly. Attempting your exam in a neat way makes the answer sheet look a bit attractive to the eyes of the checker and he might find it easy to check. 

  • Don't Leave the 90% for the Sake of 10% 

Students often have a tendency to jump on to the sample papers and other practice things without being thorough with the foundational level— the NCERT textbook. 

FAQs on Standard Form

1. Why are maths formulas considered to be important? 

Since they provide the foundation for solving any kind of mathematical problem, they are considered to be very important. Revising on memorizing these formulas might seem like a difficult task to do but memorizing them is very important, especially for the students.

2. Is the NCERT syllabus too difficult? 

NCERT prepares the syllabus keeping in mind all kinds of students. Hence, the difficulty level is not high, it is considered to be moderate. Also, the flow of the questions is from easy to difficult thus, helping the student to get in tune with it over time.

3. Who Introduced Standard Form in Mathematics?

Standard form in Mathematics was introduced by the Persian Mathematician in the 9th Century named Muhammad-Al Khwarizmi.

4. What is the Meaning of Scientific Notation?

Scientific notation is the method of writing numbers so that they can be easily evaluated and manipulated. The method is standardized to be easy to read, with a one single-digit number written before the decimal point, and the exponents showing the overall intensity of a number.


Scientific notation provides techniques to avoid mistakes that come with calculations that include very large or small numbers.

5. How Scientific Notation is Helpful?

  • The exponents written in scientific notation can be helpful to compare two large numbers quickly and accurately. For example, you can easily compare which number is larger between 3 × 108 and 3 × 1011, than to make a comparison between 200000000 and 200000000000.

  • Most of the zeros written in a number such as 5023 000 000 000 000 000 000 are completely meaningless. This can be written precisely in  standard form. Writing the number as 5.023 × 1021 shows that precision is only to four significant figures.

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