Standard Form

What is standard form?

A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules.

Different forms of numbers have different standard forms. For example, there are different rules for writing whole numbers in standard form, decimal numbers in standard form etc. Let us learn about some of them.

Standard Form of Whole Numbers

We know that the numbers, 0, 1, 2, 3 and so on are whole numbers. To represent these numbers we have rules of place values. Place value can be defined as the value represented by a digit in a number on the basis of its position in the number.  So, if we wish to write the number “Thirty five thousand, four hundred and seven” it will be written as 35407. Is this number in its standard form? Let us find out.

To represent whole numbers in their standard form, we have a system of writing numbers, known as the International system of writing numbers. The International place value chart is given below:

The chart can be extended to its left to include more than 9 places. The period just to the left of the millions period is called the billions period. Then we have the trillions period. The places left to the hundred billion places are called one trillion places, ten trillion places and hundreds trillion places.

Note that each period consists of three digits.  We place a comma after every 3 digits. In the modern convention system, instead of putting a comma, we leave a space after every 3 digits.

So, how do we use the above system to write the whole numbers in their standard form? The answer lies in the commas being placed in the right place. Therefore, the number “Thirty five thousand, four hundred and seven” it will be not be written as 35407. The standard form of the number 35407 will be 35, 407 or 35 407. Let us understand this with some more examples.

Example: Write the number 62509842 in its standard form.

Solution:        

We have been given the number 62509842. Start counting the number of digits from the right. After every three digits, place a comma to separate the periods. We will have 62,509,842. Therefore the number 62509842 in its standard form will be62,509,842 which in words would be written as “Sixty two million, five hundred and nine thousand, eight hundred and forty two.

Example: Write the number 712984301 in its standard form.

Solution              

We have been given the number 712984301. Start counting the number of digits from the right. After every three digits, place a comma to separate the periods. We will have 712,984,301. Therefore the number 712984301 in its standard form will be712,984,301 which in words would be written as “Seven hundred and twelve million, nine hundred and eighty four thousand, three hundred and one.”

Standard Form of Decimals

We know that a decimal consists of a whole number and a fractional part separated by a decimal. Before we understand how to write the decimals in their standard form, it is important to understand the decimal number system.

Decimal Place Value System

The decimal place value system is an extension of the number system of whole numbers that we have just learnt. In this system, the digits of the fraction part have been defined according to their place values apart from the existing whole number writing system.

The Fractional part of the decimal is represented as:

Decimal (.)TenthHundredthThousandth

So, in a number 32.15 is the digit 1 is at the tenth place while the digit 5 is at the hundredths place. But, is this number in the standard form? Let us find out.

Standard Form of a decimal

By definition, any number that can be written as the decimal number, between the numbers 1.01.0 and 10.010.0, and then multiplied by the power of the number 1010, is known to be in the standard form. For example, the number 5410000 in its standard form will be 5.4 x 106.

How did we obtain this standard form?

It involves expressing a given decimal number by its first digit followed by a decimal point and its remaining digits, multiplied by a power of 10 such that it is equivalent to the original value.

Converting a decimal number into standard form

Converting a decimal number into standard form mostly just requires an understanding of the decimal place value system. We just need to multiply by the correct power of 10. We just need to use the following steps to rewrite a number in its standard form-

  1. Power to the 10 shows the exact place of decimal to be moved.
  2. To find the place of decimal you should first check whether the power is greater or less than 10.
  3. If the power on 10 is greater than or equal to 10, the decimal point is moved to the left side. This case implies only if the power of 10 is positive. For example, 102.
  4. If the power on 10 is lesser than 1, the decimal point is moved to the right side. This case implies only if the power of 10 is negative. For example, 10−2.

Let us understand it with an example.

Example: Write the number 650000 in its standard form.

Solution              

We have been given the number 650000. To convert this number in its standard form, we first count the number of digits after the first digit. We have 5 digits after the first digit, i.e. 6. Therefore, we will place a decimal after 6 and before 5 and multiply the given number by 106. We will get 6.5 x 106

Hence, the standard form of the number 650000 will be 6.5 x 106.

Example: Write the number 0.000568

Solution              

We have been given the number 0.000568. To convert this number in its standard form, we must first count the number of digits that lie between the decimal point and the first number from the left, i.e 5. The total number of digits is 4. We will, therefore, place a decimal between the first two numbers, other than 0, and multiply the number by 10-4. We will get, 5.68 x 10-4

Hence, the standard form of the number 0.000568 will be 5.68 x 10-4

The standard form of decimal numbers is also known as scientific notation.

Standard Form of Polynomials

We know that a polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In other words, a polynomial is an algebraic sum in which no variables appear in denominators or under radical signs and all variables do appear area raised only to positive integral powers. The term that does not contain any variable is called a constant term. So, how do we write a polynomial in its standard form? Let us find out.

The first and foremost rule in writing the polynomials in standard form is that polynomials need to be written with the exponents in decreasing order. This means that for a polynomial to be in standard form, its highest degree is placed first, followed by the term which has the next highest degree, and so on. What do we mean by the degree? The degree of a polynomial is the value of the largest exponent in the polynomial.

In other words, the degree of a polynomial is the sum of all the exponents of the variable in a term. The highest degree of all the terms that appear with non-zero coefficients in a polynomial is called its degree.

Another term that needs to be revisited before we go ahead with understanding the standard form of polynomials is the like and unlike terms.

Like Terms – The terms having the same variable and the powers are called the like terms. For example, 5x and 7x are like terms.

Unlike Terms – The terms that do not have the same variable and the powers are called, unlike terms. For example, 5x and 7xy are like terms.

Now we will learn about the steps that need to be followed to write a polynomial in its standard form.

  1. First, write all the given terms of the polynomial.
  2. Group all the like terms by writing them next to each other.
  3. Find the value of the degree of each term.
  4. The terms with the highest exponent (degree) need to be written first.
  5. Write the remaining terms in the decreasing order of their degree.
  6. The constant term will be written at the end.

Let us understand this through an example.

Example: Write 7x2 + 4x8 + 5x5 + x9 + 3 in the standard form.

Solution              

We have been given the polynomial 7x2 + 4x8 + 5x5 + x9 +3

First, let us write the degree of each term separately

7x2 – degree 2

4x8 – degree 8

5x5 – degree 5

x9 – degree 9

3 – constant term, degree 0

Since there are no like terms in the given polynomial, we will write the terms in the decreasing order of their degrees. We will get

x9 + 4x8 + 5x5 + 7x2 + 3

Hence, the standard form of the polynomial 7x2 + 4x8 + 5x5 + x9 +3 will be x9 + 4x8 + 5x5 + 7x2 + 3

Standard Form of a Rational Number

We know that a number of the form $\frac{p}{q}$ or a number that can be expressed in the form of $\frac{p}{q}$, where p and q are integers and q ≠ 0 is called a rational number. So how do we define the standard form of a rational number?

Definition – A rational number $\frac{p}{q}$ is said to be in the standard form if q is positive and the integers p and q have no common divisor other than 1.

In order to express a given rational number in the standard form, the following steps should be followed:

Step 1 – Check whether the given number is in the form of $\frac{p}{q}$. i.e. a rational number.

Step 2 – See whether the denominator of the rational number is positive or not. If it is negative, multiply or divide the numerator as well as the denominator by -1 so that the denominator becomes positive.

Step 3 – Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step 4 – Divide the numerator and the denominator of the given rational number by the GCD (HCF) obtained in step III. The rational number so obtained is the standard form of the given rational number.

Let us understand the above steps with the help of some examples.

Example 1: Express the following rational numbers in standard form

a) $\frac{-8}{28}$

b) $\frac{-12}{-30}$

c) $\frac{14}{-49}$

d) $\frac{-16}{-56}$

Solution

a) We have been given the rational number $\frac{-8}{28}$ and we need to express it in its standard form.

Let us find our answer using the above steps. We can see that the number is given to us in the form of $\frac{p}{q}$. Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is 28 which, is a positive number. In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 8 and 28.

The greatest common divisor of 8 and 28 is 4.

Therefore,

Dividing the numerator and the denominator of $\frac{-8}{28}$ by 4, we get

$\frac{-8}{28}=\frac{-8/4}{28/4}=\frac{-2}{7}$

Hence, the standard form of $\frac{-8}{28}$ is $\frac{-2}{7}$.

b) The number is given to us $\frac{-12}{-30}$.

Again, we will find our answer using the above steps. We can see that the number is given to us in the form of $\frac{p}{q}$. Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is -30 which, is a negative number.

Therefore, first, we will have to make the denominator positive. In order to do so, we will multiply the numerator as well as the denominator by -1.

Hence, multiplying the numerator and the denominator by -1 we get,

$\frac{-12}{-30}=\frac{(-12)\times (-1)}{(-30)\times (-1)}=\frac{12}{30}$                   

Now, we have a positive denominator; therefore let us move to the next step.

In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 12 and 30.

The greatest common divisor of 12 and 30 is 6.

Therefore,

Dividing the numerator and the denominator of $ \frac{12}{30}$ by 6, we get

$\frac{12}{30}=\frac{12/6}{30/6}=\frac{2}{5}$

Hence, the standard form of $\frac{-12}{-30}$ is $\frac{2}{5}$.

c) The number given to us $\frac{14}{-49}$.

As done above, we will find our answer using the stated steps. We can see that the number is given to us in the form of $\frac{p}{q}$. Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is -49, which is a negative number.

Therefore, first, we will have to make the denominator positive. In order to do so, we will multiply the numerator as well as the denominator by -1.

Hence, multiplying the numerator and the denominator by -1 we get,

$\frac{14}{-49}=\frac{14\times (-1)}{(-49)\times (-1)}=\frac{-14}{49}$

Now, we have a positive denominator; therefore let us move to the next step.

In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 14 and 49.

The greatest common divisor of 14 and 49 is 7.

Therefore,

Dividing the numerator and the denominator of $\frac{-14}{49}$ by 7, we get

$\frac{-14}{49}=\frac{-14/7}{49/7}=\frac{-2}{7}$

Hence, the standard form of $\frac{14}{-49}$ is $\frac{-2}{7}$.

d) The number is given to us $\frac{-16}{-56}$.

We can see that the number is given to us in the form of $\frac{p}{q}$. Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is -56 which, is a negative number.

Therefore, first, we will have to make the denominator positive. In order to do so, we will multiply the numerator as well as the denominator by -1.

Hence, multiplying the numerator and the denominator by -1 we get,

$\frac{-16}{-56}=\frac{(-16)\times (-1)}{(-56)\times (-1)}=\frac{16}{56}$                      

Now, we have a positive denominator; therefore let us move to the next step.

In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 16 and 56.

The greatest common divisor of 16 and 56 is 8.

Therefore,

Dividing the numerator and the denominator of $\frac{16}{56}$ by 8, we get:

$\frac{16}{56}=\frac{16/8}{56/8}=\frac{2}{7}$

Hence, the standard form of $\frac{-16}{-56}$ is $\frac{2}{7}$.

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