STANDARD FORM OF A NUMBER 

What is the standard form of a number?

A number is cosidered to be in standard form, when it can be read or written easily.

For example, consider the decimal number 0.3. You can read or write 0.3 easily. But, if you have 0.0000003, it is litlle difficult to read or write. Using power of 10, we can write 0.0000003 as 3x10⁶. The standard form of 0.0000003 is 3x10⁶.

And also, the standard form of the number 500000000 is 5x10⁸.

How to Express Large Numbers in Standard Form

Consider the number 63700000000. The standard form of this number can be obtained as follows.

Step 1 :

Take the decimal point right after the first digit '6'

6.3500000

Step 2 :

Count the number of digits to the right of the decimal point. Since there are seven digits to the right of the decimal point, 7 has to be taken as exponent for 10 and 10⁷ can be used to write the given number in standard form.  

Step 3 :

To the right of the decimal point, the last non zero digit is '5'. Leave all the zeros to the right of '5'.

Hence, the standatd form the number 63700000000 is

6.37x10⁷

How to Express Decimal Numbers in Standard Form

Consider the number 23.549. The standard form of this decimal number can be obtained as follows.

Step 1 :

237.54

Take the decimal point right after the first nonzero number.

2.3754

Step 2 :

Count the number of digits that the decimal point is shifted. The decimal point is shifted to the left by two digits. Take 2 as exponent for 10 and 10² can bed used to write the given decimal number in standard form.

Note :

In case, the decima point point is shifted to the right, the exponent of 10 has to be negative. For example, if the decimal point is moved to the right by 3 digits, -3 has to be taken as exponent for 10.

Hence, the standatd form the number 237.54 is

2.3754x10²

Numbers in Expanded Form

Consider the number 75823. 

The first digit 7 is in ten thousands place, the next digits 5 is in thousands place, 8 is in hundreds place, 2 is in tens place and 3 is in ones place.

The expanded form of 75823 :

75823 = 7x10000 + 5x1000 + 8x100 + 2x10 + 3x1

Expanded Form of Decimal Numbers

In decimal number, the first digit to the right of the decimal place is in tenths place (¹⁄₁₀), the second digit to the right of the decimal place is in hundredths place (¹⁄₁₀₀) and so on. 

For example, the positions of digits in the decimal number 0.4873 :

4 ----> tenths place

8 ----> hundreths place

7 ----> thousandths place

3 ----> ten thousandths place

The expanded form of 0.4873 :

0.4873 = 4(¹⁄₁₀) + 8(¹⁄₁₀₀) + 7(¹⁄₁₀₀₀) + 3(¹⁄₁₀₀₀₀)

= ⁴⁄₁₀ + ⁸⁄₁₀₀ + ⁷⁄₁₀₀₀ + ³⁄₁₀₀₀₀

= 0.4 + 0.08 + 0.007 + 0.0003

Standard Form a Rational Number

Consider the rational numbers given below : 

½, ⅞, ⁻⁴⁄₉, ⅕, ⁻¹³⁄₂₅

In all the rational numbers above, both numerator and denominator are integers. The denominators of the rational numbers are all positive integers. There is no common divisor other than 1 for the numerator and the denominator of each of them. The negative sign occurs only in the numerator.

Such rational numbers are said to be in standard form.

Note :

In a rational number, if there is a negative sign in denominator, it can be moved to the numerator.

For example,

ˣ⁄₋y = ⁻ˣ⁄y

Solved Examples

Example 1 :

Write 274.35 in exponential form.

Solution :

274.35 :

= 2x100 + 7x10 + 4x1 + 3x⅒ + 5x¹⁄₁₀₀

= 2x10² + 7x10¹ + 4x10⁰ + 3x10^-1 + 5x10^-2

Example 2 :

Write the average size of a blood cell in standard form.

Solution :

Average size ofa blood cell is 0.000015 m.

0.000015 = 1.5 x 10^-5 m

When numbers are written standard form, it is easier to read write and understand.

Example 3 :

Write 150000000000 in standard form.

Solution :

150000000000 = 1.5 x 10¹¹ m

Example 4 :

Write the following addition in standard form.

(9.7 x 10^-3) + (8.4 x 10^-3)

Solution :

= (9.7 x 10^-3) + (8.4 x 10^-3)

Factor 10^-3.

= (9.7 + 8.4)10^-3

= 18.1 x 10^-3

= 1.81 x 10¹ x 10^-3

= 1.81 x 10^{1 - 3}

= 1.81 x 10^-2

Example 5 :

Write the following product in standard form.

(6.2 x 10⁵)(3.1 x 10^-3)

Solution :

= (6.2 x 10⁵)(3.1 x 10^-3)

= (6.2 x 3.1)(10⁵ x 10^-3)

= 19.22 x 10^{5 + (-3)}

= 1.922 x 10¹ x 10^{5 - 3}

= 1.922 x 10¹ x 10²

= 1.922 x 10^{1 + 2}

= 1.922 x 10³

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