3.04 Standard form

In scientific fields like astronomy, medicine and computer science it is common to work with very large or very small numbers. 

For example,

• The mass of the sun is approximately $1988000000000000000000000000000$1988000000000000000000000000000 kg. 
• The approximate diameter of a red blood cell is $0.000007$0.000007 m. 

In economics too, large numbers are common, with some monetary values in the trillions of dollars.

• The gross domestic product (GDP) of the United States in 2017 was $19390000000000$19390000000000 USD ($19.39$19.39 trillion).

Because these numbers have so many digits, recording them or inputing them into a device, can easily lead to errors.

 

Standard form

Standard form (also known as scientific notation) is a more convenient way of representing numbers that contain a lot of digits.

Decimal form	Standard form
$1988000000000000000000000000000$1988000000000000000000000000000	$1.988\times10^{30}$1.988×1030
$0.000007$0.000007	$7\times10^{-6}$7×10−6

To represent a number in standard form, we place a decimal point directly after the first non-zero digit. This gives us a number between $1$1 (inclusive), and $10$10. We then consider what power of $10$10 we need to multiply this value by, to give the original number.

 

Although standard form is used mainly for writing very large or very small numbers, any non-zero number can be written using standard form.

 

 

Worked examples

Example 1

Express $63300$63300 in standard form.

Think: In standard form, a number is represented by a value between $1$1 and $10$10, that is multiplied by an appropriate power of $10$10. 

Do: Starting from the left, the first non-zero digit in $63300$63300 is $6$6, so we place a decimal point directly after the $6$6 and remove all trailing zeros. This gives $6.33$6.33, which is a value between $1$1 and $10$10.

We then work out what power of $10$10 we must multiply $6.33$6.33 by, to get $63300$63300. We can think about this in terms of how many decimal places we have to move to the right. In this case we move $4$4 decimal places to the right, which is the same as multiplying $6.33$6.33 by $10000$10000, or $10^4$104.

Therefore, we write,

$63300$63300

$=$=

$6.33\times10^4$6.33×104

 

Example 2

Write $0.000004981$0.000004981 in standard form.

Think: In standard form, a number is represented by a value between $1$1 and $10$10, that is multiplied by an appropriate power of $10$10. 

Do: Starting from the left, the first non-zero digit in $0.000004981$0.000004981 is $4$4, so we place a decimal point directly after the $4$4 and remove all leading zeros. This gives $4.981$4.981, which is a value between $1$1 and $10$10.

We then work out what power of $10$10 we must multiply $4.981$4.981 by, to get $0.000004981$0.000004981. We can think about this in terms of how many decimal places we have to move to the left. In this case we move $6$6 decimal places to the left, which is the same as dividing $4.981$4.981 by $1000000$1000000, or $10^6$106. Here we recall from the index laws (see final panel below), that dividing by $10^6$106 is the same as multiplying by $10^{-6}$10−6.

Therefore, we write,

$0.000004981$0.000004981

$=$=

$4.981\times10^{-6}$4.981×10−6

 

Practice Questions

Question 1

Question 2

 

Powers of ten and the index laws

As we have seen, representing a number in standard form involves a power of $10$10 with an index (exponent) that is an integer, (i.e. a whole number that can be positive, negative or zero).

• A positive index is used to represent large numbers, that is, numbers greater than or equal to $10$10.
   
• A negative index is used to represent small numbers, that is, numbers between $0$0 and $1$1.
   
• A zero index indicates a number greater than or equal to $1$1, but less than $10$10.

 

 

Standard form on the calculator

Calculators usually display a limited number of digits on their screens at any one time, so when very large or small numbers are entered into a calculator, they will often be displayed in standard form. 

Some calculators can display standard form the same way it appears in a textbook, i.e. $3.05\times10^{-8}$3.05×10−8.

Calculators that don't support textbook display, may use the letter $E$E in place of $\times$×$10$10. The $E$E stands for 'exponent'. This means a number like $3.05\times10^{-8}$3.05×10−8 may display as $3.05$3.05$E$E$-8$−8.  

Numbers can also be entered in standard form, directly into the calculator, using a dedicated button. The button will be labelled something like $\editable{\times10^x}$×10x or $\editable{\text{EXP }}$EXP , but may appear differently, depending on the model.

Here are two examples of how to enter numbers in standard form on a calculator:

standard form	How to enter the number on a calculator
$2.3\times10^{12}$2.3×1012	$\editable{2}$2$\editable{.}$.$\editable{3}$3$\editable{\times10^x}$×10x$\editable{12}$12
$1.9\times10^{-4}$1.9×10−4	$\editable{1}$1$\editable{.}$.$\editable{9}$9$\editable{\times10^x}$×10x$\editable{(-)}$(−)$\editable{4}$4

 

Standard form and significant figures

Rounding to a certain number of significant figures can also be used to round numbers in standard form. 

For example:

$123958372$123958372

$=$=

$1.24\times10^8$1.24×108 ($3$3 s.f.)

 

$0.0000029212$0.0000029212

$=$=

$2.9\times10^{-6}$2.9×10−6 ($2$2 s.f.)

 

Notice that rounding only affects the value before the multiplication sign. It doesn't change the size of the number being represented.

 

Practice Questions

Question 3

Question 4

QUESTION 5