# 3.04 Standard form

Lesson

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}$7×106

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.

Standard form

A number expressed in standard form is written in the form,

$a\times10^n$a×10n

where $1\le a<10$1a<10 and $n$n is an integer (i.e. a positive or negative whole number, or zero).

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}$106.

Therefore, we write,

 $0.000004981$0.000004981 $=$= $4.981\times10^{-6}$4.981×10−6

#### Practice Questions

##### Question 1

The distance between two stars is approximate $9\times10^7$9×107 metres.

Express this distance as a whole number.

##### Question 2

Express $3.66\times10^{-6}$3.66×106 as a decimal number.

### 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.

Powers of $10$10 review

Several powers of $10$10 are listed in the following table:

Power Equivalent to
$10^3$103 $1000$1000
$10^2$102 $100$100
$10^1$101 $10$10
$10^0$100 $1$1
$10^{-1}$101 $\frac{1}{10}$110
$10^{-2}$102 $\frac{1}{100}$1100
$10^{-3}$103 $\frac{1}{1000}$11000

Based on the index laws (see final panel below), we see that multiplying by a power of $10$10 with a negative index, is the same as dividing by the same power of $10$10 with a positive index.

 $\times$×$10^{-n}$10−n $=$= $\div$÷​$10^n$10n

### 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}$3.05×108.

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}$3.05×108 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}$1.9×104 $\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

What is the output on your calculator when you enter $2.7\times10^7$2.7×107?

##### Question 4

Use a calculator to find the value of

$\frac{7257\times3937}{0.0083}$7257×39370.0083

Answer in standard form, correct to four significant figures.

##### QUESTION 5

A light year is defined as the distance that light can travel in one year. It is measured to be $9460730000000000$9460730000000000 metres.

1. Write this using standard form.

2. How many kilometres is this? Write this using standard form.

3. How many centimetres is this? Write this using standard form.

Index laws review

Calculations using standard form, may make use of the following index laws:

 $x^a\times x^b$xa×xb $=$= $x^{a+b}$xa+b

 $x^a\div x^b$xa÷​xb $=$= $x^{a-b}$xa−b

 $\left(x^a\right)^b$(xa)b $=$= $x^{ab}$xab

 $x^{-a}$x−a $=$= $\frac{1}{x^a}$1xa​

 $x^0$x0 $=$= $1$1

### Outcomes

#### MS11-3

solves problems involving quantity measurement, including accuracy and the choice of relevant units