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# Standard Form Physics

Standard form (also known as standard index form) allows you to represent very large and very small numbers by using a system of numerical notation. It is similar to the use of SI prefixes.For example, one hundred metres can be expressed as 100 m, but it can also be expressed as $$1 \cdot 10^3$$ metres using standard form. The principle behind…

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# Standard Form Physics

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Standard form (also known as standard index form) allows you to represent very large and very small numbers by using a system of numerical notation. It is similar to the use of SI prefixes.

For example, one hundred metres can be expressed as 100 m, but it can also be expressed as $$1 \cdot 10^3$$ metres using standard form. The principle behind this equivalence is simple and involves multiplying the quantity by ten and raising it to a power that gives you the correct number. See the following two examples:

$$1000 \space grams = 1 \space kilogram = 1 \cdot 10^3 \space g$$

$$0.0000023 \space meters = 2.3 \space micrometers = 2.3 \cdot 10^{-6} m$$

The last numbers are the factor. So, for instance, if you multiply $$1 \cdot 10^3$$ g, you get 1000 grams. The standard form also helps us to reduce large numbers to a smaller notation, as in the examples below.

$$1,530,000 \space watts = 1.53 \cdot 10^6 \space watts$$

$$45,500,000 \space calories = 45.5 \cdot 10^6 \space calories$$

$$120,000 \space kg = 12 \cdot 10^4 \space kg$$

## Using standard form

Standard form is used differently, depending on the size of the number. If the number is smaller than the unit, the exponent is negative. If the number is larger than the unit, the exponent is positive.

### Small numbers

Here is an explanation of how to use the standard form for small numbers.

First, check how many decimals your number is below the unit. Lets use the example 0.0003.

To make the number 3 appear before the decimal point, you need to move the decimal point 4 places to the right.

Then you multiply three by ten. Your exponent is -4, giving you $$3 \cdot 10^{-4}$$.

### Large numbers

And here is an explanation of how to use the standard form for large numbers.

First, check how many decimals your number is above the unit. Lets use the example $$32476.0$$.

To make the number 3 appear immediately before the decimal point, you need to move the decimal point 4 places to the left.

Then you multiply three by ten. The exponent this time is 4, giving you $$3.2476 \cdot 10^4$$.

## What are the standard symbols?

The SI system allows you to exchange prefixes and standard form to symbols as and when necessary. The standard symbols are symbols used to replace factor forms and prefixes.

For example, 2.3 micrometres (prefix micro) is equal to both 2.3μm (symbol) and $$2.3 \cdot 10^{-6}$$ m (standard form).

You can find a table with the prefixes, factors, and symbols used for all units below.

### Symbols, standard form, representation, and names for large quantities

Table 3. Symbols, standard form and representation of large quantities.
Symbol
Standard form
Representation
Name
Y
$$10 ^ {24}$$
1,000,000,000,000,000,000,000,000
Septillion
Z
$$10 ^ {21}$$
1,000,000,000,000,000,000,000
Sextillion
E
$$10 ^ {18}$$
1,000,000,000,000,000,000
Quintilion
P
$$10 ^ {15}$$
1,000,000,000,000,000
T
$$10 ^ {12}$$
1,000,000,000,000
Trllion
G
$$10 ^ 9$$
1,000,000,000
Billion
M
$$10 ^ 6$$
1,000,000
million
k
$$10 ^ 3$$
1,000
Thousand
H
$$10 ^ 2$$
100
Hundred
‘there
$$10 ^ 1$$
10
Ten

### Symbols, standard form, representation, and names for small quantities

Table 4. Symbols, standard form, representation of small quantities.
Symbol
Standard form
Representation
Name
y
$$10 ^ {-24}$$
0,000,000,000,000,000,000,000,001
septillionth
z
$$10 ^ {-21}$$
0,000,000,000,000,000,000,001
sextillionth
a
$$10 ^ {-18}$$
0,000,000,000,000,000,001
quintilionth
f
$$10 ^ {-15}$$
0,000,000,000,000,001
p
$$10 ^ {-12}$$
0,000,000,000,001
trllionth
n
$$10 ^ {-9}$$
0,000,000,001
billionth
μ
$$10 ^ {-6}$$
0.000.001
millionth
m
$$10 ^ {-3}$$
0.0001
thousandth
c
$$10 ^ {-2}$$
0.01
hundredth
d
$$10 ^ {-1}$$
0.1
tenth

### Standard form examples using units

Standard form is very useful when dealing with units and calculations in physics, mathematics, or engineering. Many quantities are very small, such as the charge of an electron, its mass, or even the pressure in pascals. See the following examples of using the standard form.

Calculate the total charge in Coulombs of an alpha particle and express the result using the standard form.

An alpha particle is made of two protons and two neutrons. The only charged particles are the protons, which have a charge of $$1.602176634 \cdot 10^{−19}$$ C.

The total charge is the proton charge multiplied by two.

$$\text{Total charge} = (1.602 \cdot 10^{-19} C) \cdot 2 = 3.204 \cdot 10 ^{-19} C$$

Express the atmospheric pressure at sea level from pascals to grams per square metre using the standard form.

The accepted value for atmospheric pressure at sea level is 101325 Pa, and one pascal is equal to one newton applied over one square metre.

$$101325 \space Pa = 101325 \space N/m^2$$

We also know that a newton is equal to one kilogram per metre over a square second.

$$101325 \space N/m^2 = 101325 (kg \cdot m)/s^2m^2 = 101325 kg/s^2m$$

And we know that one kilogram is 1000 grams.

$$101325 kg/s^2 m = 101325000 g/s^2 m$$

This quantity is very large, so we can express it using standard form.

$$101325000 g/s^2m = 1.01 \cdot 10^8 g/s^2m$$

This is a much shorter and better way to express the pressure if you use grams.

## Standard Form - Key takeaways

• The SI system allows you to use compact forms to represent small and large quantities in numbers. The compact form is called standard form.
• Standard form uses exponents where the number is multiplied by factors of ten to make expressions more compact. Examples of expressing numbers in standard form are $$100 = 1 \cdot 10^2$$ and $$1000 = 1 \cdot 10^3$$.
• In standard form, quantities larger than the unit use a positive exponent, while quantities smaller than the unit use a negative exponent such as $$0.1 = 1 \cdot 10^{-1}$$
• The SI system also uses symbols to replace prefixes and factor forms.

Standard form is a way to represent large or small numbers by using exponents and powers of ten.

To write a number using standard form, you need to know how far you are from the unit, which will give you the exponent.

Multiply the number by ten and write the exponent to the number 10 on top.

If the number is larger than the unit, the exponent is positive. If the number is smaller than the unit, the exponent is negative. Some useful examples can be found in the article.

## Standard Form Physics Quiz - Teste dein Wissen

Question

What is standard form?

Standard form is a compact form to express large and small quantities.

Show question

Question

How do you use standard form to express large quantities?

You multiply by 10 and elevate to a positive exponent.

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Question

How do you use standard form to express small quantities?

You multiply by 10 and elevate to a negative exponent.

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Question

How would you express 3.1415 if you want the result to be 31415.0?

31415.0x10^-4.

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Question

How would you express 8567.0 if you want the result to be 85.67?

85.67x10^2.

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Question

10^0 is equal to what quantity?

1.

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Question

What are the standard symbols?

The standard symbols are characters or abbreviations used to replace a full quantity.

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Question

What is the standard symbol for micro?

μ.

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Question

Write the number 5*10^0 in normal form.

5.

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Question

Write $$31415.0 \cdot 10^{-4}$$ in normal form.

3.1415

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Question

Does a negative exponent in standard form move the decimal point to the left or to the right?

To the right.

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Question

Convert 1.5 million to standard form.

1.5*10^6

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Question

Write the symbol for a tera.  How would you write one tera in standard form?

T.

1*10^12.

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Question

A tera is equal to what quantity?

One trillion.

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Question

Convert 2.34 kilograms to grams using standard form.

2.340*10^3g.

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Question

Convert 2321 kilonewtons to standard form, using only one integer.

2.321*10^6 Newtons.

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Question

How much is 23.4 kilometres in metres using standard form with only one integer?

2.34*10^4 metres.

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Question

Write the symbol and the standard form for millimetres.

mm.

1*10^-3.

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