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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 100m, but it can also be expressed as 1x10^3 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:
The last numbers are the factor. So, for instance, if you multiply 1x10^3g, you get 1000 grams. The standard form also helps us to reduce large numbers to a smaller notation, as in the examples below.
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.
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. Let’s 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 3x10^-4.
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. Let’s 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 3.2476x10^4.
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 2.3x10^-6m (standard form).
You can find a table with the prefixes, factors, and symbols used for all units below.
| 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 | Quadrillion |
| 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 |
| 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 | quadrillionth |
| 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 |
Figure 1. Many instruments use unit symbols such as mm and cm as short names for the units. Source: Robbie Sproule, Flickr (CC BY 2.0).
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 × 10^−19 C.
The total charge is the proton charge multiplied by two.
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.
We also know that a newton is equal to one kilogram per metre over a square second.
And we know that one kilogram is 1000 grams.
This quantity is very large, so we can express it using standard form.
This is a much shorter and better way to express the pressure if you use grams.
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.
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