Standard Form

In many fields, such as astronomy, extremely large numbers can be encountered. On the other hand, in fields such as nuclear physics, very small numbers are frequently dealt with. The problem with these numbers is that due to their magnitude, writing them in the mathematical form you are used to is extremely long, which takes up a large amount of physical space and is less comprehensible for the human eye.

For example, the distance from the Earth to the Sun is approximately 150 million km. Written as a Number in metres, this gives us 150,000,000,000 m. This is already an extremely long number and we're only just scratching the surface; there exists many examples of much larger numbers in our universe.

How can this problem be solved? A way of writing numbers in a shortened form was invented to deal with this: standard form. This article will explain what standard form is and how to convert numbers to and from standard form.

Standard form definition

Numbers written in standard form are written in the form:

$A\times {10}^n$

Where A is any Number greater than or equal to 1 and less than 10 and n is any integer (whole Number), negative or positive.

The exponent of 10 determines how big or small the number is, as larger positive exponents result in larger numbers:

${10}^1=10$

${10}^2=10\times 10=100$

${10}^3=10\times 10\times 10=1000$

${10}^4=10\times 10\times 10\times 10=10000$

Larger negative exponents result in smaller numbers:

${10}^{-1}=1/10=0.1$

${10}^{-2}=1/100=0.01$

${10}^{-3}=1/1000=0.001$

${10}^{-4}=1/10000=0.0001$

Standard form calculations

Converting numbers into standard form

Numbers in standard form are written as a multiple of a power of 10. In the case of large numbers, the power of 10 will be large, meaning a positive exponent. For small numbers, the power of 10 will be extremely small (as multiplying a number by a decimal makes the number smaller), meaning a negative exponent.

In order to convert a number into standard form, follow these steps:

1. Move the decimal point until there is only one non-zero digit to the left of the decimal point. The number that has been formed is the value for A. For example, 5000 becomes 5.000, and we can remove the leading 0's giving us 5.
2. Count the number of times that the decimal point was moved. If the decimal point was moved to the left, the value for n in the formula will be positive. If the decimal point was moved to the right, the value for n in the formula will be negative. In the case of 5000, the decimal point was moved to the left 3 times, meaning n is equal to 3.
3. Write the number in the form $A\times {10}^n$ using your results from step 1 and step 2.

Converting numbers from standard form

In the case of converting numbers from standard form, we can simply multiply A by ${10}^n$, as standard form numbers are written as $A\times {10}^n$.

Adding and subtracting numbers in standard form

The easiest way to add or subtract numbers which are written in standard form is to convert them into Real Numbers, perform the operation and then convert the result back into standard form. If you are permitted to use a calculator, these steps are not required as the calculator can perform the operation while displaying the result in standard form.

Multiplying and dividing numbers in standard form

When multiplying and dividing numbers in standard form, the numbers can be kept in standard form, unlike with adding and subtracting. Follow these steps:

1. Perform the multiplication/division with the A of each number. This gives the A of the result.

2. If multiplying, add the exponents of 10 from each number together. If dividing, subtract the exponent of 10 from the 2^nd number from the exponent of 10 from the 1^st number. This is done because of the index laws.

3. You will now have a number in the form $A\times {10}^n$. If A is 10 or more, or less than 1, you must convert the number back into a real number, and then back into standard form, so that the number is written in the correct standard form.

Standard form examples