In certain fields such as astronomy, we frequently encounter extraordinarily large numbers, while in others like nuclear physics, we come across extremely small numbers. However, dealing with these numbers can be challenging as they are either too big or too small to be written in the typical mathematical format, making them difficult to read and occupying significant physical space.
So, how do we solve this problem? The answer lies in a technique called standard form, which enables us to express numbers in a compact format. In this article, we will explore the concept of standard form and grasp the process of converting numbers to and from this format.
Standard form is a method of writing numbers in a shorter form, whether they are large or small numbers. In standard form, numbers are written as a multiple of a power of 10.
In standard form, numbers are represented as:
A x 10^{n},
where A is any number greater than or equal to 1 and less than 10, and n is any integer, positive or negative. The exponent of 10 indicates the size of the number, with larger positive exponents resulting in bigger numbers and larger negative exponents resulting in smaller numbers. For example:
Let's test our understanding by checking if the following number is written in standard form:
Solution: The given number is not in standard form as A must be a number less than 10 and greater than or equal to 1. However, in this case, A is 12, which is greater than 10. The number in standard form would be:
1.2 x 10^{1}.
To convert a number into standard form, follow these steps:
When converting numbers from standard form, we simply need to multiply A by 10 to the power of n, as standard form numbers are expressed as A x 10^{n}. For example:
3.73 x 10^{6} = 3.73 x 1,000,000 = 3,730,000
The easiest way to add or subtract numbers in standard form is by converting them into regular numbers, performing the operation, and then converting the result back into standard form. However, if a calculator is available, this step is unnecessary as the calculator can handle operations and display the result in standard form.
When multiplying and dividing numbers in standard form, we can keep the numbers in standard form, unlike with addition and subtraction. Follow these steps:
Convert the following number to standard form: 0.0086
Solution: First, move the decimal point until there is only one non-zero digit to the left of it, resulting in 8.6 as the value for A.
Next, count the number of times the decimal point was moved, which is 3 times to the left, making the exponent n equal to -3.
Final result: 8.6 x 10^{-3}.
Standard form is a format used to represent both large and small numbers in a concise manner. In this form, numbers are expressed as a multiple of a power of ten.
To convert a number into standard form, it is important to grasp the structure of this form. It is written as Ax10^n, where A can range from 1 to 10 and n is an integer (positive or negative).
For example, let's take the number 753000. To write this in standard form, we must move the decimal point until there is only one non-zero digit to the left. In this case, moving the decimal point three places to the left gives us an A value of 7.53. Since the decimal point was moved to the left, n is positive with a value of 3. Therefore, 753000 in standard form is written as 7.53x10^3.
To convert a number written in standard form back to its ordinary form, we simply multiply A by 10 to the power of n. In the previous example, multiplying 7.53 by 10^3 yields the original number, 753000.
Now, let's look at how to add or subtract numbers in standard form. To perform these operations, we must first convert the numbers into ordinary form, perform the operation, and then convert the result back into standard form. For instance, if we have the numbers 5x10^2 and 2x10^2, we convert them to ordinary form, giving us 500 and 200. Adding these gives us 700, which is then converted back into standard form as 7x10^2, our final answer.
Multiplying or dividing numbers in standard form, on the other hand, follows a different process:
It is important to note that if the resulting A value is greater than or equal to 10 or less than 1, we must convert the number back into an ordinary number and then into standard form to ensure it is written correctly.
In summary, standard form is a convenient way to write numbers, and converting them to and from this form is a straightforward process. By following the steps outlined above, handling operations with numbers in standard form becomes effortless.
Key Takeaways:
Standard form is a convenient way of writing very large or very small numbers. However, when using these numbers in calculations or trying to grasp their value, it can be beneficial to convert them to decimal form. In this article, we will discuss the process of transforming numbers from standard form to decimal form.
The standard form of a number is written as Ax10^n, where A is a decimal number between 1 and 10 and n is an integer. To convert this number to decimal form, follow these steps: