Math Lesson 7.3.1 - The Meaning of Standard Form.

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Welcome to our Math lesson on The Meaning of Standard Form., this is the first lesson of our suite of math lessons covering the topic of Standard Form, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Meaning of Standard Form. The Relationship between Ordinary, Decomposed and Standard Form of Numbers.

In tutorial 1.7, we briefly discussed writing ordinary numbers as powers of ten. This method or writing numbers is useful when we want to highlight various placeholders and the specific weight they "occupy" in a number. Thus, the number 271 is written in the decomposed form as

Moreover, we can write 100, 10 and 1 as powers of ten, i.e.

Writing numbers in the decomposed form is very useful to understand what value each digit has. However, this method is too long. It is better to write them in a shorter way while preserving some features of expanded or decomposed form discussed above. Thus, we write only the power of the leftmost digit while the original number is written in two parts: one decimal part that gives all digits of the original number and another part that gives the powers of ten. This method of writing numbers is known as the standard form. Therefore, the standard form represents a form of writing numbers in terms of powers of ten.

Any number N written in standard form has the general structure

where A is a number between 1 and 10 (without reaching the value 10) while m is an integer that represents the highest power of the decomposed form of that number.

If a number W in the ordinary form has n digits, when written in the standard form it becomes

where A, B, … , N are the coefficients representing the place values of the original number.

Example 1

Which of the following numbers is written in the correct decomposed form? Then, write all of them in the standard form.

1. 12,478
2. 3 × 5² + 2 × 5¹ + 9 × 5⁰
3. 7 × 1000^1/3 + 4 × 100^1/2 + 3 × 7⁰
4. 6 × 10⁴ + 3 × 10¹ + 9 × 10⁰

Solution 1

1. This number is not written in the decomposed form but in the ordinary form instead. If decomposed it becomes
   12,478 = 1 × 10⁴ + 2 × 10³ + 4 × 10² + 7 × 10¹ + 8 × 10⁰
   The standard form of this number therefore is
   12,478 = 1.2418 × 10⁴
2. This number is not written in the decomposed form as it is not written in terms of the powers of ten but of powers of five instead. If we want to write it in the decomposed form, first we must concert it in an ordinary number and only then we can write it in standard form. Thus,
   3 × 5² + 2 × 5¹ + 9 × 5⁰
   = 3 × 25 + 2 × 5 + 9 × 1
   = 75 + 10 + 9
   = 94
   When written in the decomposed form, this number becomes
   94 = 9 × 10¹ + 4 × 10⁰
   and in the standard form it becomes
   94 = 9.4 × 10¹
3. The number is not in the decomposed form but with a few arrangements it can be written as such. We have
   7 × 1000^1/3 + 4 × 100^1/2 + 3 × 7⁰
   = 7 × ∛1000 + 4 × √100 + 3 × 1
   = 7 × ∛(10³ ) + 4 × √(10² ) + 3 × 1
   = 7 × 10 + 4 × 10 + 3 × 1
   = 70 + 40 + 3
   = 113
   When written in the decomposed form, this number becomes
   113 = 1 × 10² + 1 × 10¹ + 3 × 10⁰
   The standard form of this number therefore is
   113 = 1.13 × 10²
4. This number is already written in the decomposed form as all terms represent powers of ten. We can just write the missing terms for aesthetic purpose although this is not necessary, i.e.
   6 × 10⁴ + 3 × 10¹ + 9 × 10⁰
   = 6 × 10⁴ + 0 × 10³ + 0 × 10² + 3 × 10¹ + 9 × 10⁰
   This helps to write the number in the standard form without errors. Thus,
   6 × 10⁴ + 3 × 10¹ + 9 × 10⁰ = 6.0039 × 10⁴

More Standard Form Lessons and Learning Resources

Whats next?

Enjoy the "The Meaning of Standard Form." math lesson? People who liked the "Standard Form lesson found the following resources useful:

1. Meaning Feedback. Helps other - Leave a rating for this meaning (see below)
2. Powers and Roots Math tutorial: Standard Form. Read the Standard Form math tutorial and build your math knowledge of Powers and Roots
3. Powers and Roots Video tutorial: Standard Form. Watch or listen to the Standard Form video tutorial, a useful way to help you revise when travelling to and from school/college
4. Powers and Roots Revision Notes: Standard Form. Print the notes so you can revise the key points covered in the math tutorial for Standard Form
5. Powers and Roots Practice Questions: Standard Form. Test and improve your knowledge of Standard Form with example questins and answers
6. Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
7. Continuing learning powers and roots - read our next math tutorial: Surds

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