Number
Grade 4-5Standard Form
Standard form is a compact way of writing very large or very small numbers using powers of 10, and almost every mistake in this topic comes down to getting the power wrong by one, or leaving the number in the wrong format. This lesson covers writing numbers in standard form, converting back to ordinary numbers, comparing numbers, and multiplying and dividing.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Writing large numbers in standard form
A number in standard form is written as A × 10ⁿ, where A is between 1 and 10, and n is a positive whole number for large numbers. Count how many places the decimal point moves to get A between 1 and 10, and that count becomes the power.
358,000 in standard form: move the decimal point 5 places to get 3.58, so 358,000 = 3.58 × 10⁵.
Writing small numbers in standard form
For numbers smaller than 1, the power of 10 is negative. Count how many places the decimal point moves to the right to get A between 1 and 10, and use that count as a negative power.
0.000462 in standard form: move the decimal point 4 places to the right to get 4.62, so 0.000462 = 4.62 × 10⁻⁴.
Converting standard form back to an ordinary number
A positive power of 10 means the decimal point moves that many places to the right, making the number larger. A negative power means the decimal point moves that many places to the left, making the number smaller.
6.3 × 10⁴: move the decimal point 4 places to the right, giving 63,000. 4.7 × 10⁻³: move the decimal point 3 places to the left, giving 0.0047.
Comparing and calculating with standard form
To compare numbers in standard form, compare the powers of 10 first. Only compare the coefficients A if the powers of 10 are equal. To multiply, multiply the coefficients and add the powers of 10. To divide, divide the coefficients and subtract the powers of 10. Adjust the answer back into standard form if the coefficient ends up outside 1 to 10.
(5 × 10⁴) × (4 × 10²): multiply the coefficients, 5 × 4 = 20, and add the powers, 10⁴ × 10² = 10⁶, giving 20 × 10⁶. Since 20 is not between 1 and 10, adjust to 2 × 10⁷.
Worked Examples
Three exam-style questions, fully solved.
Write 358,000 in standard form.
Easy- 1.Move the decimal point until only one non-zero digit is in front of it: 3.58
- 2.Count how many places the decimal point moved: 5
Answer: 3.58 × 10⁵
Write 0.000462 in standard form.
Medium- 1.Move the decimal point to the right until only one non-zero digit is in front of it: 4.62
- 2.Count how many places the decimal point moved: 4
- 3.Since the original number is smaller than 1, use a negative power
Answer: 4.62 × 10⁻⁴
Work out (5 × 10⁴) × (4 × 10²). Give your answer in standard form.
Hard- 1.Multiply the coefficients: 5 × 4 = 20
- 2.Add the powers of 10: 10⁴ × 10² = 10⁶, giving 20 × 10⁶
- 3.Since 20 is not between 1 and 10, rewrite it as 2 × 10¹ and combine the powers: 2 × 10⁷
Answer: 2 × 10⁷
Avoid These
The most common mistakes students make.
Miscounting the number of decimal places moved, giving a power of 10 that is one too high or too low.
Leaving the coefficient outside the range 1 to 10, for example writing 35.8 × 10⁴ instead of adjusting it to 3.58 × 10⁵.
Using a positive power of 10 for a number smaller than 1, or a negative power for a number larger than 1.
When comparing numbers in standard form, comparing the coefficients first instead of comparing the powers of 10 first.
When multiplying or dividing, forgetting to adjust the final answer back into standard form if the coefficient ends up being 10 or greater.
FAQ
Questions parents and students ask.
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