Powers of Ten Notation
Interpret and compare numbers in standard form A × 10ⁿ where 1 ≤ A < 10 and n is an integer; convert between ordinary numbers and standard form
What a learner can do afterwards
- Convert large and small numbers into standard form A × 10ⁿ
- Convert numbers from standard form back to ordinary notation
- Compare and order numbers given in standard form
The lesson
Some numbers are too long to write comfortably. The distance to the sun is about 150,000,000 kilometers. A red blood cell is about 0.000007 meters wide. You already know how a small raised number works from squares and cubes, like 5² or 4³. Standard form uses that same idea. You write any number as A × 10ⁿ, where A is between 1 and 10, and n is a whole number called the exponent.
Take 3,400,000. First write the digits as a number between 1 and 10: 3.4. Next, count how many places the decimal point moves to turn 3.4 back into 3,400,000. That is six places, so 3,400,000 = 3.4 × 10⁶. To go the other way, start at 3.4 and move the decimal point six places to the right.
Small numbers work the same way, but the exponent is negative. Take 0.00045. Write it as 4.5, then count how many places the decimal point moved to the left to get there: four places. So 0.00045 = 4.5 × 10⁻⁴. A negative exponent means the number is smaller than 1, not smaller than zero.
To compare two numbers in standard form, look at the exponent first. A bigger exponent means a bigger number, as long as A is between 1 and 10. Only compare the A values if the exponents match.
Standard form writes any number as A × 10ⁿ, with 1 ≤ A < 10, so huge or tiny numbers become short to write and easy to compare.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.