science

Power of 10

Power of 10 is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Power of 10 rather than just read about it. In short: In mathematics, a power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten.

Key takeaways

Power of 10 belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Power of 10 to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Power of 10 from memory before moving on to harder problems.

Reference excerpt

In mathematics, a power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The first few non-negative powers of ten are:

1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000... (sequence A011557 in the OEIS)

Positive powers In decimal notation the nth power of ten is written as '1' followed by n zeroes. It can also be written as 10n or as 1En in E notation. See order of magnitude and orders of magnitude (numbers) for named powers of ten. There are two conventions for naming positive powers of ten, beginning with 109, called the long and short scales. Where a power of ten has different names in the two conventions, the long scale name is shown in parentheses. The positive 10 power related to a short scale name can be determined based on its Latin name-prefix using the following formula: 10[(prefix-number + 1) × 3] Examples:

billion = 10[(2 + 1) × 3] = 109 octillion = 10[(8 + 1) × 3] = 1027

For further examples, see Names of large numbers. Numbers larger than about a trillion are rarely referred to by name or written out as digits, but instead are typically described with exponent notation.

Negative powers The sequence of powers of ten can also be extended to negative powers. Similar to the positive powers, the negative power of 10 related to a short scale name can be determined based on its Latin name-prefix using the following formula: 10−[(prefix-number + 1) × 3] Examples:

billionth = 10−[(2 + 1) × 3] = 10−9 quintillionth = 10−[(5 + 1) × 3] = 10−18

Googol and googolplex

The number googol is 10100. The term was coined by 9-year-old Milton Sirotta, nephew of American mathematician Edward Kasner. It was popularized in Kasner's 1940 book Mathematics and the Imagination, where it was used to compare and illustrate very large numbers. Googolplex, a much larger power of ten (10 to the googol power, or 1010100), was also introduced in that book.

Scientific notation

Scientific notation is a way of writing numbers of very large and very small sizes compactly. A number written in scientific notation has a significand (sometime called a mantissa) multiplied by a power of ten. Sometimes written in the form:

m × 10n Or more compactly as:

10n This is generally used to denote powers of 10. Where n is positive, this indicates the number of zeros after the number, and where the n is negative, this indicates the number of decimal places before the number. As an example:

105 = 100,000 10−5 = 0.00001 The notation of mEn, known as E notation, is used in computer programming, spreadsheets and databases, but is not used in scientific papers.

See also Power of two Power of three SI prefix Cosmic View, inspiration for the film Powers of Ten Exponentiation Philip and Phylis Morrison wrote a book called "Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding Another Zero" to accompany the video of Eames. [1]

Further reading Powers of Ten (1977). Nine-minute film. US Public Broadcasting Service (PBS), made by Charles and Ray Eames. "An adventure in magnitudes. Starting at a picnic by the lakeside in Chicago, this film transports the viewer to the outer edges of the universe. Every ten seconds we view the starting point from ten times farther out until our own galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, we move inward - into the hand of the sleeping picnicker - with ten times more magnification every two seconds. Our journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell."

References

Illustrations

Worked examples

Example 1 — a first encounter with Power of 10

Start with the simplest possible case. Write down what Power of 10 claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Power of 10 before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Power of 10 ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Power of 10

In research: Power of 10 appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Power of 10 in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Power of 10 is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Orders of magnitude, Powers of ten, so understanding it makes those chapters shorter.
In everyday life: Look for Power of 10 outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Power of 10 in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Power of 10 means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Power of 10 out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Power of 10 in simple terms?

Why does Power of 10 matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Power of 10?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Power of 10.