M8MATH-2.3

Writing & Comparing Numbers in Scientific Notation

Learn to write very large and very small numbers in scientific notation, convert between forms, and compare quantities using powers of 10.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Writing & Comparing Numbers in Scientific Notation, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Scientific notation is how scientists, engineers, and economists write numbers that are too huge or too tiny to write comfortably in standard form. The distance from Earth to the sun is about 150 million kilometers — that's 150,000,000 km in standard form, but just 1.5 × 10⁸ km in scientific notation. A virus might measure 0.0000001 meters, which is 1 × 10⁻⁷ m. In this lesson, you'll learn to convert numbers to and from scientific notation, understand what each part means, and compare gigantic or microscopic quantities by figuring out how many times larger or smaller one is than another.

What Scientific Notation Is and Why It Works

Scientific notation writes a number as the product of two parts: a coefficient aa where 1≤a<101 \leq a < 10, and a power of 10. For example, 4,500 becomes 4.5×1034.5 \times 10^3, and 0.0032 becomes 3.2×10−33.2 \times 10^{-3}.

The key rule is that the coefficient must be at least 1 and less than 10. That means it's always a single nonzero digit, followed by a decimal point and any other digits needed. The exponent on 10 tells you how many places to move the decimal point, and whether to move it right (positive exponent) or left (negative exponent).

Why does this work? Because 103=100010^3 = 1000, 10−2=0.0110^{-2} = 0.01, and so on. When you multiply the coefficient by a power of 10, you're really just moving the decimal point. Scientific notation uses this fact to express any real number compactly, which is especially useful when the number has many zeros.

Converting to Scientific Notation from Standard Form

To convert a number to scientific notation:

First, place the decimal point right after the first nonzero digit. That gives you your coefficient. Then count how many places you moved the decimal point from where it started.

If you moved it left (the original number was large), the exponent is positive. If you moved it right (the original number was small), the exponent is negative.

Example: Convert 28,000 to scientific notation. The first nonzero digit is 2. Place the decimal point after it: 2.8. Now count: you moved from 28,000.0 to 2.8, which is 4 places left. So the answer is 2.8×1042.8 \times 10^4. Check: 2.8×1000=28002.8 \times 1000 = 2800. Wait, that's not right—let me recount. From 28,000, the decimal point starts at the end (28,000.0), and you move it left 4 places to get 2.8000. That's 2.8×104=28,0002.8 \times 10^4 = 28,000. Correct.

Example: Convert 0.0045 to scientific notation. The first nonzero digit is 4. Place the decimal point after it: 4.5. Count from 0.0045 to 4.5: you moved right 3 places. So the exponent is −3-3, and the answer is 4.5×10−34.5 \times 10^{-3}.

Converting from Scientific Notation Back to Standard Form

To convert from scientific notation to standard form, use the exponent to tell you which direction and how far to move the decimal point.

Positive exponent: move the decimal point right. The exponent value is the number of places.

Negative exponent: move the decimal point left. The exponent value (ignoring the minus sign) is the number of places.

Example: Convert 6.7×1056.7 \times 10^5 to standard form. The exponent is 5 (positive), so move the decimal point 5 places to the right. Start with 6.7, and add zeros as needed: 6.70000→670,0006.70000 \to 670,000.

Example: Convert 8.2×10−48.2 \times 10^{-4} to standard form. The exponent is −4-4, so move the decimal point 4 places to the left. Start with 8.2, and add zeros as needed: 0.000820.00082. The decimal point moves from after the 8 to four places left, which puts it before the 8 with three zeros in between.

Comparing Numbers in Scientific Notation

Once both numbers are in scientific notation, you can compare them and find how many times larger or smaller one is than the other.

First, compare the exponents. The number with the larger exponent is larger. If the exponents are the same, compare the coefficients.

To find how many times as large one number is as another, divide the first by the second. The quotient tells you the multiplicative relationship.

Example: Is 3×1083 \times 10^8 larger or smaller than 6×1046 \times 10^4? Yes, the first one is much larger because 8>48 > 4. To find how many times as large: 3×1086×104=36×108104=0.5×104=5×103\frac{3 \times 10^8}{6 \times 10^4} = \frac{3}{6} \times \frac{10^8}{10^4} = 0.5 \times 10^4 = 5 \times 10^3. So 3×1083 \times 10^8 is 5×1035 \times 10^3 (or 5,000) times as large as 6×1046 \times 10^4.

A common mistake is to compare only the exponents and ignore the coefficients. If one number is 9.9×1049.9 \times 10^4 and another is 1.1×1051.1 \times 10^5, they have different exponents (4 and 5), so the second is larger. But they're much closer than you might think: 1.1×105÷9.9×104≈1.111.1 \times 10^5 \div 9.9 \times 10^4 \approx 1.11, so the second is only about 1.11 times as large.

Estimation and Rounding to Scientific Notation

Sometimes you don't need an exact answer—you need a rough estimate in scientific notation form. To estimate a large calculation, round each number to one significant figure (the first nonzero digit), write it as a single digit times a power of 10, and then multiply or divide.

Example: Estimate the product (417)(58,000)(417)(58,000). Round 417 to 4×1024 \times 10^2 (because 417 is close to 400). Round 58,000 to 6×1046 \times 10^4 (because 58,000 is close to 60,000). Then (4×102)(6×104)=24×106=2.4×107(4 \times 10^2)(6 \times 10^4) = 24 \times 10^6 = 2.4 \times 10^7. The exact answer is 24,186,000, which is about 2.4×1072.4 \times 10^7—our estimate is very close.

Estimation matters because it helps you catch mistakes. If you compute something and get an answer that is orders of magnitude off from your estimate, you know something went wrong.

Key terms

Scientific notation.
A way of writing a number as the product of a coefficient (between 1 and 10) and a power of 10, for example 3.2×1053.2 \times 10^5 or 7.1×10−37.1 \times 10^{-3}.
Coefficient.
The number between 1 and 10 that is multiplied by a power of 10 in scientific notation. In 5.6×1045.6 \times 10^4, the coefficient is 5.6.
Exponent.
The power to which 10 is raised in scientific notation. In 2.3×10−52.3 \times 10^{-5}, the exponent is −5-5.
Standard form.
The ordinary way of writing a number using place value, without exponents. For example, 45,000 is the standard form of 4.5×1044.5 \times 10^4.
Significant figure.
The digits in a number that carry meaningful information about its precision. The first significant figure is the first nonzero digit from the left.
Order of magnitude.
A rough measure of size based on the power of 10. Two numbers differing by one order of magnitude means one is about 10 times as large as the other.

Worked example

A particle accelerator produces about 8,400,000,000 collisions per second. Write this number in scientific notation. Then compare it to another accelerator that produces 2.1×1092.1 \times 10^9 collisions per second, and determine how many times as many collisions the first accelerator produces compared to the second.
Step 1: Convert 8,400,000,000 to scientific notation.

Identify the first nonzero digit: 8.

Place the decimal point right after it: 8.4 (we only include the first digit and then the start of the next significant figures).

Count how many places the decimal point moves from the original position (at the end of the whole number) to the new position (after the 8). Starting position: 8,400,000,000. (decimal point here). New position: 8.4 (decimal point here). That's 9 places to the left.

Since we moved left, the exponent is positive 9.

Answer: 8.4×1098.4 \times 10^9 collisions per second.

Step 2: Compare the two numbers.

First accelerator: 8.4×1098.4 \times 10^9

Second accelerator: 2.1×1092.1 \times 10^9

Both have the same exponent (9), so compare the coefficients: 8.4 is larger than 2.1, so the first accelerator produces more collisions.

Step 3: Find how many times as many.

Divide the first by the second:8.4×1092.1×109=8.42.1×109109=4×100=4\frac{8.4 \times 10^9}{2.1 \times 10^9} = \frac{8.4}{2.1} \times \frac{10^9}{10^9} = 4 \times 10^0 = 4The first accelerator produces 4 times as many collisions per second as the second one.

Practice questions

The thickness of a human hair is about 0.000075 meters. Write this in scientific notation.
  1. 7.5×10−57.5 \times 10^{-5}
  2. 7.5×10−67.5 \times 10^{-6}
  3. 0.75×10−40.75 \times 10^{-4}
  4. 7.5×1057.5 \times 10^{5}

Answer: 7.5×10−57.5 \times 10^{-5}

Starting with 0.000075, the first nonzero digit is 7. Place the decimal after it: 7.5. Count from the original decimal point (at the far left, before all the zeros) to the new position: that's 5 places to the right, making the exponent −5-5. So 7.5×10−57.5 \times 10^{-5} meters. Choice 7.5×10−67.5 \times 10^{-6} is a common error—counting the 5 as one of the places moved, when it actually requires moving 5 places total.
Two countries have populations that can be written in scientific notation: Country A has 4.2×1074.2 \times 10^7 people, and Country B has 8.5×1068.5 \times 10^6 people. How many times as large is Country A's population compared to Country B's?

Answer: Approximately 4.94 or about 5 times as large

Divide: 4.2×1078.5×106=4.28.5×107106=4.28.5×101\frac{4.2 \times 10^7}{8.5 \times 10^6} = \frac{4.2}{8.5} \times \frac{10^7}{10^6} = \frac{4.2}{8.5} \times 10^1. Now 4.28.5≈0.494\frac{4.2}{8.5} \approx 0.494, so the result is 0.494×10=4.940.494 \times 10 = 4.94. This can also be written as 4.94×1004.94 \times 10^0, or roughly 5×100=55 \times 10^0 = 5. Country A is about 5 times as populated as Country B. A mistake here is not handling the exponents correctly: if you subtract wrong, you might get 10010^0 instead of 10110^1, leading to an answer around 0.5 instead of 5.
Convert 3.6×10−33.6 \times 10^{-3} to standard form.

Answer: 0.0036

The exponent is −3-3, so move the decimal point 3 places to the left. Start with 3.6 (which is 3.6 with the decimal after the 3). Moving left 3 places: left one is 0.36, left two is 0.036, left three is 0.0036. You can think of it as adding zeros to the left and placing the decimal point to show the exponent magnitude. The answer is 0.0036.

FAQ

Why does the coefficient have to be between 1 and 10?
If the coefficient were larger (like 45 × 10⁴), that's not standard scientific notation—you could write it as 4.5 × 10⁵ instead, which is cleaner. If it were less than 1 (like 0.8 × 10⁵), you'd write it as 8 × 10⁴. The rule ensures every number has exactly one standard form, which makes it easy to compare and communicate.
What's the difference between 10⁵ and 10⁻⁵?
105=100,00010^5 = 100,000 is a large number (100 thousand). 10−5=0.0000110^{-5} = 0.00001 is a tiny number (one hundred-thousandth). The negative exponent means you're dividing by 10 that many times, or equivalently, moving the decimal point left. Positive exponents multiply; negative ones divide.
When I compare two numbers in scientific notation, do I always need to look at both the coefficient and the exponent?
Look at the exponents first. If they're different, the bigger exponent wins, and you don't even need to check the coefficient. If the exponents are the same, then you compare the coefficients. For example, 5×1065 \times 10^6 is obviously bigger than 9.9×1059.9 \times 10^5 because 6>56 > 5, even though 9.9 is bigger than 5.
Is 1.0 × 10⁰ the same as 1?
Yes. 100=110^0 = 1, so 1.0×100=1.0×1=11.0 \times 10^0 = 1.0 \times 1 = 1. Scientific notation can represent 1, but usually we just write 1 in standard form since it's simpler.

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