MISSION · Number sense
Scientific Notation
A way to write numbers too big or too small to read
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Every number is written as a value between 1 and 10 multiplied by a power of ten, so the digits that matter are separated from the zeros that only record size.
\[4{,}500{,}000 = 4.5 \times 10^6\]The two rules
The first part, the mantissa, must be at least 1 and less than 10. Exactly one non-zero digit before the decimal point. \(45 \times 10^5\) is the right value written the wrong way; \(4.5 \times 10^6\) is correct.
The second part is a power of ten. Positive powers make the number bigger, negative ones make it smaller:
\[0.00072 = 7.2 \times 10^{-4}\]A negative exponent never means a negative number — it means a small one. \(-3 \times 10^4\) is a large negative number; \(3 \times 10^{-4}\) is a small positive one.
Converting
Count how far the decimal point has to move to leave one digit in front of it. That count is the exponent, and its sign is which way you moved.
- \(93{,}000{,}000\) → point moves 7 left → \(9.3 \times 10^7\)
- \(0.000045\) → point moves 5 right → \(4.5 \times 10^{-5}\)
Going back, a positive exponent moves the point right, a negative one moves it left.
Calculating with it
This is where the notation earns its place, because the exponent laws do the heavy lifting.
Multiply: multiply the mantissas, add the exponents. \((3 \times 10^4)(2 \times 10^5) = 6 \times 10^9\)
Divide: divide the mantissas, subtract the exponents. \(\frac{8 \times 10^7}{4 \times 10^3} = 2 \times 10^4\)
Then tidy up if the mantissa has left its range: \((5 \times 10^3)(4 \times 10^6) = 20 \times 10^9\), which becomes \(2 \times 10^{10}\).
Adding is the awkward one, because you can’t add the mantissas unless the exponents match. Convert to the same power of ten first, then add.
Orders of magnitude
The exponent alone tells you the scale, and that’s often the only comparison you need. Something at \(10^6\) is a thousand times bigger than something at \(10^3\), whatever the mantissas are.
Scientists use this constantly. A bacterium is about \(10^{-6}\) m, a person \(10^0\), the Earth \(10^7\), the distance to the sun \(10^{11}\). Those four numbers span seventeen orders of magnitude, and comparing them in ordinary decimals would be unreadable.
Common mix-ups
- Leaving the mantissa outside 1 to 10. \(0.45 \times 10^7\) isn’t standard form either.
- Counting the zeros instead of the decimal moves. They agree for whole numbers and disagree for decimals.
- Reading a negative exponent as a negative number.
- Adding exponents when adding numbers. That law is for multiplication.
Check yourself
Write \(0.00031\) in scientific notation, and work out \((6 \times 10^8) \div (3 \times 10^{-2})\).
\(3.1 \times 10^{-4}\). For the division, \(6 \div 3 = 2\) and \(8 - (-2) = 10\), giving \(2 \times 10^{10}\).
LEVEL 3 · REFLECT
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