Scientific notation
A blue whale can weigh about $180.000$ kilograms. A red blood cell is barely $0,0000075$ metres in diameter. Counting zeros is slow and it is easy to make mistakes. Scientists write these quantities like this: $1,8 \cdot 10^5$ kg and $7,5 \cdot 10^{-6}$ m. Shorter, clearer, fewer errors.
What scientific notation is
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Large numbers → positive exponent: $180.000 = 1,8 \cdot 10^{5}$.
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Small numbers (less than $1$) → negative exponent: $0,0000075 = 7,5 \cdot 10^{-6}$.
For a number to be correctly written in scientific notation, it must have a single non-zero digit before the decimal point. State which of the following numbers are…
For a number to be correctly written in scientific notation, it must have a single non-zero digit before the decimal point.
State which of the following numbers are correctly written in scientific notation:
- a) $12,5 \cdot 10^4$
- b) $3,07 \cdot 10^8$
- c) $0,95 \cdot 10^3$
- d) $7 \cdot 10^{11}$
The blue whale is the largest animal on the planet and can weigh up to around $180\,000$ kilograms. Write this quantity in scientific notation.
The blue whale is the largest animal on the planet and can weigh up to around $180\,000$ kilograms. Write this quantity in scientific notation.
Converting to scientific notation and back
The exponent counts how many places the decimal point moves.
Express the following astronomical distances, given in scientific notation, in ordinary decimal notation (writing all their digits): a) Distance to the Moon:…
Express the following astronomical distances, given in scientific notation, in ordinary decimal notation (writing all their digits):
- a) Distance to the Moon: $3,84 \cdot 10^5$ km
- b) Distance to the Sun: $1,5 \cdot 10^8$ km
Write the following scientific data, taken from a technology magazine, in ordinary decimal form (extended form): a) The number of transistors in an advanced…
Write the following scientific data, taken from a technology magazine, in ordinary decimal form (extended form):
- a) The number of transistors in an advanced microprocessor is $4,5 \cdot 10^9$.
- b) The transfer speed of a fibre optic cable is $2,4 \cdot 10^7$ bits per second.
- c) The storage capacity of a server is $1,25 \cdot 10^{12}$ bytes.
Comparing numbers in scientific notation
First look at the exponent: it decides. Only if the exponents are equal do you compare the mantissa.
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$9,5\cdot10^3 < 9,45\cdot10^4$: even though $9,5 < 9,45$ is not true, $10^3 < 10^4$ decides.
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$3\cdot10^7 > 2,99\cdot10^7$: same exponent, so the mantissa decides.
Complete in your notebook with the symbol $>$ (greater than) or $<$ (less than), comparing these numbers in scientific notation: a) $4,2 \cdot 10^5$ \\\\_…
Complete in your notebook with the symbol $>$ (greater than) or $<$ (less than), comparing these numbers in scientific notation:
- a) $4,2 \cdot 10^5$ ____ $1,8 \cdot 10^6$
- b) $9,5 \cdot 10^3$ ____ $9,45 \cdot 10^3$
- c) $3 \cdot 10^7$ ____ $2,99 \cdot 10^7$
Calculating in scientific notation
Multiplying and dividing: you work with the mantissas separately and with the powers of $10$ separately (adding or subtracting exponents). At the end, you adjust the mantissa if it falls outside the range from $1$ to $10$.
Adding and subtracting requires the powers of $10$ to be the same; if they are not, one of them has to be changed.
A famous influencer has reached $2,4 \cdot 10^6$ followers on their main account. If they have $850\,000$ followers on their secondary account, how many followers do…
A famous influencer has reached $2,4 \cdot 10^6$ followers on their main account. If they have $850\,000$ followers on their secondary account, how many followers do they have in total? Express the final result in scientific notation.
A computer performs approximately $5 \cdot 10^9$ operations per second. How many operations will it perform in one minute ($60$ seconds)? Express the result in…
A computer performs approximately $5 \cdot 10^9$ operations per second. How many operations will it perform in one minute ($60$ seconds)? Express the result in scientific notation.
A red blood cell has a diameter of approximately $7,5 \cdot 10^{-6}$ metres. If we could line up $100$ million red blood cells side by side, what total length would they…
A red blood cell has a diameter of approximately $7,5 \cdot 10^{-6}$ metres. If we could line up $100$ million red blood cells side by side, what total length would they reach? Express the final result in scientific notation and in kilometres.