Scientific notation makes it quick to compare very large and very small numbers, because the exponent tells you the size at a glance.
Por qué funciona
Every first factor is at least 1 and less than 10. So any number with $10^{6}$ is between one million and ten million, and any number with $10^{7}$ is at least ten million. No first factor can make up a whole power of 10, so the exponent decides first.
Idea clave
To compare two positive numbers in scientific notation, compare the exponents first; the larger exponent wins. Only when the exponents match do you compare the first factors. To find how many times as large one number is, divide.
Worked examples
Ejemplo resuelto: Exponents first
Which is greater: $9.2 \times 10^{5}$ or $1.4 \times 10^{6}$?
- Compare the exponents: 6 is greater than 5.
- In standard form: 920,000 and 1,400,000.
Respuesta: $1.4 \times 10^{6}$
Ejemplo resuelto: Negative exponents
Which is greater: $3.2 \times 10^{-4}$ or $8.7 \times 10^{-5}$?
- $-4$ is greater than $-5$ (it is closer to 0).
- In standard form: 0.00032 and 0.000087.
Respuesta: $3.2 \times 10^{-4}$
Ejemplo resuelto: How many times as large?
The United States has about $3 \times 10^{8}$ people. The whole world has about $8 \times 10^{9}$. About how many times as many people live in the world?
- Divide: $\dfrac{8 \times 10^{9}}{3 \times 10^{8}} = \dfrac{8}{3} \times 10^{9-8}$.
- $\frac{8}{3} \approx 2.7$, so the quotient is about $2.7 \times 10 = 27$.
- With these rounded values, about 27 times as many.
Respuesta: about 27 times
Ejemplo resuelto: Estimate with one digit
Estimate $4{,}870{,}000$ as a single digit times a power of 10.
- In scientific notation it is $4.87 \times 10^{6}$.
- Round 4.87 to the nearest whole number: 5.
Respuesta: about $5 \times 10^{6}$
Error común
Comparing the first factors before the exponents, and deciding that $9 \times 10^{3}$ is greater than $2 \times 10^{5}$ because 9 is greater than 2. Write them out to check: 9,000 and 200,000.