15.1: Number Talk: Non-zero Digits
Mentally decide how many non-zero digits each number will have.
$(3 \times 10^9)(2 \times 10^7)$
$ (3 \times 10^9) \div (2 \times 10^7)$
$3 \times 10^9 + 2 \times 10^7$
Let’s add and subtract using scientific notation to answer questions about animals and the solar system.
Mentally decide how many non-zero digits each number will have.
$(3 \times 10^9)(2 \times 10^7)$
$ (3 \times 10^9) \div (2 \times 10^7)$
$3 \times 10^9 + 2 \times 10^7$
Diego, Kiran, and Clare were wondering:
“If Neptune and Saturn were side by side, would they be wider than Jupiter?”
Kiran wrote $4.7 \times 10^4$ as 47,000 and $1.2 \times 10^5$ as 120,000 and added them: $$\begin{align} 120,\!000& \\ + 47,\!000& \\ \hline 167,\!000& \end{align}$$
object | diameter (km) | distance from the Sun (km) | |
---|---|---|---|
row 1 | Sun | $1.392 \times 10^6$ | $0 \times 10^0$ |
row 2 | Mercury | $4.878 \times 10^3$ | $5.79 \times 10^7$ |
row 3 | Venus | $1.21 \times 10^4$ | $1.08 \times 10^8$ |
row 4 | Earth | $1.28 \times 10^4$ | $1.47 \times 10^8$ |
row 5 | Mars | $6.785 \times 10^3$ | $2.28 \times 10^8$ |
row 6 | Jupiter | $1.428 \times 10^5$ | $7.79 \times 10^8$ |
The emcee at a carnival is ready to give away a cash prize! The winning contestant could win anywhere from \$1 to \$100. The emcee only has 7 envelopes and she wants to make sure she distributes the 100 \$1 bills among the 7 envelopes so that no matter what the contestant wins, she can pay the winner with the envelopes without redistributing the bills. For example, it’s possible to divide 6 \$1 bills among 3 envelopes to get any amount from \$1 to \$6 by putting \$1 in the first envelope, \$2 in the second envelope, and \$3 in the third envelope (Go ahead and check. Can you make \$4? \$5? \$6?).
Use the table to answer questions about different life forms on the planet.
Row 1 | creature | number | mass of one individual (kg) |
---|---|---|---|
Row 2 | humans | $7.5 \times 10^9$ | $6.2 \times 10^1$ |
Row 3 | cows | $1.3 \times 10^9$ | $4 \times 10^2$ |
Row 4 | sheep | $1.75 \times 10^9$ | $6 \times 10^1$ |
Row 5 | chickens | $2.4 \times 10^{10}$ | $2 \times 10^0$ |
Row 6 | ants | $5 \times 10^{16}$ | $3 \times 10^{\text -6}$ |
Row 7 | blue whales | $4.7 \times 10^3$ | $1.9 \times 10^5$ |
Row 8 | antarctic krill | $7.8 \times 10^{14}$ | $4.86 \times 10^{\text -4}$ |
Row 9 | zooplankton | $1 \times 10^{20}$ | $5 \times 10^{\text -8}$ |
Row 10 | bacteria | $5 \times 10^{30}$ | $1 \times 10^{\text -12}$ |
When we add decimal numbers, we need to pay close attention to place value. For example, when we calculate $13.25 + 6.7$, we need to make sure to add hundredths to hundredths (5 and 0), tenths to tenths (2 and 7), ones to ones (3 and 6), and tens to tens (1 and 0). The result is 19.95.
We need to take the same care when we add or subtract numbers in scientific notation. For example, suppose we want to find how much further the Earth is from the Sun than Mercury. The Earth is about $1.5 \times 10^8$ km from the Sun, while Mercury is about $5.8 \times 10^7$ km. In order to find $$1.5\times 10^8 - 5.8 \times 10^7$$ we can rewrite this as $$1.5 \times 10^8 - 0.58 \times 10^8$$ Now that both numbers are written in terms of $10^8$, we can subtract 0.58 from 1.5 to find $$0.92 \times 10^8$$ Rewriting this in scientific notation, the Earth is $$9.2 \times 10^7$$ km further from the Sun than Mercury.