Different Exponents

Which is greater: $3.1 \times 10^6$ or $9.8 \times 10^5$?

Check that both are normalized: Both coefficients, $3.1$ and $9.8$, are between $1$ and $10$ = Ready to compare

Compare the exponents: $6 > 5$ = $10^6$ is ten times $10^5$

Decide (the coefficients do not matter here): $3.1 \times 10^6 = 3{,}100{,}000$ and $9.8 \times 10^5 = 980{,}000$ = $3.1 \times 10^6 > 9.8 \times 10^5$

Answer: $3.1 \times 10^6$ is greater, even though its coefficient is smaller.

Equal Exponents

Compare $4.7 \times 10^8$ and $4.2 \times 10^8$.

Compare the exponents: Both are $8$ = Equal exponents

Compare the coefficients: $4.7 > 4.2$ = The first coefficient is greater

Write the answer: Same power of ten, larger coefficient = $4.7 \times 10^8 > 4.2 \times 10^8$

Answer: $4.7 \times 10^8 > 4.2 \times 10^8$

Normalize First

Compare $45 \times 10^3$ and $4.6 \times 10^4$.

Spot the number that is not normalized: $45$ is not between $1$ and $10$ = $45 \times 10^3$ must be rewritten

Normalize it: $45 = 4.5 \times 10^1$, so $45 \times 10^3 = 4.5 \times 10^{1+3}$ = $4.5 \times 10^4$

Compare the exponents: $4$ and $4$ = Equal, so compare coefficients

Compare the coefficients: $4.5 < 4.6$ = $45 \times 10^3 < 4.6 \times 10^4$

Answer: $45 \times 10^3 < 4.6 \times 10^4$ ($45{,}000 < 46{,}000$). Comparing $45$ with $4.6$ directly would give the wrong answer.

Negative Exponents

Which is greater: $2 \times 10^{-3}$ or $9 \times 10^{-5}$?

Check that both are normalized: $2$ and $9$ are between $1$ and $10$ = Ready to compare

Compare the exponents like negative numbers: $-3$ is to the right of $-5$ on the number line = $-3 > -5$

Decide: $2 \times 10^{-3} = 0.002$ and $9 \times 10^{-5} = 0.00009$ = $2 \times 10^{-3} > 9 \times 10^{-5}$

Answer: $2 \times 10^{-3}$ is greater. With negative exponents, the exponent closer to zero gives the larger number.

Ordering Sizes in the Universe

Order from smallest to largest (in meters): Earth's diameter $1.27 \times 10^7$, a hydrogen atom $1 \times 10^{-10}$, the Earth-Sun distance $1.5 \times 10^{11}$, a red blood cell $70 \times 10^{-7}$. Then find how many times wider the red blood cell is than the atom.

Normalize the red blood cell: $70 \times 10^{-7} = 7 \times 10^{1} \times 10^{-7}$ = $7 \times 10^{-6}$

List the exponents: Earth $7$, atom $-10$, Sun distance $11$, cell $-6$ = $-10 < -6 < 7 < 11$

Order the numbers by their exponents: No two exponents are equal, so the coefficients are not needed = atom, cell, Earth, Earth-Sun distance

Compare the powers of ten of cell and atom: $10^{-6} \div 10^{-10} = 10^{-6-(-10)} = 10^4$ = $10^4 = 10{,}000$

Include the coefficients: $\frac{7}{1} \times 10^4 = 7 \times 10^4$ = $70{,}000$ times

Answer: $1 \times 10^{-10} < 7 \times 10^{-6} < 1.27 \times 10^7 < 1.5 \times 10^{11}$. The red blood cell is $70{,}000$ times as wide as the hydrogen atom.

Mistake: Comparing coefficients first: thinking $9.8 \times 10^5 > 3.1 \times 10^6$ because $9.8 > 3.1$

Why: The power of ten decides the size. $10^6$ is ten times $10^5$, and no coefficient below $10$ can make up for that.

Correct: Compare exponents first: $6 > 5$, so $3.1 \times 10^6 > 9.8 \times 10^5$. Look at the coefficients only when the exponents are equal.

Mistake: Comparing without normalizing: thinking $45 \times 10^3 > 4.6 \times 10^4$ because $45 > 4.6$

Why: $45 \times 10^3$ is not in scientific notation, so its exponent does not tell its real size.

Correct: Normalize first: $45 \times 10^3 = 4.5 \times 10^4$. Then $4.5 < 4.6$, so $45 \times 10^3 < 4.6 \times 10^4$.

Mistake: Thinking $10^{-5} > 10^{-3}$ because $5 > 3$

Why: Negative exponents are compared like negative numbers: $-5$ is less than $-3$.

Correct: $10^{-5} = 0.00001$ and $10^{-3} = 0.001$, so $10^{-3} > 10^{-5}$. The exponent closer to zero wins.

Mistake: Thinking $10^6$ is twice $10^3$, or $3$ more than $10^3$

Why: The exponents differ by $3$, but each step in the exponent multiplies by $10$, it does not add.

Correct: $\frac{10^6}{10^3} = 10^{6-3} = 10^3$, so $10^6$ is $1000$ times $10^3$.

Biology: Bacteria and Viruses

Microbiologists compare tiny lengths written with negative exponents to choose the right microscope.

A light microscope shows things down to about $2 \times 10^{-7}$ m. A bacterium of $2 \times 10^{-6}$ m is larger ($-6 > -7$), so it can be seen. Most viruses are smaller and need an electron microscope.

Computing: Storage Sizes

Storage is measured in bytes: a gigabyte is $10^9$ bytes and a terabyte is $10^{12}$ bytes. Comparing exponents tells you how many files fit.

A photo of $3 \times 10^6$ bytes is much smaller than a video of $3 \times 10^9$ bytes: same coefficient, and the exponents differ by $3$, so the video is $1000$ times larger.

Physics: Laser Wavelengths

Light wavelengths are tiny and often written in different forms, for example in nanometers ($10^{-9}$ m). To compare them, normalize first.

Blue light has a wavelength of about $4.5 \times 10^{-7}$ m and red light about $7 \times 10^{-7}$ m. Same exponent, so $7 > 4.5$ shows red light has the longer wavelength.

Q: Why does the exponent matter more than the coefficient?

A: A normalized coefficient is always less than $10$, while one step up in the exponent multiplies by $10$. So $9.99 \times 10^5 = 999{,}000$ is still less than $1 \times 10^6 = 1{,}000{,}000$.

Q: What if one number is not in proper scientific notation?

A: Normalize it before comparing. Move the decimal point until the coefficient is between $1$ and $10$ and change the exponent to match: $0.8 \times 10^5 = 8 \times 10^4$, $45 \times 10^3 = 4.5 \times 10^4$.

Q: Is a number with a negative exponent always smaller than one with a positive exponent?

A: Yes, for positive numbers. A normalized number with a negative exponent is less than $1$ (for example $9.9 \times 10^{-1} = 0.99$), and one with exponent $0$ or more is at least $1$.

Q: What does "two orders of magnitude bigger" mean?

A: It means about $10^2 = 100$ times bigger. The order of magnitude is roughly the exponent, so a difference of $2$ in the exponents means a factor of about $100$.