How much is 7×10²⁷? Huge numbers without fear

♣ Newbie

Take one grain of sand. Put ten in a row: you have 10. Lay ten rows side by side and you get a slab of 100. Stack ten slabs and you have a cube of 1,000 grains.

Each time you do that, you multiply by ten, and the number grows one zero. That’s the whole trick: every extra zero is ×10.

So instead of writing the zeros one by one, you count them. 1,000 has three zeros, so it’s written 10³ (“ten to the third”). A million has six: 10⁶. And 7×10²⁷ means “a 7 with 27 zeros after it”.

Counting zeros avoids the silliest mistake with big numbers: skipping one without noticing, which makes you wrong by a factor of ten.

♦ Intermediate

Writing 7 followed by 27 zeros is awkward and easy to get wrong. Scientific notation fixes it: a number in front, between 1 and 10, times 10 raised to an exponent. The exponent tells you how many places the decimal point moves.

• 7,000,000 = 7×10⁶: the point jumps 6 places to the left.
• 0.0003 = 3×10⁻⁴: a negative exponent means the point jumps the other way. It’s a small number, not a negative one.

To calculate, multiplying adds the exponents and dividing subtracts them: (2×10³) × (4×10⁵) = 8×10⁸.

The decimals of the number in front tell you how much you know: 7×10²⁷ means “about 7”; 7.0×10²⁷ means “7, to a tenth”.

Calculators and programs write the same thing with an E: 7e27 is 7×10²⁷.

♠ Technical

Every real x ≠ 0 can be written uniquely as x = a·10ⁿ with 1 ≤ |a| < 10 and n ∈ ℤ. Taking logs, log₁₀|x| = n + log₁₀|a|, with 0 ≤ log₁₀|a| < 1: the exponent is the integer part of the logarithm, n = ⌊log₁₀|x|⌋, and the mantissa holds the fractional part. Scientific notation is a pocket table of logarithms, which is why a product reduces to adding exponents.

Significant figures live in the mantissa: 7×10²⁷ claims one figure and 7.0×10²⁷ claims two. Writing 7,000,000,000,000,000,000,000,000,000 seems to claim 28 exact figures, and nobody knows them.

Computers use the same idea in base 2. A double (IEEE 754) stores a 53-bit binary mantissa and an exponent: its maximum is 2¹⁰²⁴ − 2⁹⁷¹ ≈ 1.8×10³⁰⁸, and integers are only exact up to 2⁵³ − 1 (MDN). 10²⁷ fits easily, but in JavaScript 7e27 + 1 === 7e27 is true: the 1 falls below the precision.

♣ Newbie

Zeros in groups of three have names, and you already use several without thinking:

• kilo is a thousand: a kilometre is a thousand metres.
• mega is a million: a 12-megapixel photo has 12 million dots.
• giga is a billion: a gigabyte is a billion bytes.
• tera is a trillion: a one-terabyte drive holds a trillion bytes.

Each step is ×1,000.

And a trap worth knowing if you ever read Spanish: their billón is not your billion. It means a million millions, a thousand times more. When a translated news story mentions “un billón de dólares”, it may mean a trillion dollars.

♦ Intermediate

A prefix is a word stuck in front of a unit that multiplies it by a power of ten. Each has a one-letter symbol (two for deca, da). The International System (SI) has 24.

The four at the ends, ronna, quetta, ronto and quecto, were added in 2022, because the world’s data will soon go past 10²⁴.

Two details matter:

• Lower case and upper case are not the same: m is milli (a thousandth) and M is mega (a million). Between 1 mm and 1 Mm there’s a factor of a billion.
• Kilo is always 1,000. Computing sometimes uses “kilo” for 1,024, but there’s a separate prefix for that, kibi (Ki). A gibibyte versus a gigabyte is 7.4% bigger.

♠ Technical

The 24 SI prefixes are powers of 10³, except hecto, deca, deci and centi (10², 10¹, 10⁻¹, 10⁻²), which survive through use. The 27th General Conference on Weights and Measures (2022) added R, Q, r and q, for two stated reasons: data science will soon need to express quantities beyond 10²⁴, and unofficial names had to be headed off before they caught on.

The SI Brochure sets three rules that are often broken:

• The prefix symbol and the unit symbol form an inseparable symbol, and a power applies to both: 1 km² = (10³ m)² = 10⁶ m², not 10³ m². Likewise, 2.3 cm³ = 2.3×10⁻⁶ m³.
• Prefixes don’t stack: nm, not mµm.
• The kilogram is the only base unit that already has a prefix, so multiples are formed on the gram: 10⁻⁶ kg is mg, not µkg.

SI prefixes are strictly powers of 10: “one kilobit represents 1000 bits and not 1024 bits”. Powers of 2 get their own prefixes, from kibi (2¹⁰) to quebi (2¹⁰⁰), defined in IEC 80000-13. The gap grows with size: 2.4% for kibi versus kilo, 7.4% for gibi versus giga and 26.8% for quebi versus quetta… although nobody runs into that last one yet.

♣ Newbie

“A million” and “a billion” sound alike. To feel the difference, turn them into time:

• A thousand seconds is about 17 minutes: a coffee.
• A million seconds is almost 12 days: a holiday.
• A billion seconds is almost 32 years: half a lifetime.
• A trillion seconds is about 31,700 years.

Between a million and a billion there’s the same gap as between a holiday and half a lifetime. Each step of the staircase is ×1,000, however similar the words sound.

That’s why, to compare very different things, you look at how many zeros they have, not at the number itself. Two things with the same number of zeros are “the same size”, even if one is twice the other.

♦ Intermediate

The order of magnitude of a number is the nearest power of 10. A person is on the order of 10⁰ m (one metre) tall, the Earth on the order of 10⁷ m across and the observable universe on the order of 10²⁶ m. Two things one order of magnitude apart differ by a factor of ten.

To draw them together you use a log scale: each step multiplies by the same amount, say ×10 or ×1,000. On an ordinary (linear) ruler from 0 to a billion, a million lands a thousandth of the way along: stuck to zero. On a log scale, a thousand, a million and a billion are evenly spaced, because from each to the next it’s the same ×1,000.

Both rulers are useful, for different things. The linear one shows differences (“how much more”); the log one shows ratios (“how many times more”).

A 2013 study asked American adults to place large numbers on a linear ruler. About half did it right. Nearly all the others put a million around the middle, as if a thousand, a million and a billion were evenly spaced: they were following the words, not the numbers.

♠ Technical

With u = log₁₀ x, multiplying by k shifts by a fixed distance log₁₀ k. A log scale turns ratios into differences and powers into slopes: on log-log axes, y ∝ xᵖ is a straight line of slope p.

OpenStax defines the order of magnitude as round(log₁₀ x). Watch the threshold: rounding flips at √10 ≈ 3.16, not at 5. So 9,000 is of order 4 even though its scientific exponent is 3. By that rule, a proton (10⁻¹⁵ m) and the observable universe (10²⁶ m) are log₁₀(10²⁶/10⁻¹⁵) = 41 orders of magnitude apart.

Landy, Silbert and Goldin (Cognitive Science, 2013) found that about half of participants placed numbers from 10³ to 10⁹ linearly and that nearly all the rest followed a segmented linear heuristic: linear within each word (“millions”, “billions”) and evenly spaced between words. The segmented group was also more optimistic about ineffective policy strategies. The authors attribute it to the lexical structure of number names, not to poor arithmetic.

♣ Newbie

How many times does your heart beat in a whole life? You don’t need to know to work it out roughly:

1. Your heart beats about once a second.
2. 80 years is about 2.5 billion seconds.
3. Multiply: about 2.5 billion beats.

It isn’t exact. Maybe it’s 2 billion, maybe 3 billion. But it’s certainly not 20 million or 200 billion. You’ve got the zeros right, and for comparing, that’s what matters.

This is called a Fermi estimate, after the physicist Enrico Fermi. The trick is to break an impossible question into little pieces you do know, and multiply them.

♦ Intermediate

The Fermi method has four rules:

1. Break the problem into factors you can pin down.
2. If you don’t know a factor, bound it between a minimum and a maximum that make sense and take a value in between.
3. Use a single figure: you’re after the zeros, not the decimals.
4. Check against something you know at the end.

It works better than it looks because errors partly cancel out: if you overshoot on one factor and undershoot on another, the product gets closer.

Now, the question from the start: where does 7×10²⁷ come from? To count atoms you need the mole: a bundle of 6×10²³ things, the way a dozen is a bundle of 12. A mole of water weighs about 18 g, and each water molecule has 3 atoms (two hydrogen and one oxygen).

Pretend a 70 kg person is all water: 70,000 g ÷ 18 g ≈ 3,900 moles of water, × 3 ≈ 11,700 moles of atoms, × 6×10²³ ≈ 7×10²⁷ atoms. It matches the published estimate for the human body. With the real composition (more oxygen and carbon by mass, which are heavier) you get 6.7×10²⁷: the same number of zeros.

♠ Technical

A Fermi estimate is a product N = Π fᵢ, and on a log scale, a sum: log N = Σ log fᵢ. If each factor has an independent multiplicative error with standard deviation σ in log, the total grows as √n·σ, not n·σ. With five factors known “to within a factor of 2”, the result typically lands within a factor of 2^√5 ≈ 4.7: inside the order of magnitude.

For an unknown factor bounded between x_min and x_max, the geometric mean √(x_min·x_max) is the midpoint on a log scale. OpenStax uses it for the mass of a moose: between 10² and 10³ kg, ≈ 3×10² kg.

The atoms in a 70 kg body, with two independent models:

• All water: 70,000 / 18.015 × 3 × N_A = 7.0×10²⁷.
• Composition by mass (O 61%, C 23%, H 10%, N 2.6%, WebElements) with conventional atomic weights: 6.7×10²⁷. The remaining 3.4% are heavy atoms (Ca, P, K…) and add less than 1% to the count.

Both land next to Freitas’s ≈ 7×10²⁷ (Nanomedicine I, 3.1). The Avogadro constant has been exact since the 2019 SI redefinition, so all the uncertainty is in the model, not in N_A.