Operations with Scientific Notation

Handle the front numbers and the powers of ten separately, then tidy the result back into proper form.

Light travels 300 000 000 metres every second.

Read that number again and say how many zeros it had. Most people have to go back and count.

how many zeros did you just read? 300 000 000 the speed of light, in metres per second count the zeros. Now look away and say how many there were. 3 × 10⁸

Now write it as 3 × 108. The 8 does the counting for you, and an 8 cannot slip the way a row of zeros can.

It says: start at 3 and slide the decimal point eight places right.

The same trick runs the other way. A virus is 0.00000012 m across — or 1.2 × 10-7, where the minus means slide left.

This is why every number in science is written this way. Not to look clever, but because comparing two exponents is easy and comparing two rows of zeros is not.

Doubling: each step multiplies by two 22^142^282^3162^4322^5 an exponent counts factors — each step multiplies by 2, it does not add 2 2^5 = 32, not 10

The form itself

The front number must be at least 1 and less than 10. So 4.7 × 104 is correct; 47 × 103 is not, even though both equal 47,000.

A positive exponent means a big number. A negative exponent means a small one, between 0 and 1.

Converting: count how many places the decimal point has to move. Left moves make the exponent go up.

Negative exponent does not mean negative number

3 × 10-5 is 0.00003 — small, but positive. Only a minus in front of the 3 would make it negative.

Multiplying and dividing

Multiply the front numbers; add the exponents. Divide the front numbers; subtract the exponents.

(3 × 105)(2 × 107) = 6 × 1012.

If the new front number falls outside 1 to 10, fix it: 20 × 104 becomes 2 × 105.

Always tidy at the end

A front number of 20 or of 0.5 means you are not finished. Move one factor of ten across and adjust the exponent.

Adding and subtracting

You may only add the front numbers when the exponents match. 3 × 106 + 4 × 106 = 7 × 106.

If they differ, rewrite one so they agree. 3 × 106 + 4 × 105 = 3 × 106 + 0.4 × 106 = 3.4 × 106.

This is the same logic as needing a common denominator before adding fractions.

Match, add, tidy

Three steps in that order. Trying to add mismatched exponents gives answers that are wrong by factors of ten.

Comparing sizes

Compare the exponents first. A larger exponent wins outright, whatever the front numbers are.

Only when the exponents are equal do you compare the front numbers.

"How many times bigger?" means divide: 6 × 1082 × 105 = 3 × 103, so three thousand times.

Worked examples

Multiplying

Compute (5 × 106)(4 × 10-3) in scientific notation.

  1. Front numbers: 5 × 4 = 20 Multiply the coefficients.
  2. Powers: 106 · 10-3 = 103 Add the exponents: 6 + (-3) = 3.
  3. So far 20 × 103, but 20 is too big. The front must sit between 1 and 10.
  4. 20 = 2 × 101, so the answer is 2 × 104 One more factor of ten moves across.

2 × 104

Adding with different exponents

Compute 7.2 × 105 + 3 × 104.

  1. The exponents differ, so rewrite the smaller one. You cannot add until they match.
  2. 3 × 104 = 0.3 × 105 Move the decimal left one place, raise the exponent by one.
  3. 7.2 × 105 + 0.3 × 105 = 7.5 × 105 Now add the front numbers.
  4. 7.5 is between 1 and 10, so it is finished. No tidying needed.

7.5 × 105

How many times bigger?

The Sun is about 1.4 × 106 km across; the Earth about 1.3 × 104 km. How many times wider is the Sun?

  1. Divide: 1.4 × 1061.3 × 104 "How many times" means divide.
  2. Front numbers: 1.4 ÷ 1.3 ≈ 1.08 Just over one.
  3. Powers: 106 ÷ 104 = 102 Subtract the exponents.
  4. ≈ 1.08 × 102 ≈ 108 times About a hundred times wider.

About 108 times

Words to know

Scientific notation
a × 10n where 1 ≤ a < 10 and n is an integer.
Order of magnitude
The power of ten — roughly how many digits the number has.
Coefficient
The front number a.
Standard form
The ordinary way of writing a number, like 47,000.

Mistakes to watch for

Leaving the front number outside 1 to 10. 47 × 103 has the right value but is not scientific notation — and two numbers written that way cannot be compared by exponent, which was the entire purpose.

Adding the front numbers while the exponents still differ. 3 × 106 + 4 × 105 is not 7 × 106. Rewrite one so they match first — it is the same reason fractions need a common denominator.

Multiplying the exponents instead of adding them. 103 × 104 = 107, because you are stacking three tens next to four tens. Multiplying gives 1012, which is out by a factor of a hundred thousand.

Reading a negative exponent as a negative number. 3 × 10-5 is 0.00003 — small, but firmly positive. Only a minus in front of the 3 would make the number itself negative.

Archimedes wrote The Sand Reckoner around 250 BC to answer a taunt: that the number of grains of sand was beyond counting.

He invented a notation for enormous numbers, then estimated the grains needed to fill the universe as about 1063.

He had no zero, no decimal point and no exponent notation. He built the whole system from scratch just to make the point.

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