Math Vale

🔢 Number Kingdom · Number Sense

Scientific Notation

Write very large and very small numbers as a number between 1 and 10 times a power of ten, compare them by their exponents, and multiply and divide them.

In short

  • Scientific notation is one number from 1 up to (but not including) 10, multiplied by a power of ten.
  • The exponent counts the places the decimal point moves, not the zeros you can see.
  • A number smaller than 1 has a negative exponent; a number 10 or larger has a positive one.
  • Multiplying adds the exponents and dividing subtracts them, and then the front number is checked again.
  • Two numbers whose exponents differ by k differ by a factor of 10k, which is how quantities of wildly different size get compared.

A shorter name for enormous and tiny numbers

Some numbers are too long to read at a glance. 4,730,000,000 and 0.00000061 are both perfectly ordinary quantities — a population, a wavelength — but counting their zeros is slow and easy to get wrong.

Scientific notation writes any number as two parts multiplied together:

(a number from 1 up to but not including 10) × (a power of ten)

So 4,730,000,000 = 4.73 × 109 and 0.00000061 = 6.1 × 10-7.

The first part, often called the front number or the mantissa, carries the digits. The power of ten carries the size. Splitting the two apart is what makes these numbers easy to compare, multiply and say out loud.

Turning a large number into scientific notation

Move the decimal point until exactly one non-zero digit stands in front of it, then count the places it travelled.

Take 52,800. The point starts at the far right: 52800. Slide it left past 0, 0, 8, 2 — that is 4 places — and it lands as 5.28. So 52,800 = 5.28 × 104.

Two checks worth making every single time:

  • Is the front number at least 1 and less than 10? 52.8 × 103 has the right value but the wrong form, because 52.8 is bigger than 10. The point has one more place to go.
  • Does the exponent count places or zeros? In 52,800 there are only two zeros, but the point moved 4 places. Count places.

Going the other way is the same move in reverse. 3.06 × 105 asks the point to slide 5 places right: 3.06 becomes 306,000, with zeros filling every place that has no digit of its own.

Numbers smaller than 1 take a negative exponent

A negative exponent does not make a number negative. It means a reciprocal — 10-3 is 1/1000 — so multiplying by it makes a number smaller.

Take 0.00042. Slide the point right until one non-zero digit stands in front of it: 4.2, which took 4 places. Moving right made the number bigger, so the power of ten has to shrink it back down: 0.00042 = 4.2 × 10-4.

Watch the miscount that catches almost everyone. There are three zeros after the point in 0.00042, so -3 looks tempting. But the point has to travel one place further to get past the 4 itself. Count the places the point moves, not the zeros you can see.

A quick sanity check: a number smaller than 1 always has a negative exponent, and a number 10 or bigger always has a positive one. If a small number ends up with a positive power, the sign got lost somewhere.

Comparing, multiplying and dividing

Once two numbers are in scientific notation, comparing them is mostly reading the exponents.

3 × 108 and 3 × 105 have the same front number, so the difference is entirely in the tens: 108 / 105 = 103. The first is 1,000 times as large as the second — not 3 times, and not 1,000 more.

Multiplying and dividing use the exponent rules you already know, applied to each part separately:

  • Multiply: multiply the front numbers, add the exponents. (3 × 104) × (2 × 105) = 6 × 109.
  • Divide: divide the front numbers, subtract the exponents. (8 × 107) ÷ (2 × 103) = 4 × 104.

Then check the front number one last time. (4 × 106) × (5 × 103) gives 20 × 109, and 20 is not below 10, so move the point one place left and lift the exponent by one: 2 × 1010. A quotient can drift the other way — 2.5 / 5 = 0.5, and 0.5 × 107 is written 5 × 106.

Estimating with a single digit

Scientific notation is what makes rough comparison possible at all. Round the front number to one digit and you have an estimate you can hold in your head.

The distance to the Sun is about 149,600,000 km, which is 1.496 × 108, so roughly 1 × 108 km. A human hair is about 0.00007 m across, so roughly 7 × 10-5 m.

Two useful habits:

  • Round the front number the ordinary way — look at the next digit and decide. 8.6 × 109 rounds to 9 × 109; 4.2 × 107 rounds to 4 × 107.
  • Rounding the front number does not move the decimal point, so the exponent normally stays put. The one exception is a carry: 9.7 × 105 rounds to 10 × 105, which is written 1 × 106.

Comparing two estimates then takes one subtraction. 1 × 108 km against 7 × 10-5 m tells you at once that these quantities are not remotely in the same world, without a single long division.

Worked examples

Example 1

Write 0.000305 in scientific notation.

  1. The number is smaller than 1, so the point moves right and the exponent will be negative.
  2. Slide the point right until one non-zero digit sits in front of it: 0.000305 becomes 3.05.
  3. Count the places it moved: 0 → 0 → 0 → 3 is 4 places.
  4. So 0.000305 = 3.05 × 10-4. Check: 3.05 is between 1 and 10, and the exponent is negative for a small number.

Example 2

Work out (6 × 105) × (7 × 108), giving the answer in scientific notation.

  1. Rearrange into front numbers and powers of ten: (6 × 7) × (105 × 108).
  2. 6 × 7 = 42, and multiplying powers of the same base adds the exponents: 105 × 108 = 1013.
  3. That gives 42 × 1013, which has the right value but the wrong form, because 42 is not below 10.
  4. Move the point one place left and raise the exponent by one: 4.2 × 1014.

Example 3

The Sun weighs about 2 × 1030 kg and the Earth about 6 × 1024 kg. Roughly how many times as heavy as the Earth is the Sun?

  1. Divide: (2 × 1030) ÷ (6 × 1024) = (2 / 6) × (1030 / 1024).
  2. Dividing powers of the same base subtracts the exponents: 1030 / 1024 = 106.
  3. 2 / 6 is about 0.33, so the answer is about 0.33 × 106.
  4. 0.33 is below 1, so shift it: about 3.3 × 105, which is roughly 330,000 times as heavy.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Which of these is 63,000 written in scientific notation?

  1. 6.3 × 104
  2. 6.3 × 10-4
  3. 63 × 103
  4. 6.3 × 105

Answer: A. 6.3 × 104

  1. Move the decimal point in 63,000 until one digit stands in front of it: 6.3.
  2. The point moved 4 places to the left, so the exponent is 4.
  3. 63,000 = 6.3 × 104

Problem 2

Difficulty 3 of 5

Which of these is 0.000071 written in scientific notation?

  1. 7.1 × 10-5
  2. 7.1 × 105
  3. 71 × 10-6
  4. 7.1 × 10-4

Answer: A. 7.1 × 10-5

  1. Move the point right until one non-zero digit stands in front of it: 7.1.
  2. That took 5 places, and the number is smaller than 1, so the exponent is -5.
  3. 0.000071 = 7.1 × 10-5

Problem 3

Difficulty 4 of 5

Write 9.51 × 103 in standard form. (Standard form means the digits written out in full, like 45,000.)

Answer: 9510

  1. The exponent is 3, so the point moves 3 places to the right.
  2. 9.51 becomes 9,510.
  3. 9.51 × 103 = 9,510

Common mistakes

  • Leaving the front number at 10 or more: 52.8 × 103 is the right value in the wrong form, and 5.28 × 104 is the same number written properly.
  • Counting the zeros instead of the places. In 0.00042 there are three zeros after the point, but the point travels four places, so the exponent is -4.
  • Giving a small number a positive exponent. Anything below 1 needs a negative power of ten to pull it down.
  • Answering a "how many times as large?" question with the exponent difference. If the exponents differ by 3, the answer is 103 = 1,000, not 3.
  • Multiplying the front numbers but forgetting the powers of ten, or multiplying the exponents instead of adding them.

What you should be able to do

  • Convert between standard form and scientific notation, including numbers less than 1.
  • Compare two numbers in scientific notation and say how many times larger one is.
  • Multiply and divide numbers in scientific notation and renormalise the result.
  • Estimate very large or very small quantities as a single digit times a power of ten.

Where this fits in the curriculum

Common Core

  • 8.EE.A.3

    Grade 8 — Use numbers expressed as a single digit times an integer power of 10 to estimate very large or very small quantities and compare their sizes.

  • 8.EE.A.4

    Grade 8 — Perform operations with numbers expressed in scientific notation, including problems that mix decimal and scientific notation.

Ontario

  • G8.B1.1

    Grade 8 — Represent and compare very large and very small numbers, including through the use of scientific notation, and describe various ways they are used in everyday life.

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