Scientific Notation

Scientific notation is an agreement about how to write numbers, not a new operation. Everything it needs you to do is the exponent rules applied to powers of ten.

The form

A number between 1 and 10, times a power of ten.

• 4,500 is 4.5 x 10 to the 3rd.
• 0.0067 is 6.7 x 10 to the -3rd.
• 93,000,000 is 9.3 x 10 to the 7th.

"Between 1 and 10" includes 1 and excludes 10. So 9.99 x 10 to the 5th is correctly written and 10.2 x 10 to the 5th is not, even though it names a real number. A question asking which expression is in scientific notation is testing that one digit rule, and the wrong choices are usually arithmetically correct.

Converting, and the sign

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To write a large number: move the decimal point left until one digit remains in front, and the number of places is the exponent, positive.

52,700 becomes 5.27, and the point moved 4 places, so 5.27 x 10 to the 4th.

To write a small number: move the point right until one digit is in front, and the exponent is negative.

0.000315 becomes 3.15, and the point moved 4 places right, so 3.15 x 10 to the -4th.

Reading it back: a positive exponent means move the point right that many places; a negative exponent means move it left.

The sign of the exponent tells you whether the number is bigger or smaller than 1, and nothing else. It never means the number itself is negative. A negative number in scientific notation carries its minus sign out front: -3.2 x 10 to the 4th is negative forty thousand.

Multiplying and dividing

Multiply: multiply the front numbers, add the exponents.

(3 x 10 to the 4th)(2 x 10 to the 5th)

Front: 3 times 2 is 6. Exponents: 4 plus 5 is 9.

Answer: 6 x 10 to the 9th.

Divide: divide the front numbers, subtract the exponents.

(8 x 10 to the 7th) / (2 x 10 to the 3rd)

Front: 8 divided by 2 is 4. Exponents: 7 minus 3 is 4.

Answer: 4 x 10 to the 4th.

Fixing the front number

Sometimes the front number leaves the 1-to-10 range and has to be repaired.

(5 x 10 to the 6th)(4 x 10 to the 3rd)

Front: 5 times 4 is 20. Exponents: 6 plus 3 is 9. That gives 20 x 10 to the 9th, which is correct but not in scientific notation.

Twenty is 2.0 x 10 to the 1st, so 2 x 10 to the 10th.

Moving the point one place left adds one to the exponent. The two changes cancel, which is why the value is unchanged. If you ever have to do this, check the total size afterward: 20 x 10 to the 9th and 2 x 10 to the 10th are both twenty billion.

The same repair runs the other way when division leaves a front number below 1: 0.5 x 10 to the 6th becomes 5 x 10 to the 5th, moving the point right and subtracting one.

Adding and subtracting

Different rule, and it is the one people miss. The exponents must match first.

4.2 x 10 to the 5th plus 3 x 10 to the 4th

Rewrite the second with the same exponent: 3 x 10 to the 4th is 0.3 x 10 to the 5th.

Now add the front numbers: 4.2 plus 0.3 is 4.5.

Answer: 4.5 x 10 to the 5th.

Adding the front numbers without matching exponents gives 7.2 of something, and it is wrong by a factor of ten.

Comparing

Compare the exponents first. A bigger exponent means a bigger number, regardless of the front number, as long as both are properly written.

8.9 x 10 to the 4th is smaller than 1.2 x 10 to the 5th, because 5 beats 4. The 8.9 is a distraction.

Only if the exponents match do you compare the front numbers.

For negative exponents the same rule holds with the number line in mind: 10 to the -3 is larger than 10 to the -7, because -3 is greater than -7. A number with a more negative exponent is closer to zero.

What you can skip

Across the 32 questions on this topic:

• Significant figures never appear.
• Chains of operations. The questions convert, multiply or divide once; there is no need to practice long calculations in scientific notation.

Where people lose points

Moving the decimal point the wrong way, giving an exponent of the right size and the wrong sign. The check is whether the number is bigger or smaller than 1.

Reading a negative exponent as a negative number.

Adding exponents when adding numbers. Addition needs matching exponents; only multiplication adds them.

Leaving a front number outside 1 to 10 and picking a choice that is arithmetically equal but not in scientific notation.

Comparing front numbers before exponents.

Work one in under a minute

Compute (6.4 x 10 to the -3) divided by (1.6 x 10 to the 2nd).

Front: 6.4 divided by 1.6 is 4.

Exponents: -3 minus 2 is -5.

Answer: 4 x 10 to the -5.

Check the direction: dividing a small number by a larger one makes it smaller still, so a more negative exponent is right.

Where this leads

Scientific notation is how General Science writes measurements, and it is the exponent rules in their most practical form.