Scientific Notation: How to Write It and Calculate With It

Scientific notation writes a number as a value from 1 up to (but not including) 10, multiplied by a power of 10. So 4,500,000 becomes 4.5 x 10^6. The exponent counts how many places the decimal point moved. The Common Core grade 8 standards ask students to use numbers written as a single digit times an integer power of 10 to estimate very large or very small quantities, and to do arithmetic with them (8.EE.A.3 and 8.EE.A.4).

1. Writing a large number

Take 4,500,000.

1. Put the decimal point after the first nonzero digit: 4.500000, which is 4.5.
2. Count how many places the point moved. From 4,500,000. to 4.5 is 6 places.
3. Write 4.5 x 10^6.

Positive exponent means a large number. Check by moving the point back 6 places to the right: 4.5 becomes 4,500,000. NASA's powers-of-ten page reads it the same way: 3.4 x 10^2 is 340, and 2.5 x 10^-3 is 0.0025.

2. Writing a small number

Take 0.00032.

1. Move the point right until one nonzero digit is in front of it: 3.2.
2. The point moved 4 places to the right.
3. Write 3.2 x 10^-4.

A negative exponent means a number smaller than 1. Another one: 0.0000062 = 6.2 x 10^-6, because the point moves 6 places.

A quick rule: moving the point left makes the exponent positive; moving it right makes the exponent negative.

3. Is it in correct form?

The front number must be at least 1 and less than 10. So 12.5 x 10^3 is not finished. Move the point one place left and add 1 to the exponent: 1.25 x 10^4. Likewise, 0.45 x 10^5 becomes 4.5 x 10^4 (one place right, exponent down by 1).

4. Multiplying

Multiply the front numbers and add the exponents.

(3 x 10^4)(2 x 10^5) = 6 x 10^9

If the product of the front numbers reaches 10, fix it:

(4 x 10^6)(5 x 10^3) = 20 x 10^9 = 2 x 10^10

5. Dividing

Divide the front numbers and subtract the exponents.

(8.4 x 10^7) / (2.1 x 10^3) = 4 x 10^4

Check: 84,000,000 / 2,100 = 40,000. It matches.

6. Adding and subtracting

Add and subtract only after both numbers have the same power of 10.

3.2 x 10^5 + 4.5 x 10^4

1. Rewrite the second number with 10^5: 4.5 x 10^4 = 0.45 x 10^5.
2. Add the front numbers: 3.2 + 0.45 = 3.65.
3. Result: 3.65 x 10^5, which is 365,000.

Check: 320,000 + 45,000 = 365,000.

A common error is to add the front numbers and the exponents separately, which gives nonsense such as 7.7 x 10^9.

7. Comparing two numbers

Compare the exponents first. The larger exponent is the larger number, as long as both are in correct form. If the exponents match, compare the front numbers.

To say how many times larger one number is, divide. For 6.4 x 10^6 and 8 x 10^3:

(6.4 x 10^6) / (8 x 10^3) = 0.8 x 10^3 = 800

The first number is 800 times as large. Fix the form at the end: 0.8 x 10^3 is 8 x 10^2.

For small numbers, compare 2.5 x 10^-3 and 4 x 10^-4. These are 0.0025 and 0.0004. The first is bigger. The ratio is 2.5 x 10^-3 / 4 x 10^-4 = 0.625 x 10^1 = 6.25, so it is 6.25 times as large.

8. A word problem

A cell is about 2 x 10^-5 meters wide. How many cells fit across a line 1 x 10^-3 meters long (one millimeter)?

(1 x 10^-3) / (2 x 10^-5) = 0.5 x 10^2 = 50 cells.

The width and the length are made-up values for practice, not measurements of a particular cell. The method is the point: divide the front numbers, subtract the exponents, then fix the form.

9. Reading calculator output

The grade 8 standard also mentions interpreting scientific notation that has been generated by technology. A calculator that shows 3.2E5 or 3.2e+5 means 3.2 x 10^5. The E is not a variable. It stands for "times ten to the power."

10. Common mistakes

• Wrong exponent sign. Big numbers get a positive exponent; numbers below 1 get a negative one.
• Front number outside 1 to 10. Adjust the point and the exponent together.
• Adding without matching powers. Convert one number first.
• Counting zeros instead of places. Count how many places the decimal point moves.

11. Key takeaways

• Scientific notation is a number from 1 to under 10 times a power of 10.
• Multiplying adds exponents; dividing subtracts them.
• Add or subtract only after matching the powers of 10.
• Finish by checking that the front number is at least 1 and less than 10.
• A calculator's E5 or e+5 means times 10^5.