Examples · Basic Math
Scientific Notation Examples: 8 Worked Problems
Eight conversions, ordered so each one adds exactly one idea to the last. Every answer is checked by moving the decimal point back.
Large numbers: a positive exponent
The first three worked examples are big numbers, so each ends with a positive power of 10.
- 5280the number, decimal point after the last digit
- 5.280 → 5.28put the point after the first non-zero digit; the trailing zero adds nothing
- 5.28 × 10³the point moved 3 places left, so the exponent is +3
- 93000000the number, commas removed
- 9.3put the point after the 9
- 9.3 × 10⁷the point moved 7 places left past the 3 and six zeros: exponent +7
- 380000000the number
- 3.8put the point after the 3
- 3.8 × 10⁸8 places left, so the exponent is +8
In every one the exponent is the number of places the point travelled, not the number of zeros. Example 2 has six zeros but its exponent is 7.
Small numbers: a negative exponent
- 0.0306the number
- 3.06put the point after the first non-zero digit, the 3; the zero between 3 and 6 stays
- 3.06 × 10⁻²the point moved 2 places right, so the exponent is −2
- 0.00000048the number
- 4.8put the point after the 4
- 4.8 × 10⁻⁷7 places right, so the exponent is −7
Numbers already near 1
- 12.5the number
- 1.25put the point after the 1
- 1.25 × 10¹one place left, exponent +1
A number from 1 up to 10 needs no move, so its exponent is 0: 8 is 8 × 10⁰. A zero exponent is valid, it just says the number is of ordinary size.
Converting back to a standard number
- 6.022 × 10²³the number in scientific notation
- positive exponent: move right 23 placesa positive exponent means a big number
- 602200000000000000000000the three digits 022 fill 3 of the 23 places; zeros fill the other 20
- 7.1 × 10⁻⁶the number in scientific notation
- negative exponent: move left 6 placesa negative exponent means a small number
- 0.0000071five zeros in front of the 71, after the decimal point
Going back is the same move in reverse: read the sign of the exponent, move the point that many places, and fill with zeros.
Checking every result the same way
Move the decimal point back by the exponent and confirm you recover the original number.
| Example | Result | Check by moving the point back |
|---|---|---|
| 1 | 5.28 × 10³ | 5.28 → 5280 ✓ |
| 4 | 3.06 × 10⁻² | 3.06 → 0.0306 ✓ |
| 7 | 6.022 × 10²³ | 6.022 → 23 places right → 602200000000000000000000 ✓ |
This is the check the Scientific Notation Calculator on this site relies on: it moves the point on the digits themselves, so a long number keeps every digit.
What all eight examples have in common
Every example is the same two decisions. First, where does the decimal point go: after the first non-zero digit, always. Second, which way did it travel, and how far: left for a big number, right for a small one. The coefficient always keeps every significant figure of the original, and the exponent only records the move.
The two directions are worth practising together. A very large number such as 380,000,000 and a very small number such as 0.00000048 are the same idea with opposite signs on the exponent. Writing a value in standard form, the ordinary way, and then writing it in scientific notation is one move; to convert back, undo that move exactly. Throughout, the coefficient holds all the significant figures, so none of them is lost in either direction.
Questions about scientific notation examples
What is 93,000,000 in scientific notation?
9.3 × 10⁷. Put the decimal point after the 9 and count 7 places to the left.
What is 0.00000048 in scientific notation?
4.8 × 10⁻⁷. The point moves 7 places right to sit after the 4, so the exponent is negative.
How do I turn 6.022 × 10²³ back into a standard number?
Move the decimal point 23 places to the right, filling with zeros: 602200000000000000000000.
How do I check a scientific notation answer?
Move the decimal point back by the exponent. If you recover the original number, the conversion is right.