Scientific Notation Guide 2026: Convert, Multiply and Examples
Table of Contents
Conversion Rules
Large: 93,000,000 = 9.3 x 10^7 (move decimal 7 left).
Small: 0.00000052 = 5.2 x 10^-7 (move decimal 7 right).
Calculator E: 9.3E7 means 9.3 x 10^7 - same thing, compact display.
Small-number conversion is just decimal-hop counting. For 0.00072, move the decimal right until one nonzero digit sits left of it: 7.2, which took 4 hops, so 7.2 x 10^-4. Each hop is one power of ten, and moving right makes the exponent negative. Counting hops beats memorizing zeros - 0.00000052 needs 7 hops, giving 5.2 x 10^-7, with no zero-counting required.
The E on your calculator is scientific notation in disguise: 6.02E23 means 6.02 x 10^23. Use the EE or EXP key to enter it - typing 6.02 x 10 ^ 23 as separate keystrokes evaluates as (6.02 x 10)^23 on many calculators because ^ binds before the implied multiplication. That single keystroke error turns Avogadro's number into nonsense, and it is the most common scientific-notation calculator mistake.
Arithmetic With Powers of 10
- Multiply: Multiply N's, add exponents. (3 x 10^4)(2 x 10^5) = 6 x 10^9.
- Divide: Divide N's, subtract exponents. (8 x 10^9)/(2 x 10^3) = 4 x 10^6.
- Add/subtract: Match exponents first. 3 x 10^5 + 2 x 10^4 = 30 x 10^4 + 2 x 10^4 = 32 x 10^4 = 3.2 x 10^5.
- Powers: (2 x 10^3)^2 = 4 x 10^6.
Mixed-sign exponents follow the same add-the-exponents rule. (4 x 10^6)(3 x 10^-2): multiply coefficients 4 x 3 = 12, add exponents 6 + (-2) = 4, giving 12 x 10^4, which renormalizes to 1.2 x 10^5. Three distinct steps - coefficients, exponents, renormalize - and the negative exponent changes nothing about the procedure.
Powers of ten turn unit conversion into exponent addition. 1 km = 10^3 m = 10^6 mm, since each step down multiplies by 10^3. Light speed 3 x 10^8 m/s becomes 3 x 10^5 km/s by subtracting 3 from the exponent. Once lengths, masses, and times are all powers of ten, converting is just integer arithmetic on exponents.
Scale of the Universe
| Thing | Size/Count |
|---|---|
| Hydrogen atom | 1.2 x 10^-10 m |
| Human cells | 3 x 10^13 |
| Earth to Sun | 1.5 x 10^11 m |
| Stars (observable) | 2 x 10^23 |
| Seconds since Big Bang | 4.3 x 10^17 |
Comparing giants means subtracting exponents and ignoring the coefficients. About 10^19 atoms in a grain of sand versus 2 x 10^23 stars in the observable universe: the stars win by a factor of roughly 2 x 10^4, or 20,000 times. When magnitudes differ by orders of ten, the leading digits are decoration - the exponent carries the comparison.
The doubling penny is exponential intuition in one story. Double a penny daily for 30 days: 2^30 is about 1.07 x 10^9 pennies, or $10.7 million. Yet on day 20 the total is only $5,243 - more than 99.9% of the growth lands in the final ten days. Exponents back-load everything, which is why compound interest feels broken until it suddenly does not.
Engineering Notation Cousin
Engineering notation restricts exponents to multiples of 3, matching SI prefixes:
- 4.7 x 10^3 = 4.7 k (kilo)
- 2.2 x 10^-6 = 2.2 u (micro)
- 5 x 10^9 = 5 G (giga)
Electronics lives here: resistors in k-ohms, capacitors in microfarads, frequencies in GHz.
Engineering notation walks down the SI prefix ladder in steps of three. 0.0047 F = 4.7 x 10^-3 F = 4.7 mF = 4,700 uF. Electronics catalogs list capacitors in microfarads and resistors in kilohms, so engineering notation reads component values directly off the datasheet - scientific notation would force a translation step every time.
The multiples-of-three restriction exists because SI prefixes come in threes: 33,000 ohms is 33 k-ohms, 0.0000022 F is 2.2 uF, 2.4 x 10^9 Hz is 2.4 GHz. Scientific notation would write 3.3 x 10^4 ohms, which matches no standard component label. Engineers chose notation that speaks the language of the parts bin.
Frequently Asked Questions
Writing numbers as N x 10^n with 1 <= N < 10. It compactly handles very large and very small numbers.
Move the decimal to make a number between 1 and 10; the moves count becomes the exponent (left = positive, right = negative).
E = 'times 10 to the power of.' 6.02E23 = 6.02 x 10^23.
First match the exponents by adjusting one number, then add the N values and keep the exponent.
Engineering notation uses only exponents divisible by 3 (matching kilo, mega, micro prefixes); scientific uses any exponent with 1 <= N < 10.