Convert either way and combine two numbers with the exponent law shown.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Scientific notation writes every number as one digit, a decimal part, and a power of ten: 4500 is 4.5 × 10³, and 0.00032 is 3.2 × 10⁻⁴. The coefficient always sits between 1 and 10, which is what makes two numbers comparable at a glance - and the exponent is nothing more mysterious than the number of places the decimal point had to travel to get there.
That is why the sign works the way it does. Moving the point LEFT makes a number smaller, so the power has to make it bigger again: the exponent is positive. Moving RIGHT makes it bigger, so the power shrinks it back: the exponent is negative. Every mistake with scientific notation is a mistake about that trade, which is why this page shows the point moving rather than just printing the answer.
The notation earns its keep in arithmetic. Multiplying two numbers in scientific notation is one small multiplication and one ADDITION of exponents - that is the whole reason the notation exists. Adding two of them is harder, because the powers of ten have to match first, and that is the step almost everybody skips.
In 4500 that is the 4; in 0.00032 it is the 3. Everything before it is packaging.
4500 becomes 4.5 and 0.00032 becomes 3.2. That is the coefficient, and it is now between 1 and 10 by construction.
From 4500. to 4.5 the point moved 3 places left. From 0.00032 to 3.2 it moved 4 places right.
Left is positive, right is negative: 4.5 × 10³ and 3.2 × 10⁻⁴. Check it by moving the point back - if you do not land on the number you started with, the sign is the thing to look at.
A positive exponent moves the point right and a negative one moves it left, padding with zeros: 6.7 × 10⁵ is 670000, and 8.1 × 10⁻³ is 0.0081.
Ordinary, scientific, E notation, and engineering - which forces the exponent to a multiple of three so it matches the SI prefixes.
| Number | Scientific | E notation | Engineering |
|---|---|---|---|
| 4500 | 4.5 × 10³ | 4.5e+3 | 4.5 × 10³ |
| 45000 | 4.5 × 10⁴ | 4.5e+4 | 45 × 10³ |
| 0.00032 | 3.2 × 10⁻⁴ | 3.2e-4 | 320 × 10⁻⁶ |
| 299792458 | 2.99792458 × 10⁸ | 2.99792458e+8 | 299.792458 × 10⁶ |
| 1 | 1 × 10⁰ | 1e+0 | 1 × 10⁰ |
| 0.5 | 5 × 10⁻¹ | 5e-1 | 500 × 10⁻³ |
| 0.000000001 | 1 × 10⁻⁹ | 1e-9 | 1 × 10⁻⁹ |
| 602000000000000000000000 | 6.02 × 10²³ | 6.02e+23 | 602 × 10²¹ |
4500The first non-zero digit is 4, so the point goes after it: 4.5. Getting there from 4500 moved the point 3 places left, so the exponent is +3 and the answer is 4.5 × 10³. Two significant figures, because the trailing zeros were only placeholders.
0.00032The first non-zero digit is 3. Moving the point to just after it travels 4 places to the RIGHT, so the exponent is negative: 3.2 × 10⁻⁴. In engineering notation that is 320 × 10⁻⁶ - 320 micro-units, which is how an engineer would say it out loud.
4.5e3Typed in scientific notation, so the page goes the other direction: a power of +3 moves the point 3 places right, padding with a zero, giving 4500. No direction switch to set - the form you typed says which way you are going.
3e4 × 2e5Multiply the coefficients: 3 · 2 = 6. Add the exponents: 4 + 5 = 9. Answer 6 × 10⁹, and that is the whole method. Notice you never wrote out 30000 or 200000 - which is exactly what the notation buys you.
5e3 × 4e3The coefficients give 20 and the exponents give 6, so the intermediate answer is 20 × 10⁶ - a correct value, but not scientific notation, because 20 is not between 1 and 10. Move the point once more: 2 × 10⁷.
3e5 + 4e4The exponents differ, so nothing can be added yet. Rewrite the smaller at the larger's power: 4 × 10⁴ is 0.4 × 10⁵. Now the coefficients add: 3 + 0.4 = 3.4, giving 3.4 × 10⁵. Adding the exponents here would have been the classic error.
Put the decimal point just after the first non-zero digit, then count how many places it moved. Moving left gives a positive exponent, moving right a negative one. 4500 becomes 4.5 × 10³ and 0.00032 becomes 3.2 × 10⁻⁴.
They are the same thing under two names - "standard form" is the usual British term and "scientific notation" the American one. Both mean a coefficient between 1 and 10 times a power of ten. Engineering notation is a different variant, where the exponent is forced to a multiple of three.
It is scientific notation written on one line, the way a calculator, spreadsheet or programming language shows it: 4.5e3 means 4.5 × 10³ = 4500. The e stands for exponent, not for Euler's number, and 4.5e-3 means 4.5 × 10⁻³ = 0.0045.
Multiply the coefficients and ADD the exponents. (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹. If the coefficients multiply to 10 or more, move the point once more and raise the exponent to match - 20 × 10⁶ is written 2 × 10⁷.
Make the exponents match first, then add only the coefficients. For 3 × 10⁵ + 4 × 10⁴, rewrite the second as 0.4 × 10⁵ and add: 3.4 × 10⁵. This is the step that gets skipped, and adding the exponents instead is the classic mistake.
It is scientific notation with the exponent forced to a multiple of three, so it lines up with the SI prefixes: 10³ kilo, 10⁶ mega, 10⁻³ milli, 10⁻⁶ micro. 0.00032 becomes 320 × 10⁻⁶, which an engineer reads directly as 320 microunits.
Because it has no first significant digit to put the decimal point after, and no power of ten multiplied by a coefficient between 1 and 10 can ever equal zero. The honest answer is that zero is a special case, which is why this page says so rather than printing 0 × 10⁰.
As many as the coefficient shows. That is one of the notation's quiet advantages: 4500 is ambiguous about whether the zeros are measured or merely placeholders, while 4.5 × 10³ claims two significant figures and 4.500 × 10³ claims four.
