One digit before the point, a power of ten behind it. The form makes two numbers comparable at a glance and turns multiplication into an addition of exponents; addition is the operation it does not simplify, and that is worth knowing before you attempt one.
Core ideas
This is the exponent laws of “Integer Exponents”, applied to one base, plus a rule about where the decimal point goes.
Exactly one nonzero digit in front of the point. Two numbers in this form are compared by their exponents first, and only by their mantissas if the exponents tie — which is why 8.9 × 10⁻⁵ is smaller than 1.2 × 10⁻⁴ despite the larger-looking front.
A positive exponent moves the point right and increases the magnitude; a negative one moves it left and decreases the magnitude. For a negative mantissa that means farther from or nearer to zero, not larger or smaller in the number-line order. "n zeros" is a description that works only when the mantissa is a single digit, and it fails on 9.3 × 10⁵, which is 930000.
Multiply the mantissas and add the exponents. Adding requires both numbers to carry the same power of ten first, which is the same manoeuvre as finding a common denominator, and it is the step people skip.
The material
What normalised means, how to convert either way, and why one of the four operations behaves differently from the others.
602000000000000000000000 and 0.00000000000000000016 are unreadable, and the failure is not aesthetic: nobody can count 21 zeros reliably, so the notation itself introduces errors. Scientific notation replaces the zeros with a number.
A number is in scientific notation when it is written a × 10ⁿ with 1 ≤ |a| < 10 and n an integer. a is the mantissa and n the exponent.
0 has no such form, since no mantissa with |a| ≥ 1 can produce it. Every other real number has exactly one.
12.5 × 10³ is a perfectly correct way to write 12500, and it is not in scientific notation, because 12.5 is not below 10. Those are two different criticisms and only the second one applies. Renormalised it is 1.25 × 10⁴, and the value never changed.
Move the decimal point until exactly one nonzero digit stands in front of it, count the places you moved, and that count is the exponent. Moving left raises the exponent; moving right lowers it.
Then check the sign against the magnitude rather than against your count: a number with absolute value at least 1 has n ≥ 0, and a nonzero number with absolute value below 1 has n < 0. That wording also covers negative values such as −93000, whose exponent is positive although the number itself is less than 1.
Mantissas multiply, exponents add — the product law of “Integer Exponents” with base 10. For division, mantissas divide and exponents subtract.
The result may not be normalised, and that is expected rather than wrong. If the mantissa reaches 10, move the point one place left and add one to the exponent; if it falls below 1, move it right and subtract one. (5 × 10⁻⁷)(2 × 10⁻³) is 10 × 10⁻¹⁰, which renormalises to 1 × 10⁻⁹.
Addition needs a shared power of ten before the mantissas can be touched at all:
The rewritten term 0.34 × 10⁵ is not in scientific notation and does not need to be; it is an intermediate line. Only the final answer is held to the form.
One electron carries about 1.6 × 10⁻¹⁹ coulombs, so one coulomb is 1 / (1.6 × 10⁻¹⁹) electrons. The mantissas give 1/1.6 = 0.625 and the exponents give 0 − (−19) = 19, so the answer is 0.625 × 10¹⁹, renormalised to 6.25 × 10¹⁸.
Interactive
Normalised exactly, and arithmetic done in it
Method
Four moves, and the fourth is the one that catches a reversed sign.
Leading zeros are placeholders and are not part of the mantissa. In 0.00072 the first nonzero digit is the 7, and in 93000 it is the 9.
That produces the mantissa: 7.2 and 9.3. Trailing zeros after the last nonzero digit are dropped unless you are deliberately recording precision.
The count is the size of the exponent. Four places in both examples above, in opposite directions.
Absolute value at least 1 means a non-negative exponent; nonzero absolute value below 1 means a negative one. Then confirm the mantissa satisfies 1 ≤ |a| < 10, which is the condition the form imposes.
The two directions of renormalising, one in each calculation.
In the first the mantissa was too large and the exponent went up by one; in the second it was too small and the exponent went down by one. In both cases the move to the mantissa is paid for by the exponent, which is the whole content of renormalising.
Neither 32 × 10⁻³ nor 0.25 × 10⁻⁴ is a wrong answer to the arithmetic. They are the right numbers in the wrong form, and the distinction is worth keeping: the form is a convention about presentation, and the value is not.
Fade the arithmetic with (2.5 × 10⁻³)(4 × 10⁶). Multiplying the mantissas and adding exponents first gives 10 × 10³; only then normalise to 1 × 10⁴. 10 × 10³ has the right value but not the required form, while 1 × 10³ moved the point without paying for it.
Write −0.000083 and −720000 in scientific notation, then multiply and divide them by handling mantissas and exponents separately. Use magnitude, not signed order, to justify each exponent, and check that the final mantissa satisfies 1 ≤ |a| < 10.
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