Scientific Notation

Scientists deal with numbers like 150,000,000,000 metres, where counting the zeros is error-prone and comparing two such numbers at a glance is impossible. Scientific notation fixes that by writing every number as a single digit, a decimal part, and a power of ten: 1.5 × 10¹¹. The form is always a × 10ⁿ, where a is at least 1 but less than 10 — meaning exactly one non-zero digit sits before the decimal point. That last condition is a rule, not a suggestion. Writing 15 × 10¹⁰ gives the right value but is not scientific notation, because 15 is not under 10.

For a large number, put the decimal point after the first non-zero digit and count how many places it moved. In 5,600,000 the point starts at the far right; moving it to sit after the 5 takes six places, so the answer is 5.6 × 10⁶. A number bigger than 10 always gets a positive index, and that index is simply the number of places moved. Sense-check the result: 10⁶ is a million, and 5.6 million matches the original ✓. Notice the trailing zeros disappear — they were only holding place value, which is exactly the job the power of ten now does.

For a number smaller than 1, the decimal point moves the other way — to the right — and the index becomes negative. In 0.00042, moving the point to sit after the 4 takes four places right, giving 4.2 × 10⁻⁴. The essential point is that a negative index does not make the number negative. 4.2 × 10⁻⁴ is a small positive number: 0.00042. A negative number would have the minus in front of the a instead — −3.5 × 10⁴ is negative thirty-five thousand. So the index tells you the size, and any minus before the a tells you the direction.

Going back to an ordinary number reverses the process. A positive index moves the point right: 3.2 × 10⁵ moves five places to give 320,000, filling the empty places with zeros. A negative index moves it left: 7.5 × 10⁻⁴ moves four places to give 0.00075. Always sense-check against the index, because it tells you the rough size before you start. 10⁵ is a hundred thousand, so the answer must be a few hundred thousand. 10⁻⁴ is a ten-thousandth, so the answer must begin 0.000-something.

Scientific notation makes two jobs easy. Comparing: look at the index first, because it dominates. 3.1 × 10⁷ beats 9.8 × 10⁶ even though 3.1 is smaller than 9.8, simply because 10⁷ is ten times 10⁶. Calculating: handle the number parts and the indices separately, using the index laws. For (6 × 10⁵) ÷ (1.5 × 10⁻²): the numbers give 6 ÷ 1.5 = 4, and the indices give 5 minus negative 2, which is 7 — so 4 × 10⁷. Watch that subtraction of a negative; it adds. And if the number part comes out as 10 or more, adjust it back into range: 12 × 10⁵ becomes 1.2 × 10⁶.