IB Mathematics Applications & Interpretation SL 1.1—using Standard Form Notes

Why Standard Form Exists

1. Applications and Interpretation is a course about real quantities, and real quantities are often either enormous or minute: the mass of the Earth is about $5\,970\,000\,000\,000\,000\,000\,000\,000\,000$ kg, and the diameter of an atom is a tiny fraction of a metre.
2. Writing those numbers out in full is slow, and it is very easy to lose count of the zeros, which quietly changes the value by a factor of ten.
3. Standard form (also called scientific notation) fixes this by splitting any number into a simple decimal and a power of ten, so the size is captured by one small exponent instead of a long string of zeros.

The Structure: A Single-Digit Number Times A Power Of Ten

$$ a \times 10^{k}, \quad 1 \le a < 10, \; k \in \mathbb{Z} $$

1. The first part $a$ is the significand: it carries the digits of the number and must have exactly one non-zero digit in front of the decimal point, which is what the rule $1 \le a < 10$ guarantees.
2. The second part $10^{k}$ is the scale: the exponent $k$ is a whole number ($k \in \mathbb{Z}$) that records how far the decimal point has moved.
3. A positive $k$ makes the number large and a negative $k$ makes it small, because multiplying by $10^{k}$ simply shifts the decimal point $k$ places.
4. The condition $1 \le a < 10$ matters because it makes the form unique: without it the same value could be written in many different ways.

Converting Between Standard Form And Ordinary Numbers

1. Each factor of ten moves the decimal point one place, so the exponent is really just a place-counter.
2. To write an ordinary number in standard form, slide the decimal point until exactly one non-zero digit sits in front of it, then count how many places you moved.
   1. Moving the point left (for a large number) makes $k$ positive; moving it right (for a small number) makes $k$ negative.
3. To go the other way, read the exponent and move the decimal point that many places, filling any empty spots with zeros.

Multiplying And Dividing: Handle The Digits And The Powers Separately

1. The idea is to split the job in two: deal with the significands, then deal with the powers of ten.
2. The powers of ten obey the laws of exponents from SL 1.5: multiplying adds the exponents and dividing subtracts them, because $10^{m} \times 10^{n} = 10^{m+n}$.
3. Putting those two steps together gives the working rules: $$ (a \times 10^{m})(b \times 10^{n}) = (a \times b) \times 10^{m+n} $$ $$ \frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n} $$
4. After combining, the new significand may no longer sit between 1 and 10, so you often need one extra step.
5. That step is renormalising: rewrite the significand in standard form and fold the spare power of ten into the exponent.

Adding And Subtracting: Match The Powers First

1. Adding and subtracting is different: you can only combine the significands once the powers of ten are the same.
2. This is just like needing a common denominator before adding fractions, where the powers of ten play the role of the denominator.
3. Rewrite the term with the smaller exponent so it shares the larger exponent, then add or subtract the significands and renormalise if needed.