Math Basics: How to Read a Formula

Every formula on this site is written in the standard notation scientists use, so you can recognize it in a textbook later. This page explains that notation from zero. You do not need to read it all at once. Each formula page also explains the signs it uses in its own “Before you start” section, and links back here.

Reading a formula like a sentence

A formula is a sentence written in shorthand. The letters stand for quantities, things you can measure or count. The equals sign $=$ says the left side and the right side are the same amount. So $V = I R$ is the sentence “the voltage equals the current times the resistance”.

• Side by side means multiply. $I R$ means $I \times R$, and $2x$ means $2 \times x$. A dot means multiply too: $a \cdot b = a \times b$.
• A fraction bar means divide. $\frac{a}{b}$ means $a \div b$. You can read it as “$a$ per $b$” or “$a$ over $b$”.
• Brackets come first. In $2(3 + 4)$ you add first and then multiply, which gives 14.
• The usual order: brackets, then powers and roots, then multiplication and division, then addition and subtraction. So $2 + 3 \times 4 = 14$, not 20.

Powers

A small raised number is a power, or exponent. It says how many times to multiply a number by itself.

• $3^2 = 3 \times 3 = 9$, read “3 squared”.
• $2^3 = 2 \times 2 \times 2 = 8$, read “2 cubed” or “2 to the power 3”.
• Anything to the power 1 is itself, and anything (except 0) to the power 0 is 1: $5^0 = 1$.
• A negative power means divide instead of multiply: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Powers of ten and scientific notation

Science deals with very big and very small numbers. Writing them in full is slow and easy to get wrong, so they are written as an ordinary number times a power of ten. This is called scientific notation.

The quick rule: the exponent says how many places to move the decimal point. A positive exponent moves it to the right and makes the number bigger. A negative exponent moves it to the left and makes the number smaller.

• $5.4 \times 10^{14}$: start with 5.4 and move the point 14 places right: 540,000,000,000,000.
• $1.6 \times 10^{-19}$: start with 1.6 and move the point 19 places left: 0.00000000000000000016.

Multiplying: multiply the front numbers and add the exponents. $(2 \times 10^{3}) \times (3 \times 10^{4}) = 6 \times 10^{7}$.

Dividing: divide the front numbers and subtract the exponents. $(8 \times 10^{9}) \div (2 \times 10^{3}) = 4 \times 10^{6}$.

Tidying up: scientific notation keeps one digit before the decimal point. If you get $35.8 \times 10^{-20}$, move the point one place left and raise the exponent by one: $3.58 \times 10^{-19}$. It is the same number.

On a calculator you will often see the letter E instead: 6.6E-34 means $6.6 \times 10^{-34}$.

Roots

A square root $\sqrt{x}$ asks “which number, multiplied by itself, gives $x$?” So $\sqrt{9} = 3$ because $3 \times 3 = 9$. A square root undoes squaring. A small number on the root sign asks for a different root: $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$. A power of one half means a square root: $x^{1/2} = \sqrt{x}$.

Names on letters: subscripts and primes

A small lowered label, a subscript, tells similar quantities apart. $V_{\text{in}}$ is the voltage going in and $V_{\text{out}}$ the voltage coming out. Numbered subscripts count items in a list: $x_1, x_2, x_3$ are the first, second and third value of $x$, and $x_i$ means “the $i$-th one”, any item of the list.

A small dash, a prime, marks a changed or second version: $\lambda'$ (lambda prime) is the wavelength after something happened to it.

Greek letters

Greek letters are extra names for quantities, because the Latin alphabet runs out. They are only names; each page says what its letters stand for.

Adding up and multiplying a list: $\Sigma$ and $\Pi$

$\sum$ (capital sigma) means “add up”. The labels under and over it say what to count through.

$$\sum_{i=1}^{3} x_i = x_1 + x_2 + x_3$$

Read it as “the sum of $x_i$ for $i$ from 1 to 3”. $\prod$ (capital pi) is the same idea with multiplication: $\prod_{i=1}^{3} x_i = x_1 \times x_2 \times x_3$.

The number e, exp and logarithms

$e$ is a fixed number, about 2.71828, that shows up wherever something grows or shrinks in proportion to its own size, like money with interest or a cooling cup of tea. $e^{x}$ is also written $\exp(x)$. A negative power, $e^{-t}$, describes something fading away: it starts at 1 and keeps shrinking toward 0.

A logarithm undoes a power. $\log_{10} 1000 = 3$ because $10^{3} = 1000$. It answers “how many times do I multiply the base to reach this number?”

• $\log_2$ uses base 2. It counts halvings and doublings, and bits in computing: $\log_2 8 = 3$.
• $\ln$ is the natural logarithm, base $e$. It undoes $e^{x}$: $\ln(e^{x}) = x$.

Other signs you will meet

Rates of change: $\frac{d}{dt}$ and $\int$

$\frac{dI}{dt}$ is read “d I by d t”. It means how fast $I$ is changing at this moment, like the speedometer of a car, which shows how fast the distance is changing. If the current rises by 2 amperes in 1 second, $\frac{dI}{dt}$ is 2 amperes per second.

$\int$ is an integral. It means “add up a huge number of tiny pieces”. $\int_{a}^{b} f(x)\, dx$ is the area under the curve of $f$ between $a$ and $b$. You will see it in quantum mechanics, where it adds up probability over all positions.

Vectors

A vector is a list of numbers treated as one object, like an arrow with a length and a direction, or a list of features describing a house. Vectors are often written in bold: $\mathbf{x}$. $\lVert \mathbf{x} \rVert$ is the vector’s length, and $\mathbf{a} \cdot \mathbf{b}$, the dot product, multiplies matching entries and adds the results: $(1, 2) \cdot (3, 4) = 1 \times 3 + 2 \times 4 = 11$.

Units and prefixes

A number without a unit means little: “5” could be 5 meters or 5 kilometers. Formulas expect the base units, so convert before you calculate.

Units in a formula can be multiplied and canceled like numbers. Joule seconds times per second gives joules, because the seconds cancel. Checking that the units come out right is one of the best ways to catch a mistake.