Standard scientific notation

Scientific notation is a representation of a non-zero number as a coefficient between one and ten multiplied by an integer power of ten. We write the standard form as

.

For a negative number, place a minus sign before the coefficient.

The exponent shows how far the decimal point has moved. Positive exponents represent large magnitudes; negative exponents represent small ones. For example,

and

The condition on the coefficient matters. Although $38.4\times10^7$ is numerically correct, it isn't in standard scientific notation. Normalize it to $3.84\times10^8$.

Calculations

When multiplying, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. Then normalize the result:

Addition and subtraction work differently. Rewrite the numbers with the same power of ten first—a step students often try to skip:

Calculators may show numbers in E notation, but this is calculator shorthand, not acceptable written mathematical notation. In your final work, write $5.2\text{E}30$ as $5.2\times10^{30}$.

Scale and context

Using scientific notation, the same place-value system can handle astronomical distances and microscopic biological measurements, as well as sub-atomic scales and global financial quantities. Chemists use it for quantities such as Avogadro's constant. Physicists use powers of ten to compare orders of magnitude. In these contexts, a measured value carries precision and uncertainty, rather than being just a string of digits.

Number notation has developed across cultures, beginning with early Sumerian systems and leading to the Hindu-Arabic numerals used internationally today. Giving names to a few very large numbers may make them feel more concrete. However, notation such as $10^k$ is more systematic because it covers every scale without requiring a separate name for each one. It shows how mathematical language can shape what is easy to express and understand.

Laws of integer exponents

An exponent is a number that specifies how a base is repeatedly multiplied or otherwise extended according to the exponent laws. For a non-zero base, the main laws are

The same laws work for both numerical and algebraic expressions. For example,

and

The exponent applies to every factor inside the brackets. However, $(a+b)^n$ cannot generally be replaced by $a^n+b^n$.

Logarithms as inverse exponents

A logarithm is an exponent to which a specified positive base must be raised to produce a given positive number. An exponential statement and its equivalent logarithmic statement can be written as

A common logarithm is a logarithm whose base is ten, written $\log N$. A natural logarithm is a logarithm whose base is Euler's number, written $\ln N$:

On a calculator, use the log key for base ten and ln for base $e$. For an equation such as $10^x=37$, find $x$ by evaluating $\log 37$. The argument of a logarithm must always be positive. Neither $\log 0$ nor the logarithm of a negative real number is defined in the real number system.

Exponential and logarithmic functions undo each other. Their graphs are therefore reflections in the line where output equals input.

Applications and reflection

Logarithmic scales compress very wide ranges, which is why they are used for earthquake magnitude, sound intensity measured in decibels and chemical pH calculations, including work with buffer solutions.

Whether logarithms and the number $e$ were invented or discovered depends partly on what we mean by mathematics. Humans chose the notation and definitions. Once those choices are made, though, the resulting relationships follow consistently. Mathematical objects may feel discovered even when their language was designed, so the distinction is worth noticing.

Decimal places and significant figures

A decimal-place approximation is a rounded value whose accuracy is specified by the number of digits after the decimal point. To round to a particular decimal place, look at the next digit. Round up if it is at least five; otherwise, leave the retained digit as it is.

A significant-figure approximation is a rounded value whose accuracy is specified by the number of meaningful digits beginning with the first non-zero digit. Zeros between non-zero digits count as significant, whereas leading zeros only position the decimal point. Trailing zeros are significant when the notation clearly shows that level of precision.

The data and context determine the appropriate degree of accuracy. Giving many more digits than the measurements support suggests false precision. Currency, for example, may be rounded to the smallest commonly used unit. Meteorological reports and scientific measurements use accuracies that suit their instruments and purposes. Since alternative rounding conventions exist, the chosen convention matters when a value falls exactly halfway between two possibilities.

Bounds

A lower bound is the smallest possible value included in the interval represented by a rounded measurement. An upper bound is the endpoint above all possible values represented by that measurement. With ordinary rounding, the lower endpoint belongs to the interval, but the upper endpoint does not.

For rounding to $d$ decimal places, the half-width is

If the rounded value is $Q$, then

So a value reported as $7.3$ to one decimal place represents

The unequal treatment of the endpoints is intentional. Under ordinary rounding, $7.25$ rounds to $7.3$, while $7.35$ rounds to $7.4$.

A number-line diagram shows the interval clearly, including which endpoint is open and which is closed.

Absolute and percentage error

Absolute error is the non-negative difference between an approximate value and the exact value. Calculate it using

Percentage error is the absolute error expressed as a percentage of the exact value. It is

When the exact value isn't available but the rounding interval is known, the bounds can be used to find the largest possible error.

Take extra care with derived quantities, since an input error may become larger. For a circle,

If the radius has been rounded, substitute both radius bounds into the area formula. Then compare the reported area with the extreme areas obtained to find the maximum possible percentage error. Don’t carry the radius error straight across to the area: squaring changes its effect.

Measurement uncertainty may arise from rounding, instrument resolution or practical limitations when taking a measurement. Precision tells us how finely a value is reported. On its own, it doesn’t guarantee accuracy.

Estimation and reasonableness

Before or after calculating, estimate by rounding the inputs to convenient values and checking the order of magnitude. The result must also fit the context. A length cannot be negative and a probability cannot exceed one. An answer that differs from the estimate by many powers of ten needs investigating.

Approximation isn't just cosmetic. Too much rounding can distort scientific conclusions, financial information or public forecasts. On the other hand, unjustified extra digits may suggest a degree of certainty that the data cannot support. Mathematical reasoning establishes what follows from the assumptions chosen. Scientific reasoning must also judge how well the measurements and models represent the physical world.

Systems of linear equations

A system of linear equations is a collection of first-degree equations whose variables must satisfy every equation simultaneously. A solution of a system is an ordered set of values that makes all equations in the system true at the same time.

Technology can solve systems with up to three variables. The unknowns may be labelled $X$, $Y$ and $Z$. Each is an abstract real variable with no fixed SI unit, although in an application it takes the unit of the quantity it represents.

A dependable calculator or software procedure is:

1. Rewrite each equation in the form required by the system solver.
2. Choose the number of variables.
3. Enter the coefficients in a consistent variable order. Include zeros for any missing terms.
4. Read the solution, then substitute it into the original equations to check it.

You don’t need to use a particular algebraic method here. In assessed systems, the equations will have a unique solution. Geometrically, this happens for two linear equations in two variables when their lines meet at exactly one point. For three equations, the planes must share exactly one point.

Systems arise naturally when a model contains simultaneous constraints. In physics, for example, Kirchhoff's laws give linked linear equations for currents and potential differences in electrical circuits.

Polynomial equations

A polynomial equation is an equation formed by setting a finite sum of non-negative integer powers of a variable equal to zero. A root, also called a zero, is a value of the variable that makes the polynomial equal to zero.

For example, consider

In a contextual model, however, it may represent a quantity with an appropriate SI unit. Enter every coefficient in descending power order, using a zero coefficient whenever a power is missing. Technology may return several roots. Record every solution requested, and distinguish exact-looking outputs from decimal approximations.

On the graph of a polynomial function, real roots are the horizontal-axis intercepts. This links equation solving to quadratic and other polynomial models.

Mathematical language

Words such as root, zero, real and imaginary have precise technical meanings, which differ from their everyday uses. This specialist vocabulary helps mathematicians share results efficiently, but readers may be misled if they interpret the words informally. Clear definitions are part of mathematical knowledge, not an optional extra.

Product, quotient and power laws

Logarithm laws turn multiplication into addition, division into subtraction, and powers into multiplication:

The conditions $p>0$ and $q>0$ are essential parts of these laws. They make sure that every logarithm shown is defined over the real numbers.

You can use the laws in either direction. For an expansion,

For condensation,

Watch out for sums. No law allows $\log(p+q)$ to be written as a sum of logarithms; the product law works only when the terms in the argument are multiplied.

Scaling and applications

Logarithms are useful when quantities cover many orders of magnitude. On a logarithmic scale, multiplicative changes become additive ones, linking these laws to the scaling of extremely large and small numbers.

In chemistry, logarithm laws are used in pH and buffer calculations. They also help with rearranging linearized relationships used to estimate activation energy from experimental data.

In mathematics, a law is a proven identity derived from definitions and established results. A scientific theory is different: it provides an explanatory framework that is evaluated against observations and can be revised when the evidence changes. The words law and theory therefore carry different expectations across different areas of knowledge.

Meaning of a rational exponent

A rational exponent is an exponent expressible as a ratio of two integers and interpreted through roots and integer powers. For a positive base,

An even root requires a non-negative radicand in real-number calculations, so a positive base avoids domain difficulties in the general laws.

The denominator tells you which root to take; the numerator gives the power. For example,

With a negative rational exponent, take the reciprocal:

For instance,

Simplifying expressions

The integer exponent laws still apply to rational exponents, as long as the expressions are defined. When the positive base is the same, add exponents for multiplication and subtract them for division:

In an algebraic expression, simplify the exponent first. Then choose whichever form—radical or exponent notation—is clearer. If $u$ is positive,

Watch the domain restrictions. Squaring and taking a square root aren’t automatically inverse operations for every real input. Since the principal square root is non-negative, expressions with even roots may need extra care when a variable isn’t known to be positive.