Scientific and Engineering Notation Explained

The distance from Earth to the Sun is about 149,600,000,000 metres. The mass of a proton is 0.00000000000000000000000000167 kilograms. Writing out all those zeros is not just tedious - it is error-prone. Scientific notation was invented to make extreme numbers readable, comparable, and safe from dropped digits.

The format

Scientific notation writes every number as m × 10^n, where the coefficient m satisfies 1 ≤ |m| < 10 and n is an integer. The exponent tells you how many places to shift the decimal point.

Standard → Scientific
149,600,000,000     → 1.496 × 10^11
0.00000000167       → 1.67 × 10^-9
speed of light      → 2.998 × 10^8 m/s
Planck constant     → 6.626 × 10^-34 J·s

Positive exponents mean large numbers (move the decimal right). Negative exponents mean small numbers (move it left). The coefficient keeps only the significant digits, which avoids ambiguity about precision.

Engineering notation

Engineering notation is a variant where the exponent is always a multiple of 3. This aligns with metric prefixes: 10^3 = kilo, 10^6 = mega, 10^9 = giga, 10^-3 = milli, 10^-6 = micro.

Logarithms: the inverse of exponents

If 10^3 = 1000, then log₁₀(1000) = 3. The logarithm asks: "what exponent produces this number?" Logarithms compress enormous ranges into manageable scales. The Richter scale, decibels, and pH are all logarithmic.

Common vs natural logs

• Common log (log₁₀) - Base 10. Used in engineering, pH, decibels. log₁₀(100) = 2.
• Natural log (ln) - Base e ≈ 2.71828. Used in calculus, compound interest, and population growth. ln(e) = 1.

Converting between bases is one formula: log_b(x) = ln(x) / ln(b). This is how calculators compute logarithms in any base using only the natural log.

The quadratic formula

The equation ax² + bx + c = 0 appears in physics (projectile motion), finance (break-even analysis), and optimisation. Its solution has been known since Babylonian mathematicians circa 2000 BC, though they solved it geometrically. The algebraic formula we use today was formalised during the Islamic Golden Age:

x = (-b ± sqrt(b² - 4ac)) / (2a)

The discriminant (b² - 4ac) determines the solution type:
  > 0  →  two real roots
  = 0  →  one repeated root
  < 0  →  two complex roots (no real solutions)

Here is y = x² − 3x + 2 plotted. The parabola crosses zero at x = 1 and x = 2 - the two roots. The vertex sits at the midpoint between them.

The discriminant decides the shape of that crossing. Three parabolas, three outcomes:

Matrices: systems of equations

A matrix is a rectangular grid of numbers. Matrices are used to solve systems of linear equations, transform 3D graphics, and train machine learning models. A system like "2x + 3y = 8, x - y = 1" becomes a matrix equation Ax = b, which can be solved by row reduction or matrix inversion.

Key operations include addition (element by element), multiplication (dot products of rows and columns), and finding the determinant (a single number that tells you whether the system has a unique solution).

Number bases

We normally count in base 10, but computers use base 2 (binary), base 8 (octal), and base 16 (hexadecimal). Each system uses a different set of digits: binary uses 0 and 1, hex uses 0-9 and A-F.

Decimal 255 in different bases:
Binary:       11111111
Octal:        377
Hexadecimal:  FF

Why hex? Two hex digits represent exactly one byte (8 bits).
FF = 1111 1111 = 255

Scientific notation, logarithms, and number bases are all tools for the same job: making numbers fit the scale of the problem. The right representation turns an unreadable wall of digits into an insight.