The practical math behind tech and STEM work: numbers, algebra, functions, geometry, logic and counting, with short code exercises.

Math for Tech & STEM

## Why this matters

Almost every technical job uses the same quantity written three different ways. A lab report says a solution is 0.25 molar, a dashboard says sign-ups grew 12%, and a drawing says a bolt is 3/8 inch. If you can move between fractions, decimals and percents without hesitating, you read reports faster and you catch mistakes that other people miss.

The good news: fractions, decimals and percents are not three different ideas. They are three ways of writing one idea, a part of a whole.

## The core idea

- A **fraction** writes the part over the whole: 3/4 means "3 out of 4 equal pieces".
- A **decimal** writes the same amount in base ten: 3/4 = 0.75.
- A **percent** writes it "per hundred": 0.75 = 75/100 = 75%.

The word *percent* literally means "per hundred". That single fact gives you every conversion:

| From | To | How | Example |
|---|---|---|---|
| fraction | decimal | divide top by bottom | 3/8 = 3 ÷ 8 = 0.375 |
| decimal | percent | multiply by 100 | 0.375 × 100 = 37.5% |
| percent | decimal | divide by 100 | 37.5% ÷ 100 = 0.375 |
| percent | fraction | write over 100, then simplify | 75% = 75/100 = 3/4 |

Multiplying by 100 just moves the decimal point two places to the right; dividing by 100 moves it two places to the left.

## Simplifying fractions

A fraction is simplified when the top and bottom share no common factor other than 1. To simplify 18/24, find the greatest common factor (6) and divide both parts by it: 18/24 = 3/4. The value has not changed; you have just written it with smaller numbers.

## Worked example 1: a test pass rate

Your test suite has 48 tests and 42 pass. What is the pass rate?

1. Fraction: 42/48. Both divide by 6, so this is 7/8.
2. Decimal: 7 ÷ 8 = 0.875.
3. Percent: 0.875 × 100 = 87.5%.

## Worked example 2: percent of a quantity

A battery is rated 3,000 mAh and the phone says 35% remains. How much charge is left?

"35% of 3,000" means 0.35 × 3,000 = 1,050 mAh. Converting the percent to a decimal first turns "of" into plain multiplication.

## Worked example 3: percent change

A web page loaded in 2.5 seconds last month and 2.0 seconds now. By what percent did load time fall?

Percent change = (new − old) ÷ old × 100 = (2.0 − 2.5) ÷ 2.5 × 100 = −20%.

The negative sign means a decrease. Always divide by the **old** (starting) value. Dividing by the new value is the single most common percent mistake.

## Repeating decimals

Some fractions never end as decimals: 1/3 = 0.3333... and 2/11 = 0.181818... Computers store these as close approximations, which is why software sometimes prints 0.30000000000000004 when you add 0.1 and 0.2. In Python, the `fractions` module keeps exact values when you need them:

```python
from fractions import Fraction

print(Fraction(18, 24))          # 3/4  (simplified automatically)
print(3 / 8)                     # 0.375
print(0.375 * 100)               # 37.5
print(Fraction(1, 3) + Fraction(1, 6))   # 1/2, exactly
```

## Common mistakes

- **Forgetting the ×100.** 0.08 is 8%, not 0.08%. A tiny-looking decimal can be a big percent.
- **Adding percents of different wholes.** A 10% rise followed by a 10% fall does not return you to the start: 100 → 110 → 99.
- **Percent vs. percentage points.** Going from 4% to 6% is a rise of 2 percentage points, but a 50% increase in the rate itself.
- **Dividing by the wrong base** in percent change (see example 3).

## Quick recap

- Fraction → decimal: divide. Decimal → percent: × 100. Percent → decimal: ÷ 100.
- Simplify fractions by dividing top and bottom by their greatest common factor.
- Percent change = (new − old) ÷ old × 100.

## Try it

Open the **Do** tab. The starter code already converts fractions to decimals and percents to fractions; you write the one missing formula that turns a decimal into a percent.


Convert confidently between fractions, decimals and percents, and solve everyday percent problems.

Fractions, decimals, and percents

Converting Between Fractions, Decimals, and Percentages

Converting percent to decimal and fraction | Decimals | Pre-Algebra | Khan Academy

## Why this matters

Technical work spans enormous ranges. A data center stores petabytes (about 10^15 bytes), a transistor gate is a few nanometres wide (around 10^-9 m), and one mole of a substance contains about 6.02 × 10^23 particles. Writing those numbers out in full is slow and error-prone. Exponents and scientific notation let you write, compare and multiply them in seconds.

## Exponents

An exponent says how many times to multiply a base by itself:

- 2^3 = 2 × 2 × 2 = 8
- 10^4 = 10 × 10 × 10 × 10 = 10,000
- 5^1 = 5, and any non-zero number to the power 0 is 1 (10^0 = 1)
- A negative exponent means "one over": 10^-3 = 1 / 10^3 = 0.001

In Python, the exponent operator is `**` (not `^`, which means something else in code):

```python
print(2 ** 3)      # 8
print(10 ** -3)    # 0.001
```

## The rules you will use most

| Rule | Example |
|---|---|
| Multiply same base: add exponents | 10^3 × 10^4 = 10^7 |
| Divide same base: subtract exponents | 10^8 ÷ 10^2 = 10^6 |
| Power of a power: multiply exponents | (10^2)^3 = 10^6 |
| Power of a product | (2 × 10)^2 = 4 × 10^2 |

These rules only combine exponents when the **base is the same**. 2^3 × 3^2 cannot be merged into one power; you just compute 8 × 9 = 72.

## Scientific notation

A number in scientific notation has two parts:

**coefficient × 10^exponent**, where the coefficient is at least 1 and less than 10.

- 4,500,000 = 4.5 × 10^6 (the decimal point moved 6 places left)
- 0.00032 = 3.2 × 10^-4 (the decimal point moved 4 places right)

A quick check: big numbers get positive exponents; numbers between 0 and 1 get negative exponents.

Computers and calculators print the same thing with an `e`: `4.5e6` means 4.5 × 10^6, and `3.2e-4` means 3.2 × 10^-4. You will see this in spreadsheets, Python and lab instruments.

```python
print(4.5e6)              # 4500000.0
print(f"{0.00032:.1e}")   # 3.2e-04
```

## Worked example 1: multiplying

(3 × 10^8) × (2 × 10^-3)

1. Multiply the coefficients: 3 × 2 = 6.
2. Add the exponents: 8 + (−3) = 5.
3. Answer: 6 × 10^5.

## Worked example 2: re-normalising

(5 × 10^3) × (4 × 10^2)

1. Coefficients: 5 × 4 = 20.
2. Exponents: 3 + 2 = 5, giving 20 × 10^5.
3. 20 is not between 1 and 10, so rewrite 20 as 2.0 × 10^1 and add that exponent: **2.0 × 10^6**.

## Worked example 3: dividing

How many 2 × 10^-6 m bacteria fit end to end across 1 × 10^-2 m (one centimetre)?

(1 × 10^-2) ÷ (2 × 10^-6) = 0.5 × 10^4 = 5 × 10^3, so about 5,000.

## Orders of magnitude

The exponent alone tells you the "size class" of a number. 3 × 10^5 and 8 × 10^5 are the same order of magnitude; 3 × 10^8 is three orders larger (about 1,000 times). Engineers use this for quick sanity checks: if your answer is off by an order of magnitude, look for a units or decimal-point slip.

## Common mistakes

- **Moving the decimal the wrong way.** 0.0045 is 4.5 × 10^-3, not 4.5 × 10^3.
- **Adding exponents when adding numbers.** 10^3 + 10^3 = 2 × 10^3, not 10^6. Only multiplication adds exponents.
- **Forgetting to re-normalise** a coefficient of 10 or more (example 2).
- **Typing `^` in code.** In Python and JavaScript `^` is bitwise XOR; use `**`.

## Quick recap

- Exponents count repeated multiplication; negative exponents mean division.
- Multiply: add exponents. Divide: subtract. Power of a power: multiply.
- Scientific notation keeps the coefficient between 1 and 10; `e` notation is the same thing for computers.

## Try it

In the **Do** tab, finish a function that multiplies two numbers in scientific notation. The re-normalising step is already written for you.


Use exponent rules and scientific notation to work with very large and very small numbers.

Exponents and scientific notation

Scientific Notation - Explained!

Algebra Basics: Laws Of Exponents - Math Antics

## Why this matters

A number without a unit is only half an answer. "The pipe is 3" could mean 3 metres or 3 inches, and that difference has cost real projects money. In 1999 NASA lost the Mars Climate Orbiter partly because one team's software produced thrust data in pound-force seconds while another expected newton seconds. Dimensional analysis is the habit that prevents that kind of error: you carry units through every step and let them tell you whether the setup is right.

## The core idea: multiply by one

A **conversion factor** is a fraction whose top and bottom are equal amounts in different units, such as 1000 m / 1 km. Because the top equals the bottom, the fraction equals 1, and multiplying by 1 never changes a quantity. It only changes the units it is written in.

Write the factor so the unit you want to get rid of appears on the **opposite** side, then cancel it like a common factor.

## Worked example 1: one step

Convert 5 km to metres.

5 km × (1000 m / 1 km) = 5000 m

The "km" on top and the "km" on the bottom cancel, leaving metres.

## Worked example 2: a compound unit

Convert 90 km/h to metres per second.

```
90 km     1000 m      1 h
------ × -------- × --------  =  25 m/s
  1 h      1 km      3600 s
```

1. km cancels with km, and h cancels with h.
2. The numbers: 90 × 1000 ÷ 3600 = 25.
3. The units left are m/s, which is what we wanted. If the units had come out as h·m/s², you would know the setup was wrong before you even looked at the number.

## Worked example 3: squared units

A room is 12 m². How many square feet is that? (1 m = 3.2808 ft)

Area units are squared, so the factor must be squared too:

12 m² × (3.2808 ft / 1 m)² = 12 × 10.764 ft² ≈ 129.2 ft²

Using the factor only once would give 39.4, an answer that is wrong by a factor of 3.28. The same rule applies to volume: cube the factor.

## Worked example 4: a data rate

How many gigabytes does a 25 Mbit/s video stream use in one hour? (8 bits = 1 byte, 1000 MB = 1 GB)

25 Mbit/s × 3600 s/h × (1 byte / 8 bits) × (1 GB / 1000 MB) = 11.25 GB per hour

Notice the bits-to-bytes step. Network speeds are usually in bits; file sizes are usually in bytes.

## Doing it in code

A program can apply a chain of factors the same way: start with the value and multiply by each top/bottom pair in turn.

```python
value = 90                       # km/h
for top, bottom in [(1000, 1), (1, 3600)]:   # m/km, h/s
    value = value * top / bottom
print(value)                     # 25.0 (m/s)
```

## Useful exact factors

- 1 inch = 2.54 cm (exact, by definition)
- 1 mile = 1.609344 km (exact)
- 1 hour = 3600 s; 1 day = 86,400 s
- 1 litre = 1000 cm³ = 0.001 m³
- 1 kWh = 3.6 × 10^6 J

## Common mistakes

- **Flipping the factor.** If the unit you want to cancel is on top, the factor needs it on the bottom. Write units on every number and the mistake becomes visible.
- **Forgetting to square or cube** for area and volume.
- **Mixing bits and bytes**, or decimal (1000) and binary (1024) prefixes. Say which one you used.
- **Dropping units halfway** through a calculation. Keep them until the very end.

## Quick recap

- A conversion factor equals 1, so multiplying by it changes units, not quantity.
- Arrange factors so unwanted units cancel; the units left over check your setup.
- Square factors for area, cube them for volume.

## Try it

In the **Do** tab, finish a small converter that applies a chain of conversion factors, then use it for km/h to m/s and mph to m/s.


Convert simple and compound units by multiplying by conversion factors and cancelling units.

Units and dimensional analysis

Unit Conversion the Easy Way (Dimensional Analysis)

Scientific Notation and Dimensional Analysis

Fractions, decimals, percents, exponents, scientific notation and unit conversion: the number sense every technical role relies on.