Real numbers and scientific notation

Start again with operations and the meaning of a fraction

A fraction a/b represents a divided by b, with b different from zero. In 3/4, the denominator indicates four equal parts of the unit and the numerator indicates three of those parts. The chosen unit matters: three quarters of a metre and three quarters of a kilometre have the same numerical value but represent different lengths. Multiplying numerator and denominator by the same nonzero number preserves the quotient: 3/4 = 6/8.

To add fractions, we need parts of the same size. Thus, 2/3 + 1/4 = 8/12 + 3/12 = 11/12. To multiply, 2/3 × 1/4 = 2/12 = 1/6: we are taking two thirds of one quarter. Dividing by 3/5 is equivalent to multiplying by 5/3, because the product of these two numbers is 1. For example, (4/7) ÷ (3/5) = 20/21.

Signs describe positions or directions. The opposite of −6 is 6; subtracting −6 means adding 6. By contrast, the reciprocal of −6 is −1/6, since their product is 1. Zero has an opposite but no multiplicative inverse. Before using a calculator, identify parentheses, powers, multiplications and divisions, and only then additions and subtractions. Multiplications and divisions at the same level are evaluated from left to right; the same applies to additions and subtractions at the same level, after higher-priority operations. Chained powers require attention to exponent structure: 2^(3²) = 512, whereas (2³)² = 64. Use parentheses to make this structure unambiguous.

Multiples, factors and a meaningful common denominator

A positive integer d is a divisor of n when n = dk for some integer k. A positive multiple of d has the form d, 2d, 3d, and so on. A prime number has exactly two positive divisors: 1 and itself. The number 1 is not prime. Decomposing a positive integer greater than 1 into prime factors organises its multiplicative structure.

For example, 18 = 2 × 3² and 30 = 2 × 3 × 5. A common multiple must contain enough factors to be divisible by both. The least common multiple, LCM, uses each prime with the greatest exponent needed: LCM(18,30) = 2 × 3² × 5 = 90. Why not simply multiply 18 by 30? That product is also a common multiple, but repeats factors that only needed to be included once and is not always the smallest.

O greatest common divisor, GCD, uses the factors appearing in both numbers, with their smallest exponents: GCD(18,30) = 2 × 3 = 6. It is useful for simplifying fractions or forming equal groups without leftovers. Do not confuse the greatest divisor with the least multiple: they answer different questions.

Adding fractions requires counting parts of the same size. For 7/18 + 1/30, denominator 90 allows 35/90 + 3/90 = 38/90 = 19/45. We multiplied numerator and denominator by the same number, preserving each fraction's value. We do not directly add denominators: that would change the size of the parts without justification.

With three numbers, the reasoning is the same. For 8 = 2³, 12 = 2² × 3 and 30 = 2 × 3 × 5, the LCM is 2³ × 3 × 5 = 120. Check divisions: 120/8 = 15, 120/12 = 10 and 120/30 = 4, all integers. The greatest-exponent method also explains why this multiple is the smallest possible positive one.

Which numbers belong to the reals?

We call …, −2, −1, 0, 1, 2, … integers and represent their set by ℤ. Rational numbers, ℚ, are numbers that can be written as a/b with integer a and b and b ≠ 0. Thus, −8 is rational because −8 = −8/1. A terminating decimal is also rational: 0,045 = 45/1000 = 9/200. A repeating decimal, with a block repeating indefinitely, is rational too.

We can demonstrate this last idea in a concrete case. If x = 0,272727…, then 100x = 27,272727…. Subtracting the equalities gives 99x = 27, so x = 27/99 = 3/11. The infinite repetition is exact; writing only 0,27 would already be an approximation.

Irrational numbers are real numbers that cannot be written as a quotient of integers. √2 and π are examples. The symbol ℝ combines rational and irrational numbers. Appearance does not determine classification: √81 = 9 is rational, whereas √80 = 4√5 is irrational. Natural-number notation varies between conventions that include or exclude zero; here, when we need to count objects, we will explicitly state 0, 1, 2, … or 1, 2, 3, ….

The real number line, distance and comparison

On the real number line, every number corresponds to a point and order increases to the right. Thus, −7 < −3, although 7 > 3: a debt of seven units leaves a smaller balance than a debt of three. Absolute value |x| is the distance from x to zero and is never negative; for example, |−7| = 7. The distance between a and b is |a − b|, so between −3 and 5 there are |5 − (−3)| = 8 units.

To compare fractions with positive denominators, we can use a common denominator. We have 5/8 = 15/24 and 2/3 = 16/24, so 5/8 < 2/3. If there are negative denominators, start by placing the sign in the numerator. To compare nonnegative roots, compare their squares: since 3² < 13 < 4², it follows that 3 < √13 < 4.

An approximation may help locate a point, but does not replace a proof when the difference is small. Knowing √13 ≈ 3,606 allows a sketch; knowing 3,60² = 12,96 and 3,61² = 13,0321 justifies the interval 3,60 < √13 < 3,61. The symbol ≈ warns that rounding occurred.

Why the laws of powers work

For a positive integer exponent, aⁿ means a product of n factors equal to a. Joining two products with the same base gives a^maⁿ = a^m+n. Raising a repeated product to another power gives (a^m)ⁿ = a^mn. By contrast, adding powers does not allow exponents to be added: 2² + 2³ = 4 + 8 = 12, not 2⁵.

For a ≠ 0, we want to preserve the quotient rule. Since a³/a³ = 1, we define a⁰ = 1. Since a²/a⁵ = 1/a³, we define a⁻³ = 1/a³. A negative exponent indicates a reciprocal, not a negative result: 10⁻⁴ = 0,0001. The expression 0⁰ is not assigned a value here, and zero raised to a negative exponent does not exist.

Parentheses change the base: (−4)² = 16, whereas −4² = −16, because in the latter notation the power acts on 4 first. For real square roots, we require a nonnegative radicand and choose the principal root, also nonnegative. Therefore √(x²) = |x|, not always x: for x = −5, the root is 5.

Scientific notation and decimal scale

A positive number is in scientific notation when written as c × 10ⁿ, with 1 ≤ c < 10 and integer n. For a negative number, the coefficient's magnitude lies in that interval. The coefficient shows the significant digits and the power shows the scale. Thus, 5 700 000 = 5,7 × 10⁶ and 0,000082 = 8,2 × 10⁻⁵. Zero is simply written 0.

Moving the decimal separator is not an uncompensated operation. In 57 × 10⁵, reducing 57 to 5,7 divides the coefficient by ten; increasing the exponent to 6 multiplies the power by ten, preserving the product. In multiplication, we multiply coefficients and add exponents; in division, we divide coefficients and subtract exponents. Then we normalise the notation.

In addition, we first need the same power of ten: 3,2 × 10⁵ + 7 × 10⁴ = 3,2 × 10⁵ + 0,7 × 10⁵ = 3,9 × 10⁵. The calculator's E notation means “times ten to the power”: 8,2E−5 represents 8,2 × 10⁻⁵. It does not mean subtracting five from the coefficient.

Units, rounding and checking the result

A measurement combines a number and a unit. The equality 2,4 × 10³ m = 2,4 km preserves the length, although the number changes. To convert areas, the factor is also squared: 1 cm² = (10⁻² m)² = 10⁻⁴ m². For volumes, 1 cm³ = 10⁻⁶ m³. Writing only “multiply by a hundred” without identifying the quantity can produce errors by factors of thousands.

Rounding to hundredths means choosing the nearest multiple of 0,01. For example, 6,284 becomes 6,28. The absolute error of this approximation is 0,004. Relative error compares that error with the reference value; multiplied by one hundred it gives a percentage. Keep intermediate digits and round at the end to reduce error accumulation.

Before accepting a calculation, ask three questions: does the sign make sense, is the scale plausible and is the unit the one requested? If you divide a distance by a time, the final unit is length per time. If an object of a few centimetres appears to be thousands of metres long, check exponents and conversion. The displayed precision must also respect the precision of the measured data.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Every infinite decimal is irrational.

Repeating decimals, such as 0,333…, are rational; irrational numbers have infinite nonrepeating expansions.

A negative exponent makes the number negative.

For a positive base it gives a positive reciprocal: 10⁻³ = 1/1000.

Converting cm² to m² uses the same factor as converting cm to m.

The length factor is squared when converting areas.

√(x²) is always x.

It is |x|, because the principal square root is nonnegative.