Pure Arithmetic

The One Thing

Pure arithmetic develops exact methods for classifying numbers, simplifying powers and surds, solving exponential equations with logarithms, and converting recurring decimals into fractions. These methods depend on applying the relevant laws together with their restrictions, particularly non-zero denominators, valid logarithm domains, and correctly matched surds.

Definitions and Results

• Rational number: A number that can be written in the form $p/q$, where $p$ and $q$ are integers and $q\ne 0$. Its decimal expansion either terminates or recurs.
• Irrational number: A number that cannot be expressed as the ratio of two integers. Its decimal expansion is non-terminating and non-recurring. Examples include $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and $\pi$.
• Real number: Any number that is either rational or irrational. The rational and irrational numbers do not overlap.
• Terminating decimal: A decimal that ends after a finite number of digits, such as $0.75$.
• Recurring decimal: A decimal in which one or more digits repeat endlessly, such as $0.333\dots$, $0.272727\dots$, or $0.121212\dots$.
• Terminating-decimal criterion: A fraction in lowest terms has a terminating decimal if and only if its denominator has no prime factors other than $2$ and $5$.
• Index or exponent: The small raised number showing how many times a base is multiplied by itself.
• Base: The number repeatedly multiplied in a power expression.
• Laws of indices: For a non-zero number $a$,

$$a^m\times a^n=a^{m+n},$$ $$a^m÷a^n=a^{m-n},$$ $$(a^m{)}^n=a^{mn},$$ $$(ab{)}^n=a^n{b}^n,$$ $${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n},b\ne 0,$$ $$a^0=1,a\ne 0,$$ $$a^{-n}=\frac{1}{a^n},a\ne 0,$$ $$a^{1/n}=\sqrt[n]{a},a^{m/n}=\sqrt[n]{a^m}.$$

• Logarithm: The exponent to which a given base must be raised to obtain a specified number. In

$$a^x=N,$$ the logarithmic form is $${\log }_{a}N=x.$$

• Common logarithm: A logarithm with base $10$, written as $\log$ or ${\log }_{10}$. In particular,

$$\log 10=1,\log 100=2,\log 1=0.$$

• Characteristic: The integral part of a common logarithm.
• Mantissa: The decimal part of a common logarithm.
• Logarithm laws: For valid logarithms,

$${\log }_a(MN)={\log }_{a}M+{\log }_{a}N,$$ $${\log }_a\left(\frac{M}{N}\right)={\log }_{a}M-{\log }_{a}N,$$ $${\log }_a(M^n)=n{\log }_{a}M.$$ The base must be positive and not equal to $1$, and every logarithm argument must be positive.

• Surd: An irrational root, such as $\sqrt{2}$ or $3\sqrt{5}$, that cannot be simplified to a rational number.
• Surd simplification: Simplify each surd before adding, subtracting, multiplying, or dividing. Only like surds may be combined.
• Conjugate surds: The product

$$(a+\sqrt{b})(a-\sqrt{b})=a^2-b$$ is rational.

• Scientific notation: A number written as

$$A\times 10^n,$$ where $1\le A<10$ and $n$ is an integer.

• Recurring-decimal result: Every recurring decimal represents a rational fraction.

Worked Methods

1. Classifying rational and irrational numbers

• Determine whether the number can be expressed as $p/q$, with $p,q$ integers and $q\ne 0$.
• If it can, classify it as rational.
• If it cannot, and its decimal expansion is non-terminating and non-recurring, classify it as irrational.
• Classify all rational and irrational numbers as real numbers.

Examples of irrational numbers specified in the material are $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and $\pi$.

For a fraction in lowest terms, inspect the denominator:

• if its prime factors are only $2$ and $5$, its decimal terminates;
• otherwise, its decimal recurs.

2. Applying the laws of indices

• For multiplication with the same base, add exponents:

$$a^m\times a^n=a^{m+n}.$$

• For division with the same base, subtract exponents:

$$a^m÷a^n=a^{m-n}.$$

• For a power raised to a power, multiply exponents:

$$(a^m{)}^n=a^{mn}.$$

• Distribute an exponent across a product or quotient:

$$(ab{)}^n=a^n{b}^n,{\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}.$$

• Replace a zero exponent by $1$, provided the base is non-zero:

$$a^0=1.$$

• Rewrite a negative exponent as a reciprocal:

$$a^{-n}=\frac{1}{a^n}.$$

• Rewrite fractional exponents as roots:

$$a^{1/n}=\sqrt[n]{a},a^{m/n}=\sqrt[n]{a^m}.$$

These laws arise from repeated multiplication and extend to zero, negative, and fractional exponents subject to their restrictions.

3. Working with logarithms

• Identify the base, exponent, and result in an equation of the form

$$a^x=N.$$

• Rewrite it as

$${\log }_{a}N=x.$$

• Use the logarithm laws where products, quotients, or powers occur:

$${\log }_a(MN)={\log }_{a}M+{\log }_{a}N,$$ $${\log }_a(M/N)={\log }_{a}M-{\log }_{a}N,$$ $${\log }_a(M^n)=n{\log }_{a}M.$$

• For common logarithms, use base $10$:

$$\log 10=1,\log 100=2,\log 1=0.$$

• Interpret a common logarithm as characteristic plus mantissa: the characteristic is the integral part and the mantissa is the decimal part.

Throughout, the logarithm base must be positive and not equal to $1$, and the argument must be positive.

4. Converting a pure recurring decimal into a fraction

For $0.777\dots$:

• Let

$$x=0.777\dots$$

• Multiply by $10$, since one digit recurs:

$$10x=7.777\dots$$

• Subtract the original equation:

$$10x-x=7.777\dots -0.777\dots$$

• Hence,

$$9x=7.$$

• Therefore,

$$x=\frac{7}{9}.$$

Similarly, $$0.121212\dots =\frac{12}{99}=\frac{4}{33}.$$

5. Converting a mixed recurring decimal

• Let the mixed recurring decimal equal $x$.
• Multiply by suitable powers of $10$ so that the repeating parts align.
• Subtract the resulting equations.
• Solve the resulting linear equation for $x$.
• Simplify the fraction obtained.

The powers of $10$ must account separately for the non-repeating part and the repeating block.

6. Simplifying surds

• Factor the number inside the root to identify any perfect-square factors.
• Extract the square root of each perfect-square factor.
• Repeat for every surd in the expression.
• Combine only like surds when adding or subtracting.
• For products or quotients, simplify each factor and apply the relevant algebraic operation.

For conjugate surds: $$(a+\sqrt{b})(a-\sqrt{b})=a^2-b,$$ so the irrational terms cancel and the result is rational.

7. Using scientific notation

• Move the decimal point so that exactly one non-zero digit remains to the left.
• Write the resulting number as $A$, with $1\le A<10$.
• Count the number of places moved.
• Use a positive exponent when the original number is large and a negative exponent when it is small.
• Express the result as

$$A\times 10^n.$$

Where It Goes Wrong

• Treating a non-terminating decimal as automatically irrational; recurring decimals are rational.
• Forgetting that a terminating decimal criterion applies to a fraction in lowest terms, whose denominator must have no prime factors other than $2$ and $5$.
• Applying index laws without respecting restrictions such as $a\ne 0$ and $b\ne 0$, especially for $a^0$, negative exponents, and quotients.
• Combining unlike surds instead of simplifying each surd first and combining only like surds.
• Using logarithms with a base equal to $1$, a non-positive base, or a non-positive argument.
• Failing to align the repeating parts when subtracting equations for a mixed recurring decimal.

What Gets Asked

• Classify given numbers as rational, irrational, or real, including $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and $\pi$.
• Decide whether a fraction has a terminating or recurring decimal using the prime factors of its denominator.
• Simplify expressions using the laws of indices, including zero, negative, and fractional exponents.
• Convert between exponential and logarithmic forms and apply the basic logarithm laws.
• Evaluate or interpret common logarithms, including characteristic and mantissa, using $\log 10=1$, $\log 100=2$, and $\log 1=0$.
• Convert pure recurring decimals such as $0.777\dots$ and $0.121212\dots$ into exact fractions.
• Convert mixed recurring decimals by aligning repeating parts and subtracting equations.
• Simplify, combine, and rationalise expressions involving surds, including conjugate products.
• Express numbers in scientific notation as $A\times 10^n$, where $1\le A<10$.