Number Systems and Operations
What this domain tests
The Number systems and operations domain tests the structure of the real and complex number systems, properties of operations, order of operations, absolute value, scientific notation, and complex arithmetic. CLEP College Algebra questions often ask you to classify a number or identify which property justifies a step.
Real number subsets and properties
- Natural numbers, whole numbers, integers, rational numbers, and irrational numbers form nested subsets of the real numbers. A rational number can be written as $p/q$ with integers $p$ and $q$, $q\ne0$.
- The commutative and associative properties apply to addition and multiplication, not to subtraction or division. The distributive property connects multiplication with addition or subtraction.
- The additive identity is $0$; the multiplicative identity is $1$. The additive inverse of $a$ is $-a$; the multiplicative inverse is $1/a$ for $a\ne0$.
- A rational plus an irrational is irrational, but an irrational times an irrational can be rational, as with $\sqrt2\cdot\sqrt2=2$.
Order of operations and absolute value
- Evaluate grouping symbols first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right.
- Absolute value gives distance from zero: $|a-b|$ is the distance between $a$ and $b$. Absolute value is never negative.
- In scientific notation $a\times10^n$, keep $1\le|a|<10$. Moving the decimal point changes the exponent in the opposite direction.
Complex numbers
- Write complex numbers as $a+bi$, where $i^2=-1$. Add and subtract like terms, and multiply by distributing and replacing $i^2$ with $-1$.
- To divide, multiply numerator and denominator by the conjugate of the denominator. Powers of $i$ repeat in the cycle $i,-1,-i,1$.
Worked example
Simplify $\frac{3+2i}{1-4i}$.
Multiply numerator and denominator by the conjugate $1+4i$: the denominator becomes $1^2+4^2=17$. The numerator becomes $(3+2i)(1+4i)=3+12i+2i+8i^2=3+14i-8=-5+14i$. Therefore the quotient is $\frac{-5+14i}{17}=-\frac{5}{17}+\frac{14}{17}i$.
Traps that cost marks
- Labeling $\sqrt2$ as rational or treating every square root as irrational.
- Mishandling the sign in $i^2=-1$, especially in products such as $8i^2$.
- Distributing an exponent over a sum, as in $(a+b)^2=a^2+b^2$.
- Forgetting that division by zero is undefined, or that absolute value cannot equal a negative number.