Numbers and Numerals: Key Concepts for Exam

Hindu-Arabic vs. Roman Numerals

The Hindu-Arabic numeral system uses the digits $0,1,2,3,4,5,6,7,8,9$ and is base-10, with zero as a key part of the system. Roman numerals use I, V, X, L, C, D, M and express values by addition, with subtractive notation when a smaller numeral precedes a larger one (for example, $IV=4, IX=9, XC=90, IC=99$ as shown in the lesson).

Base systems and binary overview

In base-10, place values correspond to powers of 10: ones $10^0$, tens $10^1$, hundreds $10^2$, etc., and decimals use negative powers like $10^{-1}$ (tenths) and $10^{-2}$ (hundredths). The base-2 system uses digits $0$ and $1$, with places given by powers of two like $2^0, 2^1, 2^2, \dots$; this is important for computers. A binary number such as $1011<em>2$ equals $11</em>{10}$. Color depth examples include $16 ext{-bit}$ and 32 ext{-bit}}, corresponding to color possibilities $2^{16}$ and $2^{32}$.

History and symbols in arithmetic

Arithmetic and algebra use symbols (plus, minus, times, division, parentheses) that were developed mainly in the 15th–17th centuries. The plus and minus signs also serve as symbols for positive and negative numbers; mathematics is described as a symbolic language for efficient communication in science.

Types of numbers and number lines

Real numbers include all positive and negative numbers with or without decimals and encompass rational and irrational numbers. Rational numbers can be written as a ratio of integers (e.g., frac{10}{2}=5, frac{7}{5}=1.4ar{2}); irrational numbers have nonrepeating, nonterminating decimals (e.g., $\pi$, $\sqrt{2}$). Imaginary or complex numbers arise from square roots of negative numbers. Number lines are tools to visualize signed numbers: zero in the middle, negatives to the left, positives to the right.

Squares, square roots, and memorization

For any number $n$, $n^2$ equals the square of that number. Inverse relation: $\sqrt{n^2}=|n|$ (and in many contexts simply $\sqrt{n^2}=n$ for nonnegative $n$). Memorize squares up to $13^2=169$. Examples: $\sqrt{36}=6\,,\quad \sqrt{121}=11].</p> <h3 id="complexnumbersandsimplifyingradicals">Complex numbers and simplifying radicals</h3> <p>The square root of negative one is denoted by$i$, with$i^2=-1$. Therefore,$\sqrt{-a}=\sqrt{a}\,i$when a is positive. Examples:$\sqrt{-19}=\sqrt{19}\,i,\quad \sqrt{-16}=4i,\quad (5i)^2=-5.$These lead to complex numbers of the form$a+bi$.</p> <h3 id="placevalueandarithmeticoperationsoverview">Place value and arithmetic operations (overview)</h3> <p>Place value underpins addition, subtraction, multiplication, and division, and related fact families show how operations undo one another (addition/ subtraction, multiplication/ division). For example, a number like$153$can be decomposed as$100+50+3$. In any multi-step calculation, align by place value and carry or borrow as needed. For money problems, align dollars and cents (e.g.,$42.10$,$14.15$, resulting in$27.95$). Subtraction should always be performed with the larger value on top.</p> <h3 id="multiplicationanddecimalarithmetic">Multiplication and decimal arithmetic</h3> <p>Multiplication is a fast way to add equal groups; the product is the result of the factors. Example:$24\times 315=7560$. When multiplying decimals, ignore the decimal points during the multiplication, then place the decimal point in the product so that the total number of decimal places equals the sum of the decimal places in the factors (e.g.,$1.3\times 22.4=29.12$, two decimal places in the product).</p> <h3 id="divisionandquotientincludingremaindersanddecimals">Division and quotient (including remainders and decimals)</h3> <p>Division yields a quotient; long division can produce a remainder (e.g.,$450\div 7=64\text{ remainder }2$, i.e.,$64\frac{2}{7}$). To obtain decimal results, append zeros to continue dividing (e.g.,$64.285714…$for$64\frac{2}{7}$). A practical word problem approach: identify what is being asked, and apply division accordingly (e.g., money divided by price per item). A story problem example:$50\div 2.50=20$chickens.</p> <h3 id="quickrecapforexamreview">Quick recap for exam review</h3> <ul> <li>Base-10 with digits$0-9$; base-2 uses$0,1$; place values follow powers of the base. </li> <li>Subtractive roman notation: smaller before larger subtracts; examples include$IV=4, IX=9, XC=90, IC=99$.</li> <li>Real vs rational vs irrational vs imaginary/complex numbers; square roots can lead to complex numbers. </li> <li>Memorize squares up to$13^2=169$; know simple roots like$\sqrt{36}=6,\;\sqrt{121}=11$;$\sqrt{-1}=i$$.