The simplest category of numbers is whole numbers: rational numbers for which the denominator is 1; real numbers with no fractional part; complex numbers with no imaginary part for which the value has no fractional part - all different ways of describing a whole number. The exercises in this group, therefore, extensively test the mental abilities of students in doing the basic arithmetic operations - without the aid of calculator or other tools.

There are ten sub-categories in the sets of exercises described and indexed in this document. Each category should normally be used before the one next following in the list, since each one builds upon experience already gained in number manipulation in earlier groups. The list of groups is not, however, ordered because the needs of individual students may dictate some other manner of study.

In addition to the main groups of exercises in the lists below which require the student to evaluate anb answer, there is a separate small collection in shich the students provide the operators to give the required answer - good for speed testing!

Each of the exercises indexed below contains 20 individual examples with the aim of exercising some particular student ability. The links refer to PDF files which may be printed and handed out as appropriate. Where available there is a link to a corresponding answer sheet (again as a PDF file) which may be consulted as needed -

• Addition - the exercises here progress from single digit, single operator examples through to multiple digit and multiple operator examples -
  1. One and two digits - answers here.
  2. Two and three digits - answers here.
  3. Four digits - answers here.
  4. One and two digits, two operations - answers here.
  5. One - three digits, two operations - answers here.
  6. Two and three digits, two operations - answers here.
  7. Three and four digits, two operations - answers here.
  8. Single digits, three operations - answers here.
  9. Two digits, four operations - answers here.
  10. Three and four digits, four or more operations - answers here.
• Subtraction - as with the addition exercises, these progress from single digit, single operator to multiple digit, multiple operator examples -
  1. Single digits, one operation - answers here.
  2. One and two digits, one operation - answers here.
  3. Two digits, two operations - answers here.
  4. Three digits, one operation - answers here.
  5. Four digits, one operation - answers here.
  6. Two digits, two operations, possible negation - answers here.
  7. Two and three digits, two operations, possible negation - answers here.
  8. Four digits, two operations, possible negation - answers here.
  9. Three digits, three operations, possible negation - answers here.
  10. Mixed digits, four or more operations, possible negation - answers here.
• Terms - a group of exercises which combine addition, Subtraction and negation operators as well as introducing the use and effects of parentheses where subtraction is involved -
  1. Two digits, two operations - answers here.
  2. Two or three digits, two operations - answers here.
  3. One to four digits, two operations - answers here.
  4. Two or three digits, three operations - answers here.
  5. Three digits, three operations - answers here.
  6. One to four digits, three operations - answers here.
  7. Up to four digits, three or four operations - answers here.
  8. Two to four digits and parentheses - answers here.
  9. Multiple digits and multiple parentheses - answers here.
  10. Multiple digits, multiple parentheses and negation - answers here.
• Multiplication - beginning with single digit single operator examples these exercises progress rather more slowly to multiple digit and multiple operator examples -
  1. One/two digits, one operation simple - answers here.
  2. One/two digits, one operation - answers here.
  3. One/two digits, one operation harder - answers here.
  4. One/two digits (up to 30), one operation - answers here.
  5. One/two digits (up to 50), one operation - answers here.
  6. One/two digits (up to 99), one operation - answers here.
  7. Two digits, one operation - answers here.
  8. Two digits, one operation - answers here.
  9. One to three digits, one operation - answers here.
  10. One to three digits, two operators - answers here.
• Division - including remainder and modulus operators, which are introduced about half way through this group. It is important to notice that all division is integer division - there are no fractional results!
  1. One/two digits, one operation - answers here.
  2. One/two digits, one operation - answers here.
  3. One to three digits, one operation - answers here.
  4. One to three digits, two operators - answers here.
  5. One/two digits, one modulus operator - answers here.
  6. One/two digits, one operator, negative values - answers here.
  7. One to three digits, one operator, negative values - answers here.
  8. One to three digits, one modulus operator, negative values - answers here.
  9. One to three digits, one modulus or remainder operator, negative values - answers here.
  10. One/two digits, mixed operators, negative values, parentheses - answers here.
• Factors - involving only multiplication and division, these exercises not only involve varying number of digits and operators, but some introduce the idea of negation and the use of parentheses to control evaluation order -
  1. One/two digits, two operators - answers here.
  2. One to three digits, two operators - answers here.
  3. One to three digits, two operators, parentheses - answers here.
  4. One to three digits, two/three operators, parentheses - answers here.
  5. One to three digits, two operators - answers here.
  6. Two to four digits, two operators, cross factors - answers here.
  7. Two to four digits, two operators, cross factors, parentheses - answers here.
  8. One to three digits, three operators, parentheses, negative numbers - answers here.
  9. One to three digits, three operators, parentheses, negative numbers - answers here.
  10. One to four digits, multiple operators, etc - answers here.
• Simple Expressions - not involving powers, these exercises combine all of the simple operators above in many ways, involving negation, parentheses to control the order of evaluation, etc -
  1. One/two digits - answers here.
  2. One/two digits, parentheses - answers here.
  3. One/two digits, parentheses - answers here.
  4. One to three digits, parentheses - answers here.
  5. One to four digits, parentheses - answers here.
  6. All prime numbers, parentheses - answers here.
  7. Up to four digits, parentheses - answers here.
  8. Up to four digits, parentheses - answers here.
  9. Up to four digits, parentheses - answers here.
  10. Up to four digits, all operators - answers here.
• Powers - a group of exercises which involve powers and roots as they are used in whole number arithmetic - answers always being whole numbers -
  1. Powers as multiplication - answers here.
  2. Cubics by multiplication - answers here.
  3. One/two exponents involved - answers here.
  4. Simple evaluation - answers here.
  5. Simplify - answers here.
  6. Simplify - with negative exponents - answers here.
  7. Simplify with fractional exponents - answers here.
  8. Simplify complex expressions - answers here.
  9. Evaluate exponent expressions - answers here.
  10. Evaluate parenthesised expressions - answers here.
• Expressions - containing all of the above in arbitrary combinations. Despite their apparent complexity, they should all be readily evaluated without the aid of calculators or tables! Since they are all merely merging the simple expressions with exponent expressions, however, a lesser number are provided here. These are generally in order of increasing difficulty.
  1. General expressions easiest - answers here.
  2. General expressions - answers here.
  3. General expressions - answers here.
  4. General expressions - answers here.
  5. General expressions hardest - answers here.

The following groups of exercises is very good at teaching speed in adding, etc. All of the exercises (there are thiry examples to each) provide equations which have to be satisfied by inserting the correct operators (+ or - or * or /) on the left-hand side to give the value 1 - on the right-hand side.

• Exercise 1.
• Exercise 2.