The decimal number system is represented on the abacus using beads and columns. Each column displays a digit between 0 and 9 (depending on the bead combination). The single top bead has a value of 5 and the four bottom beads have a value of 1 each. The abacus has a 1's column, a 10's column to its left, a 100's column and so on. In the image below, the abacus is showing the number 17,539.

When a number has a small friend or a big friend, it is just a fun way of describing its complementary number. Each number 0 to 5 has a small friend number and when they are summed together they equal 5. For example, the small friend of 2 is 3 because they sum to make 5. Similarly, each number, 0 to 10, has a big friend number and when they sum together they make 10. Here are some examples: 1's big friend is 9, 4's big friend is 6 and 0's big friend is 10. A solid understanding of these basic math concepts is required to progress in abacus.

A direct movement is the most simple abacus action as it occurs when neither the small friend or big friend value are required for the calculation. The example below demonstrates the direct movements needed for summing 5 and 3 on the abacus. First, the top bead, with a value of 5, is pulled down to show number 5. Then three bottom beads, with a value of 1 each, are pulled up to give the answer 8. Direct movements can be practiced in Part 1: Sum to Nine on the Abacus where students learn to add and subtract on the abacus.

When summing two numbers on the abacus, pull-5 occurs when the top bead is pulled down and the bottom beads are subtracted simultaneously. The example below demonstrates how the pull-5 movement uses the small friend concept to calculate 2 + 4. Firstly, two bottom beads are pulled up to show the number 2 on the abacus. With only two beads left in the bottom position, adding 4 will require *pulling* the top bead down while subtracting bottom beads. This makes it a pull-5 move. The small friend of 4, which is 1, is subtracted from the bottom beads at the same time as pulling down the top bead to give the answer 6. Practice questions are available in Part 2: The Pull-5 and Break-5 Movements where kids learn to add and subtract on the abacus.

Break-5 occurs on the abacus when a calculation requires the top bead to be removed (broken) while simultaneously adding bottom beads. The example below shows how break-5 uses the small friend concept for 7 - 3. To start, the top bead and two bottom beads are used to represent the number 7. As there are not enough remaining bottom beads to subtract 3, break-5 is required. The small friend of 3 is 2; therefore, two bottom beads are added while the top bead is broken. The abacus now displays the answer 4. Practice questions are available in Part 2: The Pull-5 and Break-5 Movements where kids learn to add and subtract on the abacus.

The carry movement is used on the abacus when two numbers are added and the answer flows over into the next column. The image below demonstrates how the carry movement uses the big friend concept for 9 + 3. First, 9 is added by pulling down the top bead and pulling up four bottom beads. Adding 3 will cause an overflow into the 10's column so carry is required. The big friend of 3 is 7 so 7 is subtracted from the 1's column while one bottom bead is added to the 10's column. This gives the answer 12. Practice questions are available in Part 3: The Borrow and Carry Movements where kids learn to add and subtract on the abacus.

When subtracting two numbers on the abacus and there are not enough available beads to subtract from one column, the borrow movement is used. In the example below, the borrow movement uses the big friend concept to calculate 14 - 5. To show 14 on the abacus, four bottom beads are pulled up in the 1's column and one bottom bead is pulled down in the 10's column. Now there is no top bead to subtract 5 so the borrow movement is used. The big friend of 5 is 5; therefore, the top bead is pulled in the 1's column and one bottom bead is subtracted from the 10's column. This leaves the answer 9 on the abacus. Practice questions are available in Part 3: The Borrow and Carry Movements where kids learn to add and subtract on the abacus.