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Essential Concepts

• Counting and keeping track of numbers have been important to humans for a very long time, but we’re not exactly sure how it all started. People used things like sticks and notches to represent the numbers and keep count of things, and eventually, they developed spoken words and written symbols to represent numbers, which became more complex over time.
• The Incas had a simpler way of doing math compared to other ancient civilizations. They used a system called quipu, which used cords and knots to count and record quantities. They also had a special counting board made of stone, where different levels and compartments determined the value of pebbles placed on them for counting.
• The quipu was a system used by the Incas to store and convey information using cords and knots. Different types of knots and their placement on the cords represented specific numbers, with single knots for tens, hundreds, and thousands, figure-eight knots for “one,” and long knots for other numbers.
• The Hindu-Arabic number system we use today has ten symbols ([latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex], [latex]6[/latex], [latex]7[/latex], [latex]8[/latex], [latex]9[/latex]) and is based on powers of ten. The position of a symbol in a number affects its value, and this system originated primarily from India.
• Roman numerals were a way of writing numbers in ancient Rome and Europe. They used letters to represent numbers, with symbols like I, V, X, L, C, D, and M having values of [latex]1[/latex], [latex]5[/latex], [latex]10[/latex], [latex]50[/latex], [latex]100[/latex], [latex]500[/latex], and [latex]1,000[/latex] respectively. Roman numerals were gradually replaced by Hindu-Arabic numerals, but they are still used in some places today, and numbers were formed by combining symbols and adding their values.
• Number bases refer to the way numbers are represented, and the most important aspect is the place value system. Different civilizations, like the Babylonians, Greeks, Chinese, and Indians, had their own number bases, with the Indian system adopting our familiar positional system where the position of a symbol determines its value, including the use of zero as a symbol. This system made calculations easier and more accurate.
• A base system is a way of counting using different symbols. In our base-[latex]10[/latex] system, we use ten symbols ([latex]0[/latex]–[latex]9[/latex]) and the position of a digit tells us its value. But we can use other bases too, like base-[latex]5[/latex], where we only use the symbols [latex]0[/latex]–[latex]4[/latex], and the positions represent different values, like ones, fives, and twenty-fives.

Glossary

absolute value of a number

its distance from [latex]0[/latex] on the number line

base system

a structure within which we count

counting numbers

start with [latex]1[/latex] and continue

Hindu-Arabic system

a base-ten system composed of the ten symbols {[latex]0[/latex],[latex]1[/latex],[latex]2[/latex],[latex]3[/latex],[latex]4[/latex],[latex]5[/latex],[latex]6[/latex],[latex]7[/latex],[latex]8[/latex],[latex]9[/latex]}

integers

counting numbers, their opposites, and zero

one to one correspondence

when items that are being counted are uniquely linked with some counting tool

opposite of a number

the number that is the same distance from zero on the number line but on the opposite side of zero

Roman numerals

a counting system where numbers in the system are represented by combinations of letters from the Latin alphabet

the quipu

a collection of cords with knots in them used as a permanent recording of quantities or computations

whole numbers

counting numbers and zero

Key Equations

Commutative Property of Addition

[latex]a+b=b+a[/latex]

Identity Property of Addition

[latex]\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}[/latex]