Britannica Educational Publishing - Arithmetic: The Foundation of Mathematics

ARITHMETIC

THE FOUNDATION OF
MATHEMATICS

Published in 2015 by Britannica Educational Publishing (a trademark of Encyclopdia Britannica, Inc.) in association with The Rosen Publishing Group, Inc.

29 East 21st Street, New York, NY 10010

Copyright 2015 by Encyclopdia Britannica, Inc. Britannica, Encyclopdia Britannica, and the Thistle logo are registered trademarks of Encyclopdia Britannica, Inc. All rights reserved.

Rosen Publishing materials copyright 2015 The Rosen Publishing Group, Inc. All rights reserved.

Distributed exclusively by Rosen Publishing.

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First Edition

Britannica Educational Publishing

J.E. Luebering: Director, Core Reference Group

Anthony L. Green: Editor, Comptons by Britannica

Rosen Publishing

Hope Lourie Killcoyne: Executive Editor

Garrett Gladle: Editor

Nelson S: Art Director

Brian Garvey: Designer

Cindy Reiman: Photography Manager

Introduction and supplementary material by Barbara Gottfried Hollander.

Library of Congress Cataloging-in-Publication Data

Arithmetic: the foundation of mathematics/edited by Garrett Gladle.First edition.

pages cm(The story of math, core principles of mathematics)

Includes bibliographical references and index.

ISBN 978-1-6227-5519-6 (eBook)

1. ArithmeticFoundations. I. Gladle, Garrett, editor.

QA248.A765 2015

513dc23

2014017958

Photo credits: Cover and interior pages Fuse/Getty Images; cover, p. 3 (top) Indivision 07 Grow B/Getty Images.

Contents

I magine a world without numbers. How would someone tell time, determine how far it is to drive to a concert in the next city, find the total cost of a new smartphone, or figure out a bank balance and the interest it has earned? Numbers are part of daily life. Arithmetic is the science of calculating with numbers. People use numbers to measure time, distance, monetary values, and more. Rather than being a course in arithmetic, this resource focuses on its operations.

Consider several goals of a high school student, like making a budget, planning to pay for college, and keeping track of job earnings. All these goals require arithmetic. Budgets involve adding income and subtracting expenses. Finding the cost of college means subtracting financial aid from a schools sticker price. Calculating a bank balance with job wage deposits and interest may use all six basic operations of arithmetic.

The six operations are addition, subtraction, multiplication, division, raising to powers, and finding roots. An example of raising to powers is the use of scientific notation. This notation is a simplified way to write very small or very large numbers, such as expressing distances in the solar system or the number of acres in a town. One sees the use of finding roots in geometry. For example, given the volume of a cube, the length of a side of the cube is found with finding roots.

Arithmetic also encompasses one-to-one correspondence. Putting on shoes is an example of one-to-one correspondence. There is one foot that corresponds, or goes with, one shoe. Making muffins also involves one-to-one correspondence. Bakers put one muffin liner in each muffin tins cup. In math, a function shows a relationship between one or more variables. A function has one-to-one correspondence because each function output value corresponds to only one input value.

When making muffins or cupcakes, the baker puts a muffin liner in each cup of the muffin tin, which is an example of one-to-one correspondence. elenaburn/Shutterstock.com

One-to-one correspondence means matching things, like a foot with a shoe. It can also match two numerical values, like in a function. Counting is an arithmetic process that matches things with numbers, like matching pencils in a classroom with numbers that indicate their amounts. The results of this counting exercise help determine if there are enough pencils for each student. People use the counting numbers, or whole numbers beginning at 0, to perform the six basic operations of arithmetic.

Many students use social media. Arithmetic is used to determine how many people visit these sites. When students vote in school, local, state, or national elections, arithmetic finds the winners. Arithmetic is also used to tally scores in sports, like baseball, as well as team activities, like chess tournaments.

Many global organizations provide people in poverty with needed food, medical treatment, housing, and education. These organizations also use arithmetic for such things as finding the number of people who need food or the number of life-saving vaccinations to send.

Arithmetic is one of the most useful of all sciences. It is used throughout the world and our daily lives, making its importance clear. This resource highlights the basic operations in arithmetic, a reminder that all of the simple operations done with numbers would be virtually impossible without it.

T he term arithmetic comes from arithmos, the Greek word for number. People began doing arithmetic long before the Greeks invented the word, however, and even before anyone invented numbers.

ONE-TO-ONE CORRESPONDENCE

Historians believe that as early as ten thousand years ago, when prehistoric people started farming, they began to use arithmetic. They needed to know such things as how many sheep they owned, or how many rows of grain they planted, or how long it would be before harvest season arrived.

According to historians, prehistoric farmers devised an ingenious method for keeping track of things; they used a process of matching that mathematicians call one-to-one correspondence. A shepherd, for example, could keep track of his flock by dropping a pebble into a pile or by cutting a notch in a twig for every sheep that went to pasture in the morning. He could make sure that all his sheep returned home by matching them, one by one, to the pebbles in his pile or the notches on his twig.

Historians think that prehistoric shepherds figured out a smart system of matching that used pebbles or notches to keep track of every last sheep in their flock. Photos.com/Thinkstock

This process of matching pebbles or notches with objects was the first step in the development of arithmetic. In fact, two of the words that people use to describe doing arithmetic, calculating and tallying, come from the Latin words for pebble (calculus) and notch (taleus). One-to-one correspondence is still the most basic arithmetic process, and it is so simple that even very young children use it. Before they learn to count, children can often put the right number of forks on the dinner table by laying out one for mama, one for daddy, one for brother, one for sister, and one for me.

COUNTING

It is just a short step from one-to-one correspondence to counting, which is the process of matching objects with the names of numbers. Counting is the second simplest arithmetic process. The earliest mathematicians, thousands of years ago, probably learned to count in much the same way as little children do todaywith their fingers. When a young child wants to show how many people are in his family, he or she might hold up a finger for each family member. The child begins to count by reciting a specific number name for each consecutive finger he or she holds up: one, two, three, four.