What Is i Squared? Imaginary Numbers
An imaginary number is a number that gives a negative answer when you multiply it by itself. The basic one is called i, and it is defined by i² = −1, so i × i = −1. No ordinary number can do this: 3 × 3 = 9 and (−3) × (−3) = 9, so squaring any everyday number, positive or negative, always gives zero or more.
One helpful way to picture it uses arrows on a number line. Multiplying by −1 turns an arrow around to face the other way, which is a half turn. Doing that twice brings it back to face forward, which is why a negative times a negative is positive. Multiplying by i is a quarter turn, so the arrow points straight up. Two quarter turns make one half turn, and that is why i × i = −1.
The name "imaginary" started as a joke from mathematicians who thought these numbers were silly. Today engineers use them for electricity and electronics, scientists use them to describe waves and sound, and they are a basic tool in physics and computer graphics.
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A Mystery Number🕵️ The Mystery Number ? i Here is a puzzle that stumped grown-up mathematicians for hundreds of years: ? × ? = −1 Can you find a number that, times itself, gives negative one? 🤔 Don't worry if it seems impossible. By the end, you'll meet a special number called i that solves it!
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Squaring Numbers🟪 What Is "Squaring"? Squaring a number means multiplying it by itself. We write a little 2 up high: 3² = 3 × 3 = 9. It's called "squaring" because the dots make a square! Tap a number to build one. 12 345 Tap a number above ☝️ Your turn: Which number, squared, makes 16? 248
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Negative = Turn Around↩️ Times −1 Means "Turn Around" Think of a number as an arrow on the number line. The arrow below points to 2. Multiplying by −1 spins the arrow to face the opposite way! Predict: If we do 2 × (−1), where will the arrow point? −202
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Two Turns = Forward🔁 Turn Around Twice If you turn around, then turn around again… you face forward again! 🧍➡️🔄🔄➡️ That's why a negative × negative gives a positive: (−3) × (−3) = 9 😮 So 3² = 9 and (−3)² = 9. Any normal number squared is never negative. Try it: What is (−2) × (−2)? −440 So no ordinary number squared can give −1… we need a brand-new kind of number! ✨
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Meet i: The Quarter Turn🌀 Meet i — the Quarter Turn Times −1 is a half turn (facing the other way). What if we only turn halfway of that — a quarter turn, pointing straight up? That's what multiplying by i does! The arrow leaves the number line and points to a new spot called i. 1 −1 i −i Start: 1 Mission: Tap × i to spin the arrow until it lands on the ⭐ glowing −1. Then press Check. × i ↺ Check ✔ Reset
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i × i = −1🎉 Mystery Solved: i² = −1 Two quarter turns make a half turn. So: i × i = −1 That's the mystery number from the start! We write it as i² = −1. Keep turning and the pattern goes round in a circle: Match it: tap a power on the left, then its answer on the right. (Hint: count the quarter turns!) i² i³ i⁴ 1 −1 −i
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Why It Matters🔌 "Imaginary" but Super Useful! The name imaginary is a bit of a joke — people once thought these numbers were silly. Now we use them everywhere! Tap each card to flip it. ⚡ElectricityEngineers use i to work out how electricity flows into your home. 🎵Music & SoundPhones use i-maths to clean up sounds and play your songs. 🎮GamesSpinning things on screen uses the same "turning" idea as i. Fun fact: the letter i stands for "imaginary". You'll learn more about it in secondary school and beyond! 🚀
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You Did It!🏆 Amazing Work, Number Explorer! ⭐🌟⭐ Squaring means a number times itself: 4² = 16. × (−1) turns an arrow around (a half turn). Turning around twice faces forward, so (−2)² = 4. Normal squares are never negative. × i is a quarter turn. Two quarter turns = a half turn, so i² = −1. The pattern loops: i, −1, −i, 1, i, … You just solved a puzzle that puzzled mathematicians for centuries. 🎉
Frequently asked questions
- What is i squared equal to?
- i squared equals −1. That is how i is defined: it is the number that, multiplied by itself, gives −1.
- Why can't a normal number squared be negative?
- A positive number times itself is positive, and a negative number times itself is also positive, because two 'turn arounds' bring you back to facing forward. So no ordinary number squared can give a negative answer, and that is why a new number, i, was needed.
- Are imaginary numbers taught in Singapore primary school?
- No. Imaginary numbers are not in the MOE primary Mathematics syllabus and usually come much later, in advanced secondary or pre-university mathematics. Primary learners can still understand the idea using squaring and turns on a number line, which builds on multiplication and negative numbers.
- Are imaginary numbers real or just made up?
- They are a real and useful part of mathematics, even though the name sounds pretend. Engineers use them to design electrical circuits, and scientists use them to describe waves, sound and light.
- What does multiplying by i do on a number line?
- Multiplying by i turns a number's arrow by a quarter turn, so it points straight up instead of along the line. Doing it twice makes a half turn, which is the same as multiplying by −1.
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