In India's startup arena, survival depends on balancing hard metrics with bold imagination -- where real numbers anchor dreams, and vision gives power to fundamentals. Imagine you're back in that college lecture theatre, chalk in hand, sketching the point (a+ib) on the complex plane -- and suddenly it hits you that every startup pitch today is really a play of the same two dimensions: the real and the imaginary.

In India's bustling ecosystem, where Swiggy delivers hot meals at midnight and Byju's promises to revolutionize classrooms, founders and investors often get swept away by a sparkling vision -- sometimes literally -- what we'll call "ib" -- while the steady ticks of revenue, customer churn, and unit economics -- the "a" -- linger in the background.

Numbers don't always favor wishful thinking. Ask any growth-stage company that once basked in the glow of a bumper funding round, and you'll hear a familiar refrain: "We thought the narrative was enough." Yet, when the next funding season turns chilly, founders scramble to shore up metrics: audited revenue trends, burn-rate projections, actual retention rates. After all, even the most dazzling concept can't pay salaries when the lights go out.

On the flip side, there's a reason top-tier VCs sit up at the mention of "AI-driven hyper personalization" or "democratizing finance with blockchain." That magic -- or "ib" -- drives excitement, social media buzz, and, yes, gigantic term sheets. But behavioral scientists have long warned of the illusory-truth effect, whereby repeated slogans ("We'll own Bharat's digital payments!") start to sound true simply through repetition. In boardrooms and pitch meetings alike, every slide deck iteration amplifies belief -- sometimes faster than real-world traction can catch up.

The beauty of complex numbers is that they give us a measure of overall strength: the absolute value. In business terms, it's the fusion of solid fundamentals and irresistible vision. A startup that scores high on both axes weathers market turbulence; one that inflates "ib" at the cost of "a" risks a brutal correction when runway dries up.

Think of the landmark Jio-Meta tie-up: Jio's vast subscriber base and distribution muscle multiplied Meta's social media expertise, rotating India's digital story onto a new trajectory -- something neither could have achieved alone. On the other hand, WeWork was once hailed as one of the most visionary companies of its time. Its 'ib' was massive: redefining office culture, creating 'physical social networks,' and transforming commercial real estate into a tech-driven community experience. But behind the scenes, the fundamentals were shaky.

Another powerful idea in this analogy is the mirror perspective -- or the conjugate -- the startup world's built-in hedge, the Chi of your business. Just as Chi harmonizes energy and keeps movement purposeful, this balance keeps bold bets grounded. An ambitious AI play paired with a steady services revenue stream isn't just playing it safe; it's allowing clarity to emerge. When vision aligns with stability, the real value of a business doesn't just grow -- it comes into sharp focus.

External stakeholders -- such as investors, auditors, and advisors -- bring a crucial perspective when assessing the "ib" against the "a." They explore what's proven, what's assumed, and how resilient the model is. Are projections grounded, or do they lean toward optimistic extrapolation? Whether through due diligence, annual audits, or quarterly reviews, this analytical rigor plays a key role in distinguishing sustainable businesses from fleeting hype.

Complex numbers exist because real numbers alone can't explain everything; likewise, the business world can't be captured by metrics alone. Every "ib" needs an "a." Every vision needs execution. Every pitch needs proof.

So next time you're on stage or in the investors' lounge, ask yourself: "Where does our real end and our imagination begin? If we dial down the vision by 20 percent, will our fundamentals still carry us? And if our numbers grow by 10 percent, how will that change the story we tell?"

Mastering your own complex arithmetic -- knowing exactly when to boost the "a," when to fuel the "ib," and how to compute the combined force -- may be the single most powerful skill in India's startup playbook today.

(The author is Managing Partner at Growth Vectors LLP; Views expressed are personal)